diff --git "a/BNAzT4oBgHgl3EQfv_6i/content/tmp_files/load_file.txt" "b/BNAzT4oBgHgl3EQfv_6i/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/BNAzT4oBgHgl3EQfv_6i/content/tmp_files/load_file.txt" @@ -0,0 +1,1322 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf,len=1321 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content='01716v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content='OC] 4 Jan 2023 First-order penalty methods for bilevel optimization Zhaosong Lu ∗ Sanyou Mei ∗ January 4, 2023 Abstract In this paper we study a class of unconstrained and constrained bilevel optimization problems in which the lower-level part is a convex optimization problem, while the upper-level part is possibly a nonconvex optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' In particular, we propose penalty methods for solving them, whose subproblems turn out to be a structured minimax problem and are suitably solved by a first- order method developed in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Under some suitable assumptions, an operation complexity of O(ε−4 log ε−1) and O(ε−7 log ε−1), measured by their fundamental operations, is established for the proposed penalty methods for finding an ε-KKT solution of the unconstrained and constrained bilevel optimization problems, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' To the best of our knowledge, the methodology and results in this paper are new.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Keywords: bilevel optimization, minimax optimization, penalty methods, first-order methods, opera- tion complexity Mathematics Subject Classification: 90C26, 90C30, 90C47, 90C99, 65K05 1 Introduction Bilevel optimization is a two-level hierarchical optimization in which partial or full decision variables in the upper level are also involved in the lower level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Generically, it can be written in the following form: min x,y f(x, y) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' g(x, y) ≤ 0, y ∈ Argmin z { ˜f(x, z)|˜g(x, z) ≤ 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content='1 (1) Bilevel optimization has found a variety of important applications, including adversarial training [36, 37, 46], continual learning [32], hyperparameter tuning [3, 17], image reconstruction [9], meta-learning [4, 23, 42], neural architecture search [15, 30], reinforcement learning [20, 27], and Stackelberg games [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' More applications about it can be found in [2, 8, 10, 11, 12, 44] and the references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Theoretical properties including optimality conditions of (1) have been extensively studied in the literature (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=', see [12, 13, 34, 47, 50]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Numerous methods have been developed for solving some special cases of (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' For example, constraint- based methods [19, 43], deterministic gradient-based methods [16, 17, 21, 35, 41, 42], and stochastic gradient-based methods [6, 18, 20, 24, 26] were proposed for solving (1) with g ≡ 0, ˜g ≡ 0, f, ˜f being smooth, and ˜f being strongly convex with respect to y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Besides, when all the functions involved in (1) are smooth and ˜f, ˜g are convex with respect to y, gradient-type methods were proposed by solving the mathematical program with equilibrium constraints (MPEC) resulting from replacing the lower-level optimization problem of (1) by its first-order optimality conditions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=', see [1, 33, 40]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Recently, difference-of-convex (DC) algorithms were developed in [51] for solving (1) with g ≡ 0, f being a DC function, and ˜f, ˜g being convex functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' In addition, a double penalty method [22] was proposed for (1), which solves a sequence of bilevel optimization problems of the form min x,y f(x, y) + ρkΨ(x, y) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' y ∈ Argmin z ˜f(x, z) + ρk ˜Ψ(x, z), (2) ∗Department of Industrial and Systems Engineering, University of Minnesota, USA (email: zhaosong@umn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content='edu, mei00035@umn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content='edu).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' This work was partially supported by NSF Award IIS-2211491.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' 1For ease of reading, throughout this paper the tilde symbol is particularly used for the functions related to the lower-level optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Besides, “Argmin” denotes the set of optimal solutions of the associated problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' 1 where {ρk} is a sequence of penalty parameters, and Ψ and ˜Ψ are a penalty function associated with the sets {(x, y)|g(x, y) ≤ 0} and {(x, z)|˜g(x, z) ≤ 0}, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Though problem (2) appears to be simpler than (1), there is no method available for finding an approximate solution of (2) in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Conse- quently, the double penalty method [22] is typically not implementable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' More discussion on algorithmic development for bilevel optimization can be found in [2, 8, 12, 31, 45, 47]) and the references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' It has long been known that the notorious challenge of bilevel optimization (1) mainly comes from the lower level part, which requires that the variable y be a solution of another optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Due to this, for the sake of simplicity, we only consider a subclass of bilevel optimization with the constraint g(x, y) ≤ 0 being excluded, namely, min x,y f(x, y) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' y ∈ Argmin z { ˜f(x, z)|˜g(x, z) ≤ 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' (3) Nevertheless, the results in this paper can be possibly extended to problem (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' The main goal of this paper is to develop a first-order penalty method for solving problem (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Our key observations toward this development are: (i) problem (3) can be approximately solved as a penalty problem (see (49));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' (ii) such a penalty problem is equivalent to a structured minimax problem (see (50)), which can be suitably solved by a first-order method proposed in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' As a result, these observations lead to development of a novel first-order penalty method for solving (3) (see Sections 3 and 4), which enjoys the following appealing features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' It uses only the first-order information of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Specifically, its fundamental operations consist only of evaluations of the gradient of ˜g and the smooth component of f and ˜f and also the proximal operator of the nonsmooth component of f and ˜f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Thus, it is suitable for solving large-scale problems (see Sections 3 and 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' It has theoretical guarantees on operation complexity, which is measured by the aforementioned fundamental operations, for finding an ε-KKT solution of (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' In particular, when ˜g ≡ 0, it enjoys an operation complexity of O(ε−4 log ε−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Otherwise, it enjoys an operation complexity of O(ε−7 log ε−1) (see Theorems 4 and 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' It is applicable to a broader class of problems than existing methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' For example, it can be applied to (3) with f, ˜f being nonsmooth and ˜f, ˜g being nonconvex with respect to x, which is however not suitable for existing methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' To the best of our knowledge, the methodology and results in this paper are new.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' The rest of this paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' In Subsection 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content='1 we introduce some notation and terminology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' In Section 2 we propose a first-order method for solving a nonconvex-concave minimax problem and study its complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' In Sections 3 and 4, we propose first-order penalty methods for unconstrained and constrained bilevel optimization and study their complexity, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' In Section 5 we present the proofs of the main results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Finally, we make some concluding remarks in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content='1 Notation and terminology The following notation will be used throughout this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Let Rn denote the Euclidean space of dimension n and Rn + denote the nonnegative orthant in Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' The standard inner product and Euclidean norm are denoted by ⟨·, ·⟩ and ∥ · ∥, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' For any v ∈ Rn, let v+ denote the nonnegative part of v, that is, (v+)i = max{vi, 0} for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' For any two vectors u and v, (u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' v) denotes the vector resulting from stacking v under u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Given a point x and a closed set S in Rn, let dist(x, S) = minx′∈S ∥x′ − x∥ and IS denote the indicator function associated with S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' A function or mapping φ is said to be Lφ-Lipschitz continuous on a set S if ∥φ(x)−φ(x′)∥ ≤ Lφ∥x−x′∥ for all x, x′ ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' In addition, it is said to be L∇φ-smooth on S if ∥∇φ(x) − ∇φ(x′)∥ ≤ L∇φ∥x − x′∥ for all x, x′ ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' For a closed convex function p : Rn → R ∪ {∞},2 the proximal operator associated with p is denoted by proxp, that is, proxp(x) = arg min x′∈Rn �1 2∥x′ − x∥2 + p(x′) � ∀x ∈ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' (4) 2For convenience, ∞ stands for +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' 2 Given that evaluation of proxγp(x) is often as cheap as proxp(x), we count the evaluation of proxγp(x) as one evaluation of proximal operator of p for any γ > 0 and x ∈ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' For a lower semicontinuous function φ : Rn → R∪{∞}, its domain is the set dom φ := {x|φ(x) < ∞}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' The upper subderivative of φ at x ∈ dom φ in a direction d ∈ Rn is defined by φ′(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' d) = lim sup x′ φ →x, t↓0 inf d′→d φ(x′ + td′) − φ(x′) t , where t ↓ 0 means both t > 0 and t → 0, and x′ φ→ x means both x′ → x and φ(x′) → φ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' The subdifferential of φ at x ∈ dom φ is the set ∂φ(x) = {s ∈ Rn��sT d ≤ φ′(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' d) ∀d ∈ Rn}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' We use ∂xiφ(x) to denote the subdifferential with respect to xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' In addition, for an upper semicontinuous function φ, its subdifferential is defined as ∂φ = −∂(−φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' If φ is locally Lipschitz continuous, the above definition of subdifferential coincides with the Clarke subdifferential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Besides, if φ is convex, it coincides with the ordinary subdifferential for convex functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Also, if φ is continuously differentiable at x , we simply have ∂φ(x) = {∇φ(x)}, where ∇φ(x) is the gradient of φ at x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' In addition, it is not hard to verify that ∂(φ1 + φ2)(x) = ∇φ1(x) + ∂φ2(x) if φ1 is continuously differentiable at x and φ2 is lower or upper semicontinuous at x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' See [7, 49] for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Finally, we introduce two types of approximate solutions for a general minimax problem Ψ∗ = min x max y Ψ(x, y), (5) where Ψ(·, y) : Rn → R ∪ {∞} is a lower semicontinuous function, Ψ(x, ·) : Rm → R ∪ {−∞} is an upper semicontinuous function, and Ψ∗ is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' A point (xǫ, yǫ) is called an ǫ-optimal solution of the minimax problem (5) if max y Ψ(xǫ, y) − Ψ(xǫ, yǫ) ≤ ǫ, Ψ(xǫ, yǫ) − Ψ∗ ≤ ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' A point (x, y) is called a stationary point of the minimax problem (5) if 0 ∈ ∂xΨ(x, y), 0 ∈ ∂yΨ(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' In addition, for any ǫ > 0, a point (xǫ, yǫ) is called an ǫ-stationary point of the minimax problem (5) if dist (0, ∂xΨ(xǫ, yǫ)) ≤ ǫ, dist (0, ∂yΨ(xǫ, yǫ)) ≤ ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' 2 A first-order method for nonconvex-concave minimax prob- lem In this section, we propose a first-order method for finding an approximate stationary point of a nonconvex-concave minimax problem, which will be used as a subproblem solver for the penalty methods proposed in Sections 3 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' In particular, we consider the minimax problem H∗ = min x max y {H(x, y) := h(x, y) + p(x) − q(y)} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' (6) Assume that problem (6) has at least one optimal solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' In addition, h, p and q satisfy the following assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Assumption 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' (i) p : Rn → R ∪ {∞} and q : Rm → R ∪ {∞} are proper convex functions and continuous on their domain, and moreover, their domain is compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' (ii) The proximal operator associated with p and q can be exactly evaluated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' (iii) h is L∇h-smooth on dom p × dom q, and moreover, h(x, ·) is concave for any x ∈ dom p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' 3 Recently, an accelerated inexact proximal point smoothing (AIPP-S) scheme was proposed in [28] for finding an approximate stationary point of a class of minimax composite nonconvex optimization problems, which includes (6) as a special case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' When applied to (6), AIPP-S requires the availability of the oracle including exact evaluation of ∇xh(x, y) and arg min x � p(x) + 1 2λ∥x − x′∥2 � , arg max y � h(x′, y) − q(y) − 1 2λ∥y − y′∥2 � (7) for any λ > 0, x′ ∈ Rn and y′ ∈ Rm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Note that h is typically sophisticated and the exact solution of the second problem in (7) usually cannot be found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' As a result, AIPP-S is generally not implementable for (6), though an oracle complexity of O(ǫ−5/2) was established in [28] for it to find an ǫ-stationary point of (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' In what follows, we first propose a modi���ed optimal first-order method for solving a strongly-convex- strongly-concave minimax problem in Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Using this method as a subproblem solver for an inexact proximal point scheme, we then propose a first-order method for (6) in Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content='2, which enjoys an operation complexity of O(ǫ−5/2 log ǫ−1), measured by the amount of evaluations of ∇h and proximal operator of p and q, for finding an ǫ-stationary point of (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content='1 A modified optimal first-order method for strongly-convex-strongly-concave minimax problem In this subsection, we consider the strongly-convex-strongly-concave minimax problem ¯H∗ = min x max y � ¯H(x, y) := ¯h(x, y) + p(x) − q(y) � , (8) where p and q satisfy Assumption 1 and ¯h satisfies the following assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' ¯h(x, y) is σx-strongly-convex-σy-strongly-concave and L∇¯h-smooth on dom p × dom q for some σx, σy > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' The goal of this subsection is to propose a modified optimal first-order method for finding an approx- imate stationary point of problem (8) and study its complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Before proceeding, we introduce some more notation below, most of which is adopted from [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Let (x∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' y∗) denote the optimal solution of (8),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' z∗ = −σxx∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' and Dp = max{∥u − v∥ ��u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' v ∈ dom p},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Dq = max{∥u − v∥ ��u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' v ∈ dom q},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' (9) ¯Hlow = min � ¯H(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' y)| � x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' y) ∈ dom p × dom q},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' (10) ˆh(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' y) = ¯h(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' y) − σx∥x∥2/2 + σy∥y∥2/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' (11) G(z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' y) = sup x {⟨x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' z⟩ − p(x) − ˆh(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' y) + q(y)},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' (12) P(z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' y) = σ−1 x ∥z∥2/2 + σy∥y∥2/2 + G(z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' y),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' (13) ϑk = η−1 z ∥zk − z∗∥2 + η−1 y ∥yk − y∗∥2 + 2¯α−1(P(zk f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' yk f) − P(z∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' y∗)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' (14) ak x(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' y) = ∇xˆh(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' y) + σx(x − σ−1 x zk g)/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' ak y(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' y) = −∇yˆh(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' y) + σyy + σx(y − yk g)/8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' where ¯α = min � 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' � 8σy/σx � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' ηz = σx/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' ηy = min {1/(2σy),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' 4/(¯ασx)},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' and yk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' yk f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' yk g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' zk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' zk f and zk g are generated at iteration k of Algorithm 1 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' By Assumptions 1 and 2, one can observe that Dp, Dq and ¯Hlow are finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' We are now ready to present a modified optimal first-order method for solving (8) in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' It is a slight modification of the novel optimal first-order method [29, Algorithm 4] by incorporating a forward- backward splitting scheme and also a verifiable termination criterion (see steps 23-25 in Algorithm 1) in order to find a τ-stationary point of (8) (see Definition 2) for any prescribed tolerance τ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' 4 Algorithm 1 A modified optimal first-order method for (8) Input: τ > 0, ¯z0 = z0 f ∈ −σxdom p,3 ¯y0 = y0 f ∈ dom q, (z0, y0) = (¯z0, ¯y0), ¯α = min � 1, � 8σy/σx � , ηz = σx/2, ηy = min {1/(2σy), 4/(¯ασx)}, βt = 2/(t + 3), ζ = � 2 √ 5(1 + 8L∇¯h/σx) �−1, γx = γy = 8σ−1 x , and ˆζ = min{σx, σy}/L2 ∇¯h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' 1: for k = 0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' do 2: (zk g , yk g) = ¯α(zk, yk) + (1 − ¯α)(zk f, yk f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' 3: (xk,−1, yk,−1) = (−σ−1 x zk g, yk g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' 4: xk,0 = proxζγxp(xk,−1 − ζγxak x(xk,−1, yk,−1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' 5: yk,0 = proxζγyq(yk,−1 − ζγyak y(xk,−1, yk,−1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' 6: bk,0 x = 1 ζγx (xk,−1 − ζγxak x(xk,−1, yk,−1) − xk,0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' 7: bk,0 y = 1 ζγy (yk,−1 − ζγyak y(xk,−1, yk,−1) − yk,0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' 8: t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' 9: while γx∥ak x(xk,t, yk,t) + bk,t x ∥2 + γy∥ak y(xk,t, yk,t) + bk,t y ∥2 > γ−1 x ∥xk,t − xk,−1∥2 + γ−1 y ∥yk,t − yk,−1∥2 do 10: xk,t+1/2 = xk,t + βt(xk,0 − xk,t) − ζγx(ak x(xk,t, yk,t) + bk,t x ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' 11: yk,t+1/2 = yk,t + βt(yk,0 − yk,t) − ζγy(ak y(xk,t, yk,t) + bk,t y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' 12: xk,t+1 = proxζγxp(xk,t + βt(xk,0 − xk,t) − ζγxak x(xk,t+1/2, yk,t+1/2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' 13: yk,t+1 = proxζγyq(yk,t + βt(yk,0 − yk,t) − ζγyak y(xk,t+1/2, yk,t+1/2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' 14: bk,t+1 x = 1 ζγx (xk,t + βt(xk,0 − xk,t) − ζγxak x(xk,t+1/2, yk,t+1/2) − xk,t+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' 15: bk,t+1 y = 1 ζγy (yk,t + βt(yk,0 − yk,t) − ζγyak y(xk,t+1/2, yk,t+1/2) − yk,t+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' 16: t ← t + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' 17: end while 18: (xk+1 f , yk+1 f ) = (xk,t, yk,t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' 19: (zk+1 f , wk+1 f ) = (∇xˆh(xk+1 f , yk+1 f ) + bk,t x , −∇yˆh(xk+1 f , yk+1 f ) + bk,t y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' 20: zk+1 = zk + ηzσ−1 x (zk+1 f − zk) − ηz(xk+1 f + σ−1 x zk+1 f ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' 21: yk+1 = yk + ηyσy(yk+1 f − yk) − ηy(wk+1 f + σyyk+1 f ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' 22: xk+1 = −σ−1 x zk+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' 23: ˆxk+1 = proxˆζp(xk+1 − ˆζ∇x¯h(xk+1, yk+1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' 24: ˆyk+1 = proxˆζq(yk+1 + ˆζ∇y¯h(xk+1, yk+1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' 25: Terminate the algorithm and output (ˆxk+1, ˆyk+1) if ∥ˆζ−1(xk+1 − ˆxk+1, ˆyk+1 − yk+1) − (∇¯h(xk+1, yk+1) − ∇¯h(ˆxk+1, ˆyk+1))∥ ≤ τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' (15) 26: end for The following theorem presents iteration and operation complexity of Algorithm 1 for finding a τ- stationary point of problem (8), whose proof is deferred to Subsection 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Theorem 1 (Complexity of Algorithm 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Suppose that Assumptions 1 and 2 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Let ¯H∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Dp,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Dq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' ¯Hlow,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' and ϑ0 be defined in (8),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' (9),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' (10) and (14),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' σx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' σy and L∇¯h be given in Assumption 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' ¯α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' ηy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' ηz,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' τ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' ˆζ be given in Algorithm 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' and ¯δ = (2 + ¯α−1)σxD2 p + max{2σy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' ¯ασx/4}D2 q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' (16) ¯K = � max � 2 ¯α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' ¯ασx 4σy � log 4 max{ηzσ−2 x ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' ηy}ϑ0 (ˆζ−1 + L∇¯h)−2τ 2 � + ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' (17) ¯N = � max � 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' � σx 2σy � log 4 max {1/(2σx),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' min {1/(2σy),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' 4/(¯ασx)}} �¯δ + 2¯α−1 � ¯H∗ − ¯Hlow �� (L2 ∇¯h/ min{σx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' σy} + L∇¯h)−2τ 2 � + × �� 96 √ 2 � 1 + 8L∇¯hσ−1 x �� + 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' (18) Then Algorithm 1 outputs a τ-stationary point of (8) in at most ¯K iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Moreover, the total 3For convenience, −σxdom p stands for the set {−σxu|u ∈ dom p}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' 5 number of evaluations of ∇¯h and proximal operator of p and q performed in Algorithm 1 is no more than ¯N, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' It can be observed from Theorem 1 that Algorithm 1 enjoys an operation complexity of O(log τ−1), measured by the amount of evaluations of ∇¯h and proximal operator of p and q, for finding a τ-stationary point of the strongly-convex-strongly-concave minimax problem (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content='2 A first-order method for problem (6) In this subsection, we propose a first-order method for finding an ǫ-stationary point of problem (6) (see Definition 2) for any prescribed tolerance ǫ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' In particular, we first add a perturbation to the max part of (6) for obtaining an approximation of (6), which is given as follows: min x max y � h(x, y) + p(x) − q(y) − ǫ 4Dq ∥y − ˆy0∥2 � (19) for some ˆy0 ∈ dom q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' We then apply an inexact proximal point method [25] to (19), which consists of approximately solving a sequence of subproblems min x max y {Hk(x, y) := hk(x, y) + p(x) − q(y)} , (20) where hk(x, y) = h(x, y) − ǫ∥y − ˆy0∥2/(4Dq) + L∇h∥x − xk∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' (21) By Assumption 1, one can observe that (i) hk is L∇h-strongly convex in x and ǫ/(2Dq)-strongly concave in y on dom p × dom q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' (ii) hk is (3L∇h + ǫ/(2Dq))-smooth on dom p × dom q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Consequently, problem (20) is a special case of (8) and we can apply Algorithm 1 to solve (20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' The resulting first-order method for (6) is presented in Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Algorithm 2 A first-order method for problem (6) Input: ǫ > 0, ǫ0 ∈ (0, ǫ/2], (ˆx0, ˆy0) ∈ dom p × dom q, (x0, y0) = (ˆx0, ˆy0), and ǫk = ǫ0/(k + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' 1: for k = 0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' do 2: Call Algorithm 1 with ¯h ← hk, τ ← ǫk, σx ← L∇h, σy ← ǫ/(2Dq), L∇¯h ← 3L∇h + ǫ/(2Dq), ¯z0 = z0 f ← −σxxk, ¯y0 = y0 f ← yk, and denote its output by (xk+1, yk+1), where hk is given in (21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' 3: Terminate the algorithm and output (xǫ, yǫ) = (xk+1, yk+1) if ∥xk+1 − xk∥ ≤ ǫ/(4L∇h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' (22) 4: end for Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' It can be observed from step 2 of Algorithm 2 that (xk+1, yk+1) results from applying Algo- rithm 1 to the subproblem (20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' As will be shown in Lemma 2, (xk+1, yk+1) is an ǫk-stationary point of (20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' We next study complexity of Algorithm 2 for finding an ǫ-stationary point of problem (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Before proceeding, we define Hlow := min {H(x, y)|(x, y) ∈ dom p × dom q} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' (23) By Assumption 1, one can observe that Hlow is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' The following theorem presents iteration and operation complexity of Algorithm 2 for finding an ǫ-stationary point of problem (6), whose proof is deferred to Subsection 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Theorem 2 (Complexity of Algorithm 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Suppose that Assumption 1 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Let H∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' H Dp,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Dq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' and Hlow be defined in (6),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' (9) and (23),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' L∇h be given in Assumption 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' ǫ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' ǫ0 and ˆx0 be given in 6 Algorithm 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' and α = min � 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' � 4ǫ/(DqL∇h) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' (24) δ = (2 + α−1)L∇hD2 p + max {ǫ/Dq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' αL∇h/4} D2 q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' (25) K = � 16(max y H(ˆx0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' y) − H∗ + ǫDq/4)L∇hǫ−2 + 32ǫ2 0(1 + 4D2 qL2 ∇hǫ−2)ǫ−2 − 1 � + ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' (26) N = �� 96 √ 2 � 1 + (24L∇h + 4ǫ/Dq) L−1 ∇h �� + 2 � max � 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' � DqL∇hǫ−1 � × � (K + 1) � log 4 max � 1 2L∇h ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' min � Dq ǫ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' 4 αL∇h �� � δ + 2α−1(H∗ − Hlow + ǫDq/4 + L∇hD2 p) � [(3L∇h + ǫ/(2Dq))2/ min{L∇h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' ǫ/(2Dq)} + 3L∇h + ǫ/(2Dq)]−2 ǫ2 0 � + + K + 1 + 2K log(K + 1) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' (27) Then Algorithm 2 terminates and outputs an ǫ-stationary point (xǫ, yǫ) of (6) in at most K + 1 outer iterations that satisfies max y H(xǫ, y) ≤ max y H(ˆx0, y) + ǫDq/4 + 2ǫ2 0 � L−1 ∇h + 4D2 qL∇hǫ−2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' (28) Moreover, the total number of evaluations of ∇h and proximal operator of p and q performed in Algo- rithm 2 is no more than N, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Since ǫ0 ∈ (0, ǫ/2], one can observe from Theorem 2 that α = O(ǫ1/2), δ = O(ǫ−1/2), K = O(ǫ−2), and N = O(ǫ−5/2 log(ǫ−1 0 ǫ−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Consequently, Algorithm 2 enjoys an operation complexity of O(ǫ−5/2 log(ǫ−1 0 ǫ−1)), measured by the amount of evaluations of ∇h and proximal operator of p and q, for finding an ǫ-stationary point of the nonconvex-concave minimax problem (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' 3 Unconstrained bilevel optimization In this section, we consider an unconstrained bilevel optimization problem4 f ∗ = min f(x, y) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' y ∈ Argmin z ˜f(x, z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' (29) Assume that problem (29) has at least one optimal solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' In addition, f and ˜f satisfy the following assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' (i) f(x, y) = f1(x, y)+f2(x) and ˜f(x, y) = ˜f1(x, y)+ ˜f2(y) are continuous on X ×Y, where f2 : Rn → R ∪ {∞} and ˜f2 : Rm → R ∪ {∞} are proper closed convex functions, ˜f1(x, ·) is convex for any given x ∈ X, and f1, ˜f1 are respectively L∇f1- and L∇ ˜f1-smooth on X × Y with X := dom f2 and Y := dom ˜f2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' (ii) The proximal operator associated with f2 and ˜f2 can be exactly evaluated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' (iii) The sets X and Y (namely, dom f2 and dom ˜f2) are compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' For notational convenience, we define Dx := max{∥u − v∥ ��u, v ∈ X}, Dy := max{∥u − v∥ ��u, v ∈ Y}, (30) ˜fhi := max{ ˜f(x, y)|(x, y) ∈ X × Y}, ˜flow := min{ ˜f(x, y)|(x, y) ∈ X × Y}, (31) flow := min{f(x, y)|(x, y) ∈ X × Y}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' (32) 4For convenience, problem (29) is referred to as an unconstrained bilevel optimization problem since its lower level part does not have an explicit constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Strictly speaking, it can be a constrained bilevel optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' For example, when part of f and/or ˜f is the indicator function of a closed convex set, (29) is essentially a constrained bilevel optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' 7 By Assumption 3, one can observe that Dx, Dy, ˜fhi, ˜flow and flow are finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' The goal of this subsection is to propose penalty methods for solving problem for solving (29).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' To this end, we observe that problem (29) can be viewed as min x,y {f(x, y)| ˜f(x, y) ≤ min z ˜f(x, z)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' (33) Notice that ˜f(x, y) − minz ˜f(x, z) ≥ 0 for all x, y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Consequently, a natural penalty problem associated with (33) is min x,y f(x, y) + ρ( ˜f(x, y) − min z ˜f(x, z)), (34) where ρ > 0 is a penalty parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' We further observe that (34) is equivalent to the minimax problem min x,y max z Pρ(x, y, z), where Pρ(x, y, z) := f(x, y) + ρ( ˜f(x, y) − ˜f(x, z)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' (35) In view of Assumption 3(i), Pρ can be rewritten as Pρ(x, y, z) = � f1(x, y) + ρ ˜f1(x, y) − ρ ˜f1(x, z) � + � f2(x) + ρ ˜f2(y) − ρ ˜f2(z) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' (36) By this and Assumption 3, one can observe that Pρ enjoys the following nice properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Pρ is the sum of smooth function f1(x, y)+ ρ ˜f1(x, y)− ρ ˜f1(x, z) with Lipschitz continuous gradient and possibly nonsmooth function f2(x)+ρ ˜f2(y)−ρ ˜f2(z) with exactly computable proximal operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Pρ is nonconvex in (x, y) but concave in z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Thanks to the nice structure of Pρ, an approximate stationary point of the minimax problem (35) can be found by Algorithm 2 proposed in Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Based on the above observations, we are now ready to propose penalty methods for the unconstrained bilevel optimization problem (29) by solving either a sequence of or a single minimax problem in the form of (35).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' In particular, we first propose an ideal penalty method for (29) by solving a sequence of minimax problems (see Algorithm 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Then we propose a practical penalty method for (29) by finding an approximate stationary point of a single minimax problem (see Algorithm 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Algorithm 3 An ideal penalty method for problem (29) Input: positive sequences {ρk} and {ǫk} with limk→∞(ρk, ǫk) = (∞, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' 1: for k = 0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' do 2: Find an ǫk-optimal solution (xk, yk, zk) of problem (35) with ρ = ρk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' 3: end for The following theorem states a convergence result of Algorithm 3, whose proof is deferred to Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Theorem 3 (Convergence of Algorithm 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Suppose that Assumption 3 holds and that {(xk, yk, zk)} is generated by Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Then any accumulation point of {(xk, yk)} is an optimal solution of problem (29).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Notice that (35) is a nonconvex-concave minimax problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' It is typically hard to find an ǫ-optimal solution of (35) for an arbitrary ǫ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Consequently, Algorithm 3 is not implementable in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' We next propose a practical penalty method for problem (29) by finding an approximate stationary point of a single minimax problem (35) with a suitable choice of ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Algorithm 4 A practical penalty method for problem (29) Input: ε ∈ (0, 1/4], ρ = ε−1, (x0, y0) ∈ X × Y with ˜f(x0, y0) ≤ miny ˜f(x0, y) + ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' 1: Call Algorithm 2 with ǫ ← ε, ǫ0 ← ε3/2, ˆx0 ← (x0, y0), ˆy0 ← y0, and L∇h ← L∇f1 + 2ε−1L∇ ˜ f1 to find an ǫ-stationary point (xǫ, yǫ, zǫ) of problem (35) with ρ = ε−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' 2: Output: (xǫ, yǫ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' 8 Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' (i) The initial point (x0, y0) of Algorithm 4 can be found by an additional procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Indeed, one can first choose any x0 ∈ X and then apply accelerated proximal gradient method [38] to the problem miny ˜f(x0, y) for finding y0 ∈ Y such that ˜f(x0, y0) ≤ miny ˜f(x0, y) + ε;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' (ii) As seen from Theorem 2, an ǫ-stationary point of (35) can be successfully found in step 1 of Algorithm 4 by applying Algorithm 2 to (35);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' (iii) For the sake of simplicity, a single subproblem of the form (35) with static penalty and tolerance parameters is solved in Algorithm 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Nevertheless, Algorithm 4 can be modified into a perhaps practically more efficient algorithm by solving a sequence of subproblems of the form (35) with dynamic penalty and tolerance parameters instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' In order to characterize the approximate solution found by Algorithm 4, we next introduce a termi- nology called an ε-KKT solution of problem (29).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Recall that problem (29) can be viewed as problem (33).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' In the spirit of classical constrained opti- mization, one would naturally be interested in a KKT solution (x, y) of (33) or equivalently (29), namely, (x, y) satisfies ˜f(x, y) ≤ minz ˜f(x, z) and moreover (x, y) is a stationary point of the problem min x′,y′ f(x′, y′) + ρ � ˜f(x′, y′) − min z′ ˜f(x′, z′) � (37) for some ρ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Yet, due to the sophisticated problem structure, characterizing a stationary point of (37) is generally difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' On another hand, notice that problem (37) is equivalent to the minimax problem min x′,y′ max z′ f(x′, y′) + ρ( ˜f(x′, y′) − ˜f(x′, z′)), whose stationary point (x, y, z) according to Definition 2 satisfies 0 ∈ ∂f(x, y) + ρ∂ ˜f(x, y) − (ρ∇x ˜f(x, z);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' 0), 0 ∈ ρ∂z ˜f(x, z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' (38) Based on this observation, we are instead interested in a (weak) KKT solution of problem (29) and its inexact counterpart that are defined below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' The pair (x, y) is said to be a KKT solution of problem (29) if there exists (z, ρ) ∈ Rm×R+ such that (38) and ˜f(x, y) ≤ minz′ ˜f(x, z′) hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' In addition, for any ε > 0, (x, y) is said to be an ε-KKT solution of problem (29) if there exists (z, ρ) ∈ Rm × R+ such that dist � 0, ∂f(x, y) + ρ∂ ˜f(x, y) − (ρ∇x ˜f(x, z);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' 0) � ≤ ε, dist � 0, ρ∂z ˜f(x, z) � ≤ ε, ˜f(x, y) − min z′ ˜f(x, z′) ≤ ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' We are now ready to present a theorem regarding operation complexity of Algorithm 4, measured by the amount of evaluations of ∇f1, ∇ ˜f1 and proximal operator of f2 and ˜f2, for finding an O(ε)-KKT solution of (29), whose proof is deferred to Subsection 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Theorem 4 (Complexity of Algorithm 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Suppose that Assumption 3 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Let f ∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' ˜f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Dx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Dy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' ˜fhi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' ˜flow and flow be defined in (29),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' (30),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' (31) and (32),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' L∇f1 and L∇ ˜ f1 be given in Assumption 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' ε,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' ρ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' x0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' y0 and zǫ be given in Algorithm 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' and �L = L∇f1 + 2ε−1L∇ ˜ f1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' ˆα = min � 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' � 4ε/(Dy�L) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' (39) ˆδ = (2 + ˆα−1)(D2 x + D2 y)�L + max � ε/Dy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' ˆα�L/4 � D2 y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' �C = 4 max � 1 2�L,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' min � Dy ε ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' 4 ˆα�L �� ��� ˆδ + 2ˆα−1(f ∗ − flow + ε−1( ˜fhi − ˜flow) + εDy/4 + �L(D2 x + D2 y)) � � (3�L + ε/(2Dy))2/ min{�L,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' ε/(2Dy)} + 3�L + ε/(2Dy) �−2 ε3 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' �K = � 16(1 + f(x0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' y0) − flow + εDy/4)�Lε−2 + 32(1 + 4D2 y�L2ε−2)ε − 1 � + ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' � N = �� 96 √ 2(1 + (24�L + 4ε/Dy)�L−1) � + 2 � max � 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' � Dy�Lε−1 � × (( � K + 1)(log �C)+ + �K + 1 + 2 �K log( �K + 1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' 9 Then Algorithm 4 outputs an approximate solution (xǫ, yǫ) of (29) satisfying dist � 0, ∂f(xǫ, yǫ) + ρ∂ ˜f(xǫ, yǫ) − (ρ∇x ˜f(xǫ, zǫ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' 0) � ≤ ε, dist � 0, ρ∂ ˜f(xǫ, zǫ) � ≤ ε, (40) ˜f(xǫ, yǫ) ≤ min z ˜f(xǫ, z) + ε � 1 + f(x0, y0) − flow + 2ε3(�L−1 + 4D2 y�Lε−2) + Dyε/4 � , (41) after at most � N evaluations of ∇f1, ∇ ˜f1 and proximal operator of f2 and ˜f2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' One can observe from Theorem 4 that �L = O(ε−1), ˆα = O(ε), ˆδ = O(ε−2), �C = O(ε−11), �K = O(ε−3), and � N = O(ε−4 log ε−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Consequently, Algorithm 4 enjoys an operation complexity of O(ε−4 log ε−1), measured by the amount of evaluations of ∇f1, ∇ ˜f1 and proximal operator of f2 and ˜f2, for finding an O(ε)-KKT solution (xǫ, yǫ) of (29) satisfying dist � 0, ∂f(xǫ, yǫ) + ρ∂ ˜f(xǫ, yǫ) − (ρ∇x ˜f(xǫ, zǫ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' 0) � ≤ ε, dist � 0, ρ∂ ˜f(xǫ, zǫ) � ≤ ε, ˜f(xǫ, yǫ) − min z ˜f(xǫ, z) = O(ε), where zǫ is given in Algorithm 4 and ρ = ε−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' 4 Constrained bilevel optimization In this section, we consider a constrained bilevel optimization problem5 f ∗ = min f(x, y) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' y ∈ Argmin z { ˜f(x, z)|˜g(x, z) ≤ 0}, (42) where f and ˜f satisfy Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Recall from Assumption 3 that X = dom f2 and Y = dom ˜f2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' We now make some additional assumptions for problem (42).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' (i) f and ˜f are Lf- and L ˜ f-Lipschitz continuous on X × Y, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' (ii) ˜g : Rn × Rm → Rl is L∇˜g-smooth and L˜g-Lipschitz continuous on X × Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' (iii) ˜gi(x, ·) is convex and there exists ˆzx ∈ Y for each x ∈ X such that ˜gi(x, ˆzx) < 0 for all i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=', l and G := min{−˜gi(x, ˆzx)|x ∈ X, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' , l} > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content='6 For notational convenience, we define ˜f ∗(x) := min z { ˜f(x, z)|˜g(x, z) ≤ 0}, (43) ˜f ∗ hi := sup{ ˜f ∗(x)|x ∈ X}, (44) ˜ghi := max{∥˜g(x, y)∥ ��(x, y) ∈ X × Y}, (45) It then follows from Assumption 4(ii) that ∥∇˜g(x, y)∥ ≤ L˜g ∀(x, y) ∈ X × Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' (46) In addition, by Assumptions 3 and 4 and the compactness of X and Y, one can observe that ˜ghi and G are finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Besides, as will be shown in Lemma 6(ii), ˜f ∗ hi is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' The goal of this subsection is to propose penalty methods for solving problem (42).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' To this end, let us introduce a penalty function for the lower level optimization problem y ∈ Argmin z { ˜f(x, z)|˜g(x, z) ≤ 0} of (42), which is given by �Pµ(x, z) = ˜f(x, z) + µ ∥[˜g(x, z)]+∥2 (47) 5For convenience, problem (42) is referred to as a constrained bilevel optimization problem since its lower level part has at least one explicit constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' 6The latter part of this assumption can be weakened to the one that the pointwise Slater’s condition holds for the lower level part of (42), that is, there exists ˆzx ∈ Y such that ˜g(x, ˆzx) < 0 for each x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Indeed, if G > 0, Assumption 4(iii) clearly holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Otherwise, one can solve the perturbed counterpart of (42) with ˜g(x, z) being replaced by ˜g(x, z) − ǫ for some suitable ǫ > 0 instead, which satisfies Assumption 4(iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' 10 for a penalty parameter µ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Observe that problem (42) can be approximately solved as the uncon- strained bilevel optimization problem f ∗ µ = min x,y � f(x, y)|y ∈ Argmin z �Pµ(x, z) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' (48) Further, by the study in Section 3, problem (48) can be approximately solved as the penalty problem min x,y f(x, y) + ρ � �Pµ(x, y) − min z �Pµ(x, z) � (49) for some suitable ρ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' One can also observe that problem (49) is equivalent to the minimax problem min x,y max z Pρ,µ(x, y, z), where Pρ,µ(x, y, z) := f(x, y) + ρ( �Pµ(x, y) − �Pµ(x, z)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' (50) In view of (47), (50) and Assumption 3(i), Pρ,µ can be rewritten as Pρ,µ(x, y, z) = � f1(x, y) + ρ ˜f1(x, y) + ρµ ∥[˜g(x, y)]+∥2 − ρ ˜f1(x, z) − ρµ ∥[˜g(x, z)]+∥2 � + � f2(x) + ρ ˜f2(y) − ρ ˜f2(z) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' (51) By this and Assumptions 3 and 4, one can observe that Pρ,µ enjoys the following nice properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Pρ,µ is the sum of smooth function f1(x, y)+ρ ˜f1(x, y)+ρµ ∥[˜g(x, y)]+∥2−ρ ˜f1(x, z)−ρµ ∥[˜g(x, z)]+∥2 with Lipschitz continuous gradient and possibly nonsmooth function f2(x) + ρ ˜f2(y) − ρ ˜f2(z) with exactly computable proximal operator;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Pρ,µ is nonconvex in (x, y) but concave in z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Due to the nice structure of Pρ,µ, an approximate stationary point of the minimax problem (50) can be found by Algorithm 2 proposed in Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Based on the above observations, we are now ready to propose penalty methods for the constrained bilevel optimization problem (42) by solving a sequence of or a single minimax problem of the form (50).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' In particular, we first propose an ideal penalty method for (42) by solving a sequence of minimax problems (see Algorithm 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Then we propose a practical penalty method for (42) by finding an approximate stationary point of a single minimax problem (see Algorithm 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Algorithm 5 An ideal penalty method for problem (42) Input: positive sequences {ρk}, {µk} and {ǫk} with limk→∞(ρk, µk, ǫk) = (∞, ∞, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' 1: for k = 0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' do 2: Find an ǫk-optimal solution (xk, yk, zk) of problem (50) with (ρ, µ) = (ρk, µk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' 3: end for To study convergence of Algorithm 5, we make the following error bound assumption on the solution set of the lower level optimization problem of (42).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' This type of error bounds has been considered in the context of set-value mappings in the literature (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=', see [14]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Assumption 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' There exists a non-decreasing function ω : R+ → R+ with limθ↓0 ω(θ) = 0 and ¯θ > 0 such that dist(z, Sθ(x)) ≤ ω(θ) for all x ∈ X, z ∈ S0(x) and θ ∈ [0, ¯θ], where Sθ(x) := Argmin z { ˜f(x, z) : ∥[˜g(x, z)]+∥ ≤ θ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' We are now ready to state a convergence result of Algorithm 5, whose proof is deferred to Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Theorem 5 (Convergence of Algorithm 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Suppose that Assumptions 3-5 hold and that {(xk, yk, zk)} is generated by Algorithm 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Then any accumulation point of {(xk, yk)} is an optimal solution of problem (42).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Notice that (50) is a nonconvex-concave minimax problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' It is generally hard to find an ǫ-optimal solution of (50) for an arbitrary ǫ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' As a result, Algorithm 5 is generally not implementable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' We next propose a practical penalty method for problem (42) by finding an approximate stationary point of (50) with a suitable choice of ρ and µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' 11 Algorithm 6 A practical penalty method for problem (42) Input: ε ∈ (0, 1/4], ρ = ε−1, µ = ε−2, (x0, y0) ∈ X × Y with �Pµ(x0, y0) ≤ miny �Pµ(x0, y) + ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' 1: Call Algorithm 2 with ǫ ← ε, ǫ0 ← ε5/2, ˆx0 ← (x0, y0), ˆy0 ← y0, and L∇h ← L∇f1 + 2ρL∇ ˜f1 + 4ρµ(˜ghiL∇˜g +L2 ˜g) to find an ǫ-stationary point (xǫ, yǫ, zǫ) of problem (50) with ρ = ε−1 and µ = ε−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' 2: Output: (xǫ, yǫ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' (i) The initial point (x0, y0) of Algorithm 6 can be found by the similar procedure as described in Remark 4 with ˜f being replaced by �Pµ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' (ii) As seen from Theorem 2, an ǫ-stationary point of (50) can be successfully found in step 1 of Algorithm 6 by applying Algorithm 2 to (50);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' (iii) For the sake of simplicity, a single subproblem of the form (50) with static penalty and tolerance parameters is solved in Algorithm 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Nevertheless, Algorithm 6 can be modified into a perhaps practically more efficient algorithm by solving a sequence of subproblems of the form (50) with dynamic penalty and tolerance parameters instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' In order to characterize the approximate solution found by Algorithm 6, we next introduce a termi- nology called an ε-KKT solution of problem (42).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' By the definition of ˜f ∗ in (43), problem (42) can be viewed as min x,y {f(x, y)| ˜f(x, y) ≤ ˜f ∗(x), ˜g(x, y) ≤ 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' (52) Its associated Lagrangian function is given by L(x, y, ρ, λ) = f(x, y) + ρ( ˜f(x, y) − ˜f ∗(x)) + ⟨λ, ˜g(x, y)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' (53) In the spirit of classical constrained optimization, one would naturally be interested in a KKT solution (x, y) of (52) or equivalently (42), namely, (x, y) satisfies ˜f(x, y) ≤ ˜f ∗(x), ˜g(x, y) ≤ 0, ρ( ˜f(x, y) − ˜f ∗(x)) = 0, ⟨λ, ˜g(x, y)⟩ = 0, (54) and moreover (x, y) is a stationary point of the problem min x′,y′ L(x′, y′, ρ, λ) (55) for some ρ ≥ 0 and λ ∈ Rl +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Yet, due to the sophisticated problem structure, characterizing a stationary point of (55) is generally difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' On another hand, notice from Lemma 6 and (53) that problem (55) is equivalent to the minimax problem min x′,y′,˜λ′ max z′ � f(x′, y′) + ρ � ˜f(x′, y′) − ˜f(x′, z′) − ⟨˜λ′, ˜g(x′, z′)⟩ � + ⟨λ, ˜g(x′, y′)⟩ + IRl +(˜λ′) � , whose stationary point (x, y, ˜λ, z) according to Definition 2 satisfies 0 ∈ ∂f(x, y) + ρ∂ ˜f(x, y) − ρ(∇x ˜f(x, z) + ∇x˜g(x, z)˜λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' 0) + ∇˜g(x, y)λ, (56) 0 ∈ ρ(∂z ˜f(x, z) + ∇z˜g(x, z)˜λ), (57) ˜λ ∈ Rl +, ˜g(x, z) ≤ 0, ⟨˜λ, ˜g(x, z)⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' (58) Based on this observation and also the fact that (54) is equivalent to ˜f(x, y) = ˜f ∗(x), ˜g(x, y) ≤ 0, ⟨λ, ˜g(x, y)⟩ = 0, (59) we are instead interested in a (weak) KKT solution of problem (42) and its inexact counterpart that are defined below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' The pair (x, y) is said to be a KKT solution of problem (42) if there exists (z, ρ, λ, ˜λ) ∈ Rm × R+ × Rl + × Rl + such that (56)-(59) hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' In addition, for any ε > 0, (x, y) is said to be an ε-KKT solution of problem (42) if there exists (z, ρ, λ, ˜λ) ∈ Rm × R+ × Rl + × Rl + such that dist � 0, ∂f(x, y) + ρ∂ ˜f(x, y) − ρ(∇x ˜f(x, z) + ∇x˜g(x, z)˜λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' 0) + ∇˜g(x, y)λ � ≤ ε, dist � 0, ρ(∂z ˜f(x, z) + ∇z˜g(x, z)˜λ) � ≤ ε, ∥[˜g(x, z)]+∥ ≤ ε, |⟨˜λ, ˜g(x, z)⟩| ≤ ε, | ˜f(x, y) − ˜f ∗(x)| ≤ ε, ∥[˜g(x, y)]+∥ ≤ ε, |⟨λ, ˜g(x, y)⟩| ≤ ε, where ˜f ∗ is defined in (43).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' 12 We are now ready to present an operation complexity of Algorithm 6, measured by the amount of evaluations of ∇f1, ∇ ˜f1, ∇˜g and proximal operator of f2 and ˜f2, for finding an O(ε)-KKT solution of (42), whose proof is deferred to Subsection 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Theorem 6 (Complexity of Algorithm 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Suppose that Assumptions 3 and 4 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Let f ∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' ˜f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' ˜g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Dx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Dy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' ˜fhi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' ˜flow,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' flow,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' ˜f ∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' ˜f ∗ hi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' and ˜ghi be defined in (29),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' (30),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' (31),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' (32),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' (43),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' (44) and (45),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' L∇f1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' L∇ ˜ f1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' L ˜ f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' L∇˜g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' L˜g and G be given in Assumptions 3 and 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' ε,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' ρ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' µ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' x0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' y0 and zǫ be given in Algorithm 6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' and ˜λ = 2ε−1[˜g(xǫ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' zǫ)]+,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' ˆλ = 2ε−3[˜g(xǫ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' yǫ)]+,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' (60) �L = L∇f1 + 2ε−1L∇ ˜ f1 + 4ε−3(˜ghiL∇˜g + L2 ˜g),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' (61) ˜α = min � 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' � 4ε/(Dy�L) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' ˜δ = (2 + ˜α−1)(D2 x + D2 y)�L + max � ε/Dy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' ˜α�L/4 � D2 y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' �C = 4 max{1/(2�L),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' min{Dyε−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' 4/(˜α�L)}} [(3�L + ε/(2Dy))2/ min{�L,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' ε/(2Dy)} + 3�L + ε/(2Dy)]−2ε5 × � ˜δ + 2˜α−1[f ∗ − flow + 2ε−1( ˜fhi − ˜flow) + ε−3˜g2 hi + εDy/4 + �L(D2 x + D2 y)] � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' �K = � 32(1 + f(x0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' y0) − flow + εDy/4)�Lε−2 + 32ε3 � 1 + 4D2 y�L2ε−2� − 1 � + ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' � N = �� 96 √ 2 � 1 + (24�L + 4ε/Dy)�L−1�� + 2 � max � 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' � Dy�Lε−1 � × [( �K + 1)(log �C)+ + �K + 1 + 2 �K log( �K + 1)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Then Algorithm 6 outputs an approximate solution (xǫ, yǫ) of (42) satisfying dist � ∂f(xǫ, yǫ) + ρ∂ ˜f(xǫ, yǫ) − ρ(∇x ˜f(xǫ, zǫ) + ∇x˜g(xǫ, zǫ)˜λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' 0) + ∇˜g(xǫ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' yǫ)ˆλ � ≤ ε,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' (62) dist � 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' ρ(∂z ˜f(xǫ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' zǫ) + ∇z˜g(xǫ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' zǫ)˜λ) � ≤ ε,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' (63) ∥[˜g(xǫ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' zǫ)]+∥ ≤ ε2G−1Dy(ε2 + L ˜ f)/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' (64) |⟨˜λ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' ˜g(xǫ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' zǫ)⟩| ≤ ε2G−2D2 y(ρ−1ǫ + L ˜ f)2/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' (65) | ˜f(xǫ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' yǫ) − ˜f ∗(xǫ)| ≤ max � ε � 1 + f(x0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' y0) − flow + 2ε5(�L−1 + 4D2 y�Lε−2) + Dyε/4 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' ε2G−2D2 yL ˜ f(ε2 + εLf + L ˜ f)/2 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' (66) ∥[˜g(xǫ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' yǫ)]+∥ ≤ ε2G−1Dy(ε2 + εLf + L ˜ f)/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' (67) |⟨ˆλ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' ˜g(xǫ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' yǫ)⟩| ≤ εG−2D2 y(ε2 + εLf + L ˜ f)2/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' (68) after at most � N evaluations of ∇f1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' ∇ ˜f1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' ∇˜g and proximal operator of f2 and ˜f2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Remark 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' One can observe from Theorem 6 that �L = O(ε−3), ˜α = O(ε2), ˜δ = O(ε−5), �C = O(ε−23), �K = O(ε−5), and � N = O(ε−7 log ε−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Consequently, Algorithm 6 enjoys an operation complexity of O(ε−7 log ε−1), measured by the amount of evaluations of ∇f1, ∇ ˜f1, ∇˜g and proximal operator of f2 and ˜f2, for finding an O(ε)-KKT solution (xǫ, yǫ) of (42) satisfying dist � 0, ∂f(xǫ, yǫ) + ρ∂ ˜f(xǫ, yǫ) − ρ(∇x ˜f(xǫ, zǫ) + ∇x˜g(xǫ, zǫ)˜λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' 0) + ∇˜g(xǫ, yǫ)ˆλ � ≤ ε, dist � 0, ρ(∂z ˜f(xǫ, zǫ) + ∇z˜g(xǫ, zǫ)˜λ) � ≤ ε, ∥[˜g(xǫ, zǫ)]+∥ = O(ε2), |⟨˜λ, ˜g(xǫ, zǫ)⟩| = O(ε2), | ˜f(xǫ, yǫ) − ˜f ∗(xǫ)| = O(ε), ∥[˜g(xǫ, yǫ)]+∥ = O(ε2), |⟨ˆλ, ˜g(xǫ, yǫ)⟩| = O(ε), where ˜f ∗ is defined in (43), ˆλ, ˜λ ∈ Rl + are defined in (60), zǫ is given in Algorithm 6 and ρ = ε−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' 5 Proof of the main results In this section we provide a proof of our main results presented in Sections 2, 3 and 4, which are particularly Theorems 1-6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' 13 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content='1 Proof of the main results in Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content='1 In this subsection we prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Before proceeding, we establish a lemma below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Suppose that Assumptions 1 and 2 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Let ¯H∗, ¯Hlow, ϑ0 and ¯δ be defined in (8), (10), (14) and (16), and ¯α be given in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Then we have ϑ0 ≤ ¯δ + 2¯α−1 � ¯H∗ − ¯Hlow � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' (69) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' By (8),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' (10),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' (11) and (12),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' one has G(¯z0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' ¯y0) (12) = sup x � ⟨x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' ¯z0⟩ − p(x) − ˆh(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' ¯y0) + q(¯y0) � (11) = max x∈dom p � ⟨x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' ¯z0⟩ − p(x) − ¯h(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' ¯y0) + σx 2 ∥x∥2 − σy 2 ∥¯y0∥2 + q(¯y0) � (8)(10) ≤ max x∈dom p � ⟨x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' ¯z0⟩ + σx 2 ∥x∥2� − σy 2 ∥¯y0∥2 − ¯Hlow = max x∈dom p σx 2 ∥x + σ−1 x ¯z0∥2 − σ−1 x 2 ∥¯z0∥2 − σy 2 ∥¯y0∥2 − ¯Hlow ≤ σxD2 p 2 − σ−1 x 2 ∥¯z0∥2 − σy 2 ∥¯y0∥2 − ¯Hlow,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' (70) where the last inequality follows from (9) and the fact that z0 ∈ −σxdom p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Recall that (x∗, y∗) is the optimal solution of (8) and z∗ = −σxx∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' It follows from (8), (11) and (12) that G(z∗, y∗) (12) = sup x � ⟨x, z∗⟩ − p(x) − ˆh(x, y∗) + q(y∗) � ≥ ⟨x∗, z∗⟩ − p(x∗) − ˆh(x∗, y∗) + q(y∗) (11) = ⟨x∗, z∗⟩ + σx 2 ∥x∗∥2 − σy 2 ∥y∗∥2 − p(x∗) − ¯h(x∗, y∗) + q(y∗) = − σ−1 x 2 ∥z∗∥2 − σy 2 ∥y∗∥2 − ¯H∗, where the last equality follows from (8), the definition of (x∗, y∗), and z∗ = −σxx∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' This together with (13) and (70) implies that P(¯z0, ¯y0) − P(z∗, y∗) = σ−1 x 2 ∥¯z0∥2 + σy 2 ∥¯y0∥2 + G(¯z0, ¯y0) − σ−1 x 2 ∥z∗∥2 − σy 2 ∥y∗∥2 − G(z∗, y∗) ≤ σxD2 p/2 − ¯Hlow + ¯H∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Notice from Algorithm 1 that z0 = z0 f = ¯z0 ∈ −σxdom p and y0 = y0 f = ¯y0 ∈ dom q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' By these, z∗ = −σxx∗, (9), (14), and the above inequality, one has ϑ0 (14) = η−1 z ∥¯z0 − z∗∥2 + η−1 y ∥¯y0 − y∗∥2 + 2¯α−1(P(¯z0, ¯y0) − P(z∗, y∗)) ≤ η−1 z σ2 xD2 p + η−1 y D2 q + 2¯α−1 � σxD2 p/2 − ¯Hlow + ¯H∗� = η−1 z σ2 xD2 p + ¯α−1σxD2 p + η−1 y D2 q + 2¯α−1 � ¯H∗ − ¯Hlow � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Hence, the conclusion follows from this, (16), ηz = σx/2 and ηy = min {1/(2σy), 4/(¯ασx)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' We are now ready to prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Suppose for contradiction that Algorithm 1 runs for more than ¯K outer itera- tions, where ¯K is given in (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' By this and Algorithm 1, one can assert that (15) does not hold for k = ¯K − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' On the other hand, by (17) and [29, Theorem 3], one has ∥(x ¯ K, y ¯ K) − (x∗, y∗)∥ ≤ (ˆζ−1 + L∇¯h)−1τ/2, (71) where (x∗, y∗) is the optimal solution of problem (8) and ˆζ is an input of Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Notice from Algorithm 1 that (ˆx ¯ K, ˆy ¯ K) results from the forward-backward splitting (FBS) step applied to the strongly monotone inclusion problem 0 ∈ (∇x¯h(x, y), −∇y¯h(x, y)) + (∂p(x), ∂q(y)) at the point (x ¯ K, y ¯ K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' It then 14 follows from this, ˆζ = min{σx, σy}/L2 ∇¯h (see Algorithm 1), and the contraction property of FBS [5, Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content='5] that ∥(ˆx ¯ K, ˆy ¯ K) − (x∗, y∗)∥ ≤ ∥(x ¯ K, y ¯ K) − (x∗, y∗)∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Using this and (71),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' we have ∥ˆζ−1(x ¯ K − ˆx ¯ K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' ˆy ¯ K − y ¯ K) − (∇¯h(x ¯ K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' y ¯ K) − ∇¯h(ˆx ¯ K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' ˆy ¯ K))∥ ≤ ˆζ−1∥(x ¯ K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' y ¯ K) − (ˆx ¯ K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' ˆy ¯ K)∥ + ∥∇¯h(x ¯ K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' y ¯ K) − ∇¯h(ˆx ¯ K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' ˆy ¯ K)∥ ≤ (ˆζ−1 + L∇¯h)∥(x ¯ K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' y ¯ K) − (ˆx ¯ K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' ˆy ¯ K)∥ ≤ (ˆζ−1 + L∇¯h)(∥(x ¯ K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' y ¯ K) − (x∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' y∗)∥ + ∥(ˆx ¯ K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' ˆy ¯ K) − (x∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' y∗)∥) ≤ 2(ˆζ−1 + L∇¯h)∥(x ¯ K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' y ¯ K) − (x∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' y∗)∥ (71) ≤ τ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' where the second inequality uses the fact that ¯h is L∇¯h-smooth on dom p × dom q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' It follows that (15) holds for k = ¯K − 1, which contradicts the above assertion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Hence, Algorithm 1 must terminate in at most ¯K outer iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' We next show that the output of Algorithm 1 is a τ-stationary point of (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' To this end, suppose that Algorithm 1 terminates at some iteration k at which (15) is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Then by (4) and the definition of ˆxk+1 and ˆyk+1 (see steps 23 and 24 of Algorithm 1), one has 0 ∈ ˆζ∂p(ˆxk+1) + ˆxk+1 − xk+1 + ˆζ∇x¯h(xk+1, yk+1), 0 ∈ ˆζ∂q(ˆyk+1) + ˆyk+1 − yk+1 − ˆζ∇y¯h(xk+1, yk+1), which yield ˆζ−1(xk+1 − ˆxk+1) − ∇x¯h(xk+1, yk+1) ∈ ∂p(ˆxk+1), ˆζ−1(yk+1 − ˆyk+1) + ∇y¯h(xk+1, yk+1) ∈ ∂q(ˆyk+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' These together with the definition of ¯H in (8) imply that ∇x¯h(ˆxk+1, ˆyk+1) + ˆζ−1(xk+1 − ˆxk+1) − ∇x¯h(xk+1, yk+1) ∈ ∂x ¯H(ˆxk+1, ˆyk+1), ∇y¯h(ˆxk+1, ˆyk+1) − ˆζ−1(yk+1 − ˆyk+1) − ∇y¯h(xk+1, yk+1) ∈ ∂y ¯H(ˆxk+1, ˆyk+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Using these and (15), we obtain dist(0, ∂x ¯H(ˆxk+1, ˆyk+1))2 + dist(0, ∂y ¯H(ˆxk+1, ˆyk+1))2 ≤ ∥ˆζ−1(xk+1 − ˆxk+1) + ∇x¯h(ˆxk+1, ˆyk+1) − ∇x¯h(xk+1, yk+1)∥2 + ∥ˆζ−1(ˆyk+1 − yk+1) + ∇y¯h(ˆxk+1, ˆyk+1) − ∇y¯h(xk+1, yk+1)∥2 = ∥ˆζ−1(xk+1 − ˆxk+1, ˆyk+1 − yk+1) − (∇¯h(xk+1, yk+1) − ∇¯h(ˆxk+1, ˆyk+1))∥2 (15) ≤ τ2, which implies that dist(0, ∂x ¯H(ˆxk+1, ˆyk+1)) ≤ τ and dist(0, ∂y ¯H(ˆxk+1, ˆyk+1)) ≤ τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' It then follows from these and Definition 2 that the output (ˆxk+1, ˆyk+1) of Algorithm 1 is a τ-stationary point of (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Finally, we show that the total number of evaluations of ∇¯h and proximal operator of p and q performed in Algorithm 1 is no more than ¯N, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Indeed, notice from Algorithm 1 that ¯α = min � 1, � 8σy/σx � , which implies that 2/¯α = max{2, � σx/(2σy)} and ¯α ≤ � 8σy/σx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' By these, one has max � 2 ¯α, ¯ασx 4σy � ≤ max � 2, � σx 2σy , � 8σy σx σx 4σy � = max � 2, � σx 2σy � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' (72) In addition, by [29, Lemma 4], the number of inner iterations performed in each outer iteration of Algorithm 1 is at most T = � 48 √ 2 � 1 + 8L∇¯hσ−1 x �� − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Then one can observe that the number of evaluations of ∇¯h and proximal operator of p and q performed 15 in Algorithm 1 is at most (2T + 3) ¯K ≤ �� 96 √ 2 � 1 + 8L∇¯hσ−1 x �� + 2 � � max � 2 ¯α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' ¯ασx 4σy � log 4 max{ηzσ−2 x ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' ηy}ϑ0 (ˆζ−1 + L∇¯h)−2τ 2 � + (72) ≤ �� 96 √ 2 � 1 + 8L∇¯hσ−1 x �� + 2 � � max � 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' � σx 2σy � log 4 max{ηzσ−2 x ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' ηy}ϑ0 (ˆζ−1 + L∇¯h)−2τ 2 � + ≤ �� 96 √ 2 � 1 + 8L∇¯hσ−1 x �� + 2 � × � max � 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' � σx 2σy � log 4 max{1/(2σx),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' min {1/(2σy),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' 4/(¯ασx)}} ϑ0 (L2 ∇¯h/ min{σx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' σy} + L∇¯h)−2τ 2 � + (69)(18) ≤ ¯N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' where the second last inequality follows from the definition of ηy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' ηz and ˆζ in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Hence, the conclusion holds as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content='2 Proof of the main results in Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content='2 In this subsection we prove Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Before proceeding, let {(xk, yk)}k∈K denote all the iterates generated by Algorithm 2, where K is a subset of consecutive nonnegative integers starting from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Also, we define K − 1 = {k − 1 : k ∈ K}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' We first establish two lemmas and then use them to prove Theorem 2 subsequently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Suppose that Assumption 1 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Let {(xk, yk)}k∈K be generated by Algorithm 2, H∗, Dp, Dq, Hlow, α, δ be defined in (6), (9), (23), (24) and (25), L∇h be given in Assumption 1, ǫ, ǫk be given in Algorithm 2, and Nk = �� 96 √ 2 � 1 + (24L∇h + 4ǫ/Dq) L−1 ∇h �� + 2 � × � max � 2, � DqL∇h ǫ � × log 4 max � 1 2L∇h , min � Dq ǫ , 4 αL∇h �� � δ + 2α−1(H∗ − Hlow + ǫDq/4 + L∇hD2 p) � [(3L∇h + ǫ/(2Dq))2/ min{L∇h, ǫ/(2Dq)} + 3L∇h + ǫ/(2Dq)]−2 ǫ2 k � + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' (73) Then for all 0 ≤ k ∈ K−1, (xk+1, yk+1) is an ǫk-stationary point of (20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Moreover, the total number of evaluations of ∇h and proximal operator of p and q performed at iteration k of Algorithm 2 for generating (xk+1, yk+1) is no more than Nk, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Let (x∗, y∗) be an optimal solution of (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Recall that H, Hk and hk are given in (6), (20) and (21), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Then we have Hk,∗ := min x max y Hk(x, y) = min x max y � H(x, y) − ǫ 4Dq ∥y − ˆy0∥2 + L∇h∥x − xk∥2 � ≤ max y {H(x∗, y) + L∇h∥x∗ − xk∥2} (6)(9) ≤ H∗ + L∇hD2 p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' (74) Moreover, by (9) and (23), one has Hk,low := min (x,y)∈dom p×dom q Hk(x, y) = min (x,y)∈dom p×dom q � H(x, y) − ǫ 4Dq ∥y − ˆy0∥2 + L∇h∥x − xk∥2 � (23) ≥ Hlow − max y∈dom q ǫ 4Dq ∥y − ˆy0∥2 (9) ≥ Hlow − ǫDq/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' (75) In addition, by Assumption 1 and the definition of hk in (21), it is not hard to verify that hk(x, y) is L∇h-strongly-convex in x, ǫ/(2Dq)-strongly-concave in y, and (3L∇h + ǫ/(2Dq))-smooth on its domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Also, recall from Remark 2 that (xk+1, yk+1) results from applying Algorithm 1 to problem (20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' The conclusion of this lemma then follows by using (74) and (75) and applying Theorem 1 to (20) with τ = ǫk, σx = L∇h, σy = ǫ/(2Dq), L∇¯h = 3L∇h + ǫ/(2Dq), ¯α = α, ¯δ = δ, ¯Hlow = Hk,low, and ¯H∗ = Hk,∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' 16 Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Suppose that Assumption 1 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Let {xk}k∈K be generated by Algorithm 2, H, H∗ and Dq be defined in (6) and (9), L∇h be given in Assumption 1, and ǫ, ǫ0 and ˆx0 be given in Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Then for all 0 ≤ K ∈ K − 1, we have min 0≤k≤K ∥xk+1 − xk∥ ≤ maxy H(ˆx0, y) − H∗ + ǫDq/4 L∇h(K + 1) + 2ǫ2 0(1 + 4D2 qL2 ∇hǫ−2) L2 ∇h(K + 1) , (76) max y H(xK+1, y) ≤ max y H(ˆx0, y) + ǫDq/4 + 2ǫ2 0 � L−1 ∇h + 4D2 qL∇hǫ−2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' (77) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' For convenience of the proof, let H∗ ǫ (x) = max y � H(x, y) − ǫ∥y − ˆy0∥2/(4Dq) � , (78) H∗ k(x) = max y Hk(x, y), yk+1 ∗ = arg max y Hk(xk+1, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' (79) One can observe from these, (20) and (21) that H∗ k(x) = H∗ ǫ (x) + L∇h∥x − xk∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' (80) By this and Assumption 1, one can also see that H∗ k is L∇h-strongly convex on dom p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' In addition, recall from Lemma 2 that (xk+1, yk+1) is an ǫk-stationary point of problem (20) for all 0 ≤ k ∈ K − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' It then follows from Definition 2 that there exist some u ∈ ∂xHk(xk+1, yk+1) and v ∈ ∂yHk(xk+1, yk+1) with ∥u∥ ≤ ǫk and ∥v∥ ≤ ǫk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Also, by (79), one has 0 ∈ ∂yHk(xk+1, yk+1 ∗ ), which together with v ∈ ∂yHk(xk+1, yk+1) and ǫ/(2Dq)-strong concavity of Hk(xk+1, ·), implies that ⟨−v, yk+1 − yk+1 ∗ ⟩ ≥ ǫ∥yk+1 − yk+1 ∗ ∥2/(2Dq).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' This and ∥v∥ ≤ ǫk yield ∥yk+1 − yk+1 ∗ ∥ ≤ 2ǫkDq/ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' (81) In addition, by u ∈ ∂xHk(xk+1, yk+1), (20) and (21), one has u ∈ ∇xh(xk+1, yk+1) + ∂p(xk+1) + 2L∇h(xk+1 − xk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' (82) Also, observe from (20), (21) and (79) that ∂H∗ k(xk+1) = ∇xh(xk+1, yk+1 ∗ ) + ∂p(xk+1) + 2L∇h(xk+1 − xk), which together with (82) yields u + ∇xh(xk+1, yk+1 ∗ ) − ∇xh(xk+1, yk+1) ∈ ∂H∗ k(xk+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' By this and L∇h-strong convexity of H∗ k, one has H∗ k(xk) ≥ H∗ k(xk+1) + ⟨u + ∇xh(xk+1, yk+1 ∗ ) − ∇xh(xk+1, yk+1), xk − xk+1⟩ + L∇h∥xk − xk+1∥2/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' (83) Using this,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' (80),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' (81),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' (83),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' ∥u∥ ≤ ǫk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' and the Lipschitz continuity of ∇h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' we obtain H∗ ǫ (xk) − H∗ ǫ (xk+1) (80) = H∗ k(xk) − H∗ k(xk+1) + L∇h∥xk − xk+1∥2 (83) ≥ ⟨u + ∇xh(xk+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' yk+1 ∗ ) − ∇xh(xk+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' yk+1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' xk − xk+1⟩ + 3L∇h∥xk − xk+1∥2/2 ≥ � − ∥u + ∇xh(xk+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' yk+1 ∗ ) − ∇xh(xk+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' yk+1)∥∥xk − xk+1∥ + L∇h∥xk − xk+1∥2/2 � + L∇h∥xk − xk+1∥2 ≥ −(2L∇h)−1∥u + ∇xh(xk+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' yk+1 ∗ ) − ∇xh(xk+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' yk+1)∥2 + L∇h∥xk − xk+1∥2 ≥ −L−1 ∇h∥u∥2 − L−1 ∇h∥∇xh(xk+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' yk+1 ∗ ) − ∇xh(xk+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' yk+1)∥2 + L∇h∥xk − xk+1∥2 ≥ −L−1 ∇hǫ2 k − L∇h∥yk+1 − yk+1 ∗ ∥2 + L∇h∥xk − xk+1∥2 (81) ≥ −(L−1 ∇h + 4D2 qL∇hǫ−2)ǫ2 k + L∇h∥xk − xk+1∥2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' where the second and fourth inequalities follow from Cauchy-Schwartz inequality,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' and the third inequal- ity is due to Young’s inequality,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' and the fifth inequality follows from L∇h-Lipschitz continuity of ∇h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Summing up the above inequality for k = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=', K yields L∇h K � k=0 ∥xk − xk+1∥2 ≤ H∗ ǫ (x0) − H∗ ǫ (xK+1) + (L−1 ∇h + 4D2 qL∇hǫ−2) K � k=0 ǫ2 k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' (84) 17 In addition, it follows from (6), (9) and (78) that H∗ ǫ (xK+1) = max y � H(xK+1, y) − ǫ∥y − ˆy0∥2/(4Dq) � ≥ min x max y H(x, y) − ǫDq/4 = H∗ − ǫDq/4, H∗ ǫ (x0) = max y � H(x0, y) − ǫ∥y − ˆy0∥2/(4Dq) � ≤ max y H(x0, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' (85) These together with (84) yield L∇h(K + 1) min 0≤k≤K ∥xk+1 − xk∥2 ≤ L∇h K � k=0 ∥xk − xk+1∥2 ≤ max y H(x0, y) − H∗ + ǫDq/4 + (L−1 ∇h + 4D2 qL∇hǫ−2) K � k=0 ǫ2 k, which together with x0 = ˆx0, ǫk = ǫ0(k + 1)−1 and �K k=0(k + 1)−2 < 2 implies that (76) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Finally, we show that (77) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Indeed, it follows from (9), (78), (84), (85), ǫk = ǫ0(k + 1)−1, and �K k=0(k + 1)−2 < 2 that max y H(xK+1, y) (9) ≤ max y � H(xK+1, y) − ǫ∥y − ˆy0∥2/(4Dq) � + ǫDq/4 (78) = H∗ ǫ (xK+1) + ǫDq/4 (84) ≤ H∗ ǫ (x0) + ǫDq/4 + (L−1 ∇h + 4D2 qL∇hǫ−2) K � k=0 ǫ2 k (85) ≤ max y H(x0, y) + ǫDq/4 + 2ǫ2 0(L−1 ∇h + 4D2 qL∇hǫ−2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' It then follows from this and x0 = ˆx0 that (77) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' We are now ready to prove Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Suppose for contradiction that Algorithm 2 runs for more than K + 1 outer iterations, where K is given in (26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' By this and Algorithm 2, one can then assert that (22) does not hold for all 0 ≤ k ≤ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' On the other hand, by (26) and (76), one has min 0≤k≤K ∥xk+1 − xk∥2 (76) ≤ maxy H(ˆx0, y) − H∗ + ǫDq/4 L∇h(K + 1) + 2ǫ2 0(1 + 4D2 qL2 ∇hǫ−2) L2 ∇h(K + 1) (26) ≤ ǫ2 16L2 ∇h , which implies that there exists some 0 ≤ k ≤ K such that ∥xk+1 − xk∥ ≤ ǫ/(4L∇h), and thus (22) holds for such k, which contradicts the above assertion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Hence, Algorithm 2 must terminate in at most K + 1 outer iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Suppose that Algorithm 2 terminates at some iteration 0 ≤ k ≤ K, namely, (22) holds for such k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' We next show that its output (xǫ, yǫ) = (xk+1, yk+1) is an ǫ-stationary point of (6) and moreover it satisfies (28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Indeed, recall from Lemma 2 that (xk+1, yk+1) is an ǫk-stationary point of (20), namely, it satisfies dist(0, ∂xHk(xk+1, yk+1)) ≤ ǫk and dist(0, ∂yHk(xk+1, yk+1)) ≤ ǫk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' By these, (6), (20) and (21), there exists (u, v) such that u ∈ ∂xH(xk+1, yk+1) + 2L∇h(xk+1 − xk), ∥u∥ ≤ ǫk, v ∈ ∂yH(xk+1, yk+1) − ǫ(yk+1 − ˆy0)/(2Dq), ∥v∥ ≤ ǫk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' It then follows that u−2L∇h(xk+1−xk) ∈ ∂xH(xk+1, yk+1) and v+ǫ(yk+1−ˆy0)/(2Dq) ∈ ∂yH(xk+1, yk+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' These together with (9), (22), and ǫk ≤ ǫ0 ≤ ǫ/2 (see Algorithm 2) imply that dist � 0, ∂xH(xk+1, yk+1) � ≤ ∥u − 2L∇h(xk+1 − xk)∥ ≤ ∥u∥ + 2L∇h∥xk+1 − xk∥ (22) ≤ ǫk + ǫ/2 ≤ ǫ, dist � 0, ∂yH(xk+1, yk+1) � ≤ ∥v + ǫ(yk+1 − ˆy0)/(2Dq)∥ ≤ ∥v∥ + ǫ∥yk+1 − ˆy0∥/(2Dq) (9) ≤ ǫk + ǫ/2 ≤ ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Hence, the output (xk+1, yk+1) of Algorithm 2 is an ǫ-stationary point of (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' In addition, (28) holds due to Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' 18 Recall from Lemma 2 that the number of evaluations of ∇h and proximal operator of p and q performed at iteration k of Algorithm 2 is at most Nk, respectively, where Nk is defined in (73).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Also, one can observe from the above proof and the definition of K that |K| ≤ K + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' It then follows that the total number of evaluations of ∇h and proximal operator of p and q in Algorithm 2 is respectively no more than �|K|−2 k=0 Nk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Consequently, to complete the rest of the proof of Theorem 2, it suffices to show that �|K|−2 k=0 Nk ≤ N, where N is given in (27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Indeed,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' by (27),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' (73) and |K| ≤ K + 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' one has |K|−2 � k=0 Nk (73) ≤ K � k=0 �� 96 √ 2 � 1 + (24L∇h + 4ǫ/Dq) L−1 ∇h �� + 2 � × � max � 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' � DqL∇h ǫ � × log 4 max � 1 2L∇h ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' min � Dq ǫ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' 4 αL∇h �� � δ + 2α−1(H∗ − Hlow + ǫDq/4 + L∇hD2 p) � [(3L∇h + ǫ/(2Dq))2/ min{L∇h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' ǫ/(2Dq)} + 3L∇h + ǫ/(2Dq)]−2 ǫ2 k � + ≤ �� 96 √ 2 � 1 + (24L∇h + 4ǫ/Dq) L−1 ∇h �� + 2 � max � 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' � DqL∇h ǫ � × K � k=0 \uf8eb \uf8ed \uf8eb \uf8edlog 4 max � 1 2L∇h ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' min � Dq ǫ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' 4 αL∇h �� � δ + 2α−1(H∗ − hlow + ǫDq/4 + L∇hD2 p) � [(3L∇h + ǫ/(2Dq))2/ min{L∇h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' ǫ/(2Dq)} + 3L∇h + ǫ/(2Dq)]−2 ǫ2 k \uf8f6 \uf8f8 + + 1 \uf8f6 \uf8f8 ≤ �� 96 √ 2 � 1 + (24L∇h + 4ǫ/Dq) L−1 ∇h �� + 2 � max � 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' � DqL∇h ǫ � × � (K + 1) � log 4 max � 1 2L∇h ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' min � Dq ǫ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' 4 αL∇h �� � δ + 2α−1(H∗ − Hlow + ǫDq/4 + L∇hD2 p) � [(3L∇h + ǫ/(2Dq))2/ min{L∇h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' ǫ/(2Dq)} + 3L∇h + ǫ/(2Dq)]−2 ǫ2 0 � + + K + 1 + 2 K � k=0 log(k + 1) � (27) ≤ N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' where the last inequality is due to (27) and �K k=0 log(k + 1) ≤ K log(K + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' This completes the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content='3 Proof of the main results in Section 3 In this subsection we prove Theorems 3 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' We first establish a lemma below, which will be used to prove Theorem 3 subsequently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Suppose that Assumption 3 holds and (xǫ, yǫ, zǫ) is an ǫ-optimal solution of problem (35) for some ǫ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Let f, ˜f, f ∗, flow and ρ be given in (29), (32) and (35), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Then we have ˜f(xǫ, yǫ) ≤ min z ˜f(xǫ, z) + ρ−1(f ∗ − flow + 2ǫ), f(xǫ, yǫ) ≤ f ∗ + 2ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Since (xǫ, yǫ, zǫ) is an ǫ-optimal solution of (35), it follows from Definition 1 that max z Pρ(xǫ, yǫ, z) ≤ Pρ(xǫ, yǫ, zǫ) + ǫ, Pρ(xǫ, yǫ, zǫ) ≤ min x,y max z Pρ(x, y, z) + ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Summing up these inequalities yields max z Pρ(xǫ, yǫ, z) ≤ min x,y max z Pρ(x, y, z) + 2ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' (86) Let (x∗, y∗) be an optimal solution of (29).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' It then follows that f(x∗, y∗) = f ∗ and ˜f(x∗, y∗) = minz ˜f(x∗, z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' By these and the definition of Pρ in (35), one has max z Pρ(x∗, y∗, z) = f(x∗, y∗) + ρ( ˜f(x∗, y∗) − min z ˜f(x∗, z)) = f(x∗, y∗) = f ∗, which implies that min x,y max z Pρ(x, y, z) ≤ max z Pρ(x∗, y∗, z) = f ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' (87) 19 It then follows from (35), (86) and (87) that f(xǫ, yǫ) + ρ( ˜f(xǫ, yǫ) − min z ˜f(xǫ, z)) (35) = max z Pρ(xǫ, yǫ, z) (86)(87) ≤ f ∗ + 2ǫ, which together with ˜f(xǫ, yǫ) − minz ˜f(xǫ, z) ≥ 0 implies that f(xǫ, yǫ) ≤ f ∗ + 2ǫ, ˜f(xǫ, yǫ) ≤ min z ˜f(xǫ, z) + ρ−1 (f ∗ − f(xǫ, yǫ) + 2ǫ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' The conclusion of this lemma directly follows from these and (32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' We are now ready to prove Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Let {(xk, yk, zk)} be generated by Algorithm 3 with limk→∞(ρk, ǫk) = (∞, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' By considering a convergent subsequence if necessary, we assume without loss of generality that limk→∞(xk, yk) = (x∗, y∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' we now show that (x∗, y∗) is an optimal solution of problem (29).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Indeed, since (xk, yk, zk) is an ǫk-optimal solution of (35) with ρ = ρk, it follows from Lemma 4 with (ρ, ǫ) = (ρk, ǫk) and (xǫ, yǫ) = (xk, yk) that ˜f(xk, yk) ≤ min z ˜f(xk, z) + ρ−1 k (f ∗ − flow + 2ǫk), f(xk, yk) ≤ f ∗ + 2ǫk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' By the continuity of f and ˜f, limk→∞(xk, yk) = (x∗, y∗), limk→∞(ρk, ǫk) = (∞, 0), and taking limits as k → ∞ on both sides of the above relations, we obtain that ˜f(x∗, y∗) ≤ minz ˜f(x∗, z) and f(x∗, y∗) ≤ f ∗, which clearly imply that y∗ ∈ Argminz ˜f(x∗, z) and f(x∗, y∗) = f ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Hence, (x∗, y∗) is an optimal solution of (29) as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' We next prove Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Before proceeding, we establish a lemma below, which will be used to prove Theorem 4 subsequently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Suppose that Assumption 3 holds and (xǫ, yǫ, zǫ) is an ǫ-stationary point of (35).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Let Dy, flow, ˜f, ρ, and Pρ be given in (30), (32) and (35), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Then we have dist � 0, ∂f(xǫ, yǫ) + ρ∂ ˜f(xǫ, yǫ) − (ρ∇x ˜f(xǫ, zǫ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' 0) � ≤ ǫ, dist � 0, ρ∂ ˜f(xǫ, zǫ) � ≤ ǫ, ˜f(xǫ, yǫ) ≤ min z ˜f(xǫ, z) + ρ−1(max z Pρ(xǫ, yǫ, z) − flow).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Since (xǫ, yǫ, zǫ) is an ǫ-stationary point of (35), it follows from Definition 2 that dist � 0, ∂x,yPρ(xǫ, yǫ, zǫ) � ≤ ǫ, dist � 0, ∂zPρ(xǫ, yǫ, zǫ) � ≤ ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Using these and the definition of Pρ in (35), we have dist � 0, ∂f(xǫ, yǫ) + ρ∂ ˜f(xǫ, yǫ) − (ρ∇x ˜f(xǫ, zǫ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' 0) � ≤ ǫ, dist � 0, ρ∂ ˜f(xǫ, zǫ) � ≤ ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' In addition, by (35), we have f(xǫ, yǫ) + ρ( ˜f(xǫ, yǫ) − min z ˜f(xǫ, z)) = max z Pρ(xǫ, yǫ, z), which along with (32) implies that ˜f(xǫ, yǫ) − min z ˜f(xǫ, z) ≤ ρ−1(max z Pρ(xǫ, yǫ, z) − flow).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' This completes the proof of this lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' We are now ready to prove Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Observe from (36) that problem (35) can be viewed as min x,y max z {Pρ(x, y, z) = h(x, y, z) + p(x, y) − q(z)} , where h(x, y, z) = f1(x, y) + ρ ˜f1(x, y) − ρ ˜f1(x, z), p(x, y) = f2(x) + ρ ˜f2(y), and q(z) = ρ ˜f2(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Hence, problem (35) is in the form of (6) with H = Pρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' By Assumption 3 and ρ = ε−1, one can see that h is 20 �L-smooth on its domain, where �L is given in (39).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Also, notice from Algorithm 4 that ǫ0 = ε3/2 ≤ ε/2 due to ε ∈ (0, 1/4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Consequently, Algorithm 2 can be suitably applied to problem (35) with ρ = ε−1 for finding an ǫ-stationary point (xǫ, yǫ, zǫ) of it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' In addition, notice from Algorithm 4 that ˜f(x0, y0) ≤ miny ˜f(x0, y)+ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Using this, (35) and ρ = ε−1, we obtain max z Pρ(x0, y0, z) = f(x0, y0) + ρ( ˜f(x0, y0) − min z ˜f(x0, z)) ≤ f(x0, y0) + ρε = f(x0, y0) + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' (88) By this and (28) with H = Pρ, ǫ = ε, ǫ0 = ε3/2, ˆx0 = (x0, y0), Dq = Dy, and L∇h = �L, one has Pρ(xǫ, yǫ, zǫ) ≤ max z Pρ(x0, y0, z) + εDy/4 + 2ε3(�L−1 + 4D2 y�Lε−2) (88) ≤ 1 + f(x0, y0) + εDy/4 + 2ε3(�L−1 + 4D2 y�Lε−2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' It then follows from this and Lemma 5 with ǫ = ε and ρ = ε−1 that (xǫ, yǫ, zǫ) satisfies (40) and (41).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' We next show that at most � N evaluations of ∇f1, ∇ ˜f1, and proximal operator of f2 and ˜f2 are respectively performed in Algorithm 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Indeed, by (31), (32) and (35), one has min x,y max z Pρ(x, y, z) (35) = min x,y {f(x, y) + ρ( ˜f(x, y) − min z ˜f(x, z))} ≥ min (x,y)∈X ×Y f(x, y) (32) = flow, (89) min (x,y,z)∈X ×Y×Y Pρ(x, y, z) (35) = min (x,y,z)∈X ×Y×Y{f(x, y) + ρ( ˜f(x, y) − ˜f(x, z))} (31)(32) ≥ flow + ρ( ˜flow − ˜fhi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' (90) For convenience of the rest proof, let H = Pρ, H∗ = min x,y max z Pρ(x, y, z), Hlow = min{Pρ(x, y, z)|(x, y, z) ∈ X × Y × Y}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' (91) In view of these, (87), (88), (89), (90), and ρ = ε−1, we obtain that max z H(x0, y0, z) (88) ≤ f(x0, y0) + 1, flow (89) ≤ H∗ (87) ≤ f ∗, Hlow (90) ≥ flow + ρ( ˜flow − ˜fhi) = flow + ε−1( ˜flow − ˜fhi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Using these and Theorem 2 with ǫ = ε, ˆx0 = (x0, y0), Dp = � D2x + D2y, Dq = Dy, ǫ0 = ε3/2, L∇h = �L, α = ˆα, δ = ˆδ, and H, H∗, Hlow given in (91), we can conclude that Algorithm 4 performs at most � N evaluations of ∇f1, ∇ ˜f1 and proximal operator of f2 and ˜f2 respectively for finding an approximate solution (xǫ, yǫ) of problem (29) satisfying (40) and (41).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content='4 Proof of the main results in Section 4 In this subsection we prove Theorems 5 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Before proceeding, we define r = G−1Dy(ρ−1ǫ + L ˜ f), B+ r = {λ ∈ Rl + : ∥λ∥ ≤ r}, (92) where Dy is defined in (30), G is given in Assumption 4(iii), and ǫ and ρ are given in Algorithm 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' In addition, one can observe from (43) and (47) that min z �Pµ(x, z) ≤ ˜f ∗(x) ∀x ∈ X, (93) which will be frequently used later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' We next establish several technical lemmas that will be used to prove Theorem 5 subsequently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Suppose that Assumptions 3 and 4 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Let Dy, L ˜ f, G, ˜f ∗, ˜f ∗ hi and B+ r be given in (30), (43), (44), (92) and Assumption 4, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Then the following statements hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' (i) ∥λ∗∥ ≤ G−1L ˜ fDy and λ∗ ∈ B+ r for all λ∗ ∈ Λ∗(x) and x ∈ X, where Λ∗(x) denotes the set of optimal Lagrangian multipliers of problem (43) for any x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' 21 (ii) The function ˜f ∗ is Lipschitz continuous on X and ˜f ∗ hi is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' (iii) It holds that ˜f ∗(x) = max λ min z ˜f(x, z) + ⟨λ, ˜g(x, z)⟩ − IRl +(λ) ∀x ∈ X, (94) where IRl +(·) is the indicator function associated with Rl +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' (i) Let x ∈ X and λ∗ ∈ Λ∗(x) be arbitrarily chosen, and let z∗ ∈ Y be such that (z∗, λ∗) is a pair of primal-dual optimal solutions of (43).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' It then follows that z∗ ∈ Argmin z ˜f(x, z) + ⟨λ∗, ˜g(x, z)⟩, ⟨λ∗, ˜g(x, z∗)⟩ = 0, ˜g(x, z∗) ≤ 0, λ∗ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' The first relation above yields ˜f(x, z∗) + ⟨λ∗, ˜g(x, z∗)⟩ ≤ ˜f(x, ˆzx) + ⟨λ∗, ˜g(x, ˆzx)⟩, where ˆzx is given in Assumption 4(iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' By this and ⟨λ∗, ˜g(x, z∗)⟩ = 0, one has ⟨λ∗, −˜g(x, ˆzx)⟩ ≤ ˜f(x, ˆzx) − ˜f(x, z∗), which together with λ∗ ≥ 0, (30) and Assumption 4 implies that G l � i=1 λ∗ i ≤ ⟨λ∗, −˜g(x, ˆzx)⟩ ≤ ˜f(x, ˆzx) − ˜f(x, z∗) ≤ L ˜ f∥ˆzx − z∗∥ ≤ L ˜ fDy, (95) where the first inequality is due to Assumption 4(iii), and the third inequality follows from (30) and L ˜ f- Lipschitz continuity of ˜f (see Assumption 4(i)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' By (92), (95) and λ∗ ≥ 0, we have ∥λ∗∥ ≤ �l i=1 λ∗ i ≤ G−1L ˜ fDy and λ∗ ∈ B+ r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' (ii) Recall from Assumptions 3(i) and 4(iii) that ˜f(x, ·) and ˜gi(x, ·), i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' , l, are convex for any given x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Using this, (43) and the first statement of this lemma, we observe that ˜f ∗(x) = min z max λ∈B+ r ˜f(x, z) + ⟨λ, ˜g(x, z)⟩ ∀x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' (96) Notice from Assumption 4 that ˜f and ˜g are Lipschitz continuous on their domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Then it is not hard to observe that max{ ˜f(x, z)+⟨λ, ˜g(x, z)⟩|λ ∈ B+ r } is a Lipschitz continuous function of (x, z) on its domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' By this and (96), one can easily verify that ˜f ∗ is Lipschitz continuous on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' In addition, the finiteness of ˜f ∗ hi follows from (44), the continuity of ˜f ∗, and the compactness of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' (iii) One can observe from (43) that for all x ∈ X, ˜f ∗(x) = min z max λ ˜f(x, z) + ⟨λ, ˜g(x, z)⟩ − IRl +(λ) ≥ max λ min z ˜f(x, z) + ⟨λ, ˜g(x, z)⟩ − IRl +(λ) where the inequality follows from the weak duality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' In addition, it follows from Assumption 3 that the domain of ˜f(x, ·) is compact for all x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' By this, (96) and the strong duality, one has ˜f ∗(x) = max λ∈B+ r min z ˜f(x, z) + ⟨λ, ˜g(x, z)⟩ − IRl +(λ) ∀x ∈ X, which together with the above inequality implies that (94) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Suppose that Assumptions 3 and 4 hold and that (xǫ, yǫ, zǫ) is an ǫ-optimal solution of problem (50) for some ǫ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Let flow, f, �Pµ, f ∗ µ, ρ and µ be given in (32), (42), (47), (48) and (50), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Then we have �Pµ(xǫ, yǫ) ≤ min z �Pµ(xǫ, z) + ρ−1(f ∗ µ − flow + 2ǫ), f(xǫ, yǫ) ≤ f ∗ µ + 2ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' (97) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' The proof follows from the same argument as the one for Lemma 4 with f ∗ and ˜f being replaced by f ∗ µ and �Pµ, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Suppose that Assumptions 3-5 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Let ˜flow, f ∗, ˜f ∗ hi, f ∗ µ be defined in (31), (42), (44) and (48), and Lf, ω and ¯θ be given in Assumptions 4 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Suppose that µ ≥ ( ˜f ∗ hi − ˜flow)/¯θ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Then we have f ∗ µ ≤ f ∗ + Lfω �� µ−1( ˜f ∗ hi − ˜flow) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' (98) 22 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Let x ∈ X, y ∈ Argminz{ ˜f(x, z)|˜g(x, z) ≤ 0} and z∗ ∈ Argminz �Pµ(x, z) be arbitrarily chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' One can easily see from (47) and (93) that ˜f(x, z∗) + µ ∥[˜g(x, z∗)]+∥2 ≤ ˜f ∗(x), which together with (31) and (44) implies that ∥[˜g(x, z∗)]+∥2 ≤ µ−1( ˜f ∗ hi − ˜flow).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' (99) Since µ ≥ ( ˜f ∗ hi− ˜flow)/¯θ2, it follows from (99) that ∥[˜g(x, z∗)]+∥ ≤ ¯θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' By this relation, y ∈ Argmin z { ˜f(x, z)|˜g(x, z) ≤ 0} and Assumption 5, there exists some ˆz∗ such that ∥y − ˆz∗∥ ≤ ω(∥[˜g(x, z∗)]+∥), ˆz∗ ∈ Argmin z � ˜f(x, z) �� ∥[˜g(x, z)]+∥ ≤ ∥[˜g(x, z∗)]+∥ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' (100) In view of (47), z∗ ∈ Argminz �Pµ(x, z) and the second relation in (100), one can observe that ˆz∗ ∈ Argminz �Pµ(x, z), which along with (48) yields f(x, ˆz∗) ≥ f ∗ µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Also, using (100) and Lf-Lipschitz conti- nuity of f (see Assumption 4), we have f(x, y) − f(x, ˆz∗) ≥ −Lf∥y − ˆz∗∥ (100) ≥ −Lfω(∥[˜g(x, z∗)]+∥).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Taking minimum over x ∈ X and y ∈ Argminz{ ˜f(x, z)|˜g(x, z) ≤ 0} on both sides of this relation, and using (42), (99), f(x, ˆz∗) ≥ f ∗ µ and the monotonicity of ω, we can conclude that (98) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Suppose that Assumptions 3-5 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Let ˜flow, flow, f, ˜f, f ∗, ˜f ∗, ˜f ∗ hi, ρ and µ be given in (31), (32), (42), (43), (44) and (50), and Lf, ω and ¯θ be given in Assumptions 4 and 5, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Suppose that µ ≥ ( ˜f ∗ hi − ˜flow)/¯θ2 and (xǫ, yǫ, zǫ) is an ǫ-optimal solution of problem (50) for some ǫ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Then we have f(xǫ, yǫ) ≤ f ∗ + Lfω �� µ−1( ˜f ∗ hi − ˜flow) � + 2ǫ, ˜f(xǫ, yǫ) ≤ ˜f ∗(xǫ) + ρ−1� f ∗ − flow + Lfω �� µ−1( ˜f ∗ hi − ˜flow) � + 2ǫ � , ∥[˜g(xǫ, yǫ)]+∥2 ≤ µ−1� ˜f ∗(xǫ) − ˜flow + ρ−1� f ∗ − flow + Lfω �� µ−1( ˜f ∗ hi − ˜flow) � + 2ǫ �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' By (47), (93), and the first relation in (97), one has ˜f(xǫ, yǫ) + µ ∥[˜g(xǫ, yǫ)]+∥2 (47) = �Pµ(xǫ, yǫ) (93)(97) ≤ ˜f ∗(xǫ) + ρ−1(f ∗ µ − flow + 2ǫ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' It then follows from this and (31) that ˜f(xǫ, yǫ) ≤ ˜f ∗(xǫ) + ρ−1(f ∗ µ − flow + 2ǫ), ∥[˜g(xǫ, yǫ)]+∥2 ≤ µ−1( ˜f ∗(xǫ) − ˜flow + ρ−1(f ∗ µ − flow + 2ǫ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' In addition, recall from (97) that f(xǫ, yǫ) ≤ f ∗ µ + 2ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' The conclusion of this lemma then follows from these three relations and (98).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' We are now ready to prove Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Let {(xk, yk, zk)} be generated by Algorithm 5 with limk→∞(ρk, µk, ǫk) = (∞, ∞, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' By considering a convergent subsequence if necessary, we assume without loss of generality that limk→∞(xk, yk) = (x∗, y∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' We now show that (x∗, y∗) is an optimal solution of problem (42).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Indeed,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' since (xk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' yk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' zk) is an ǫk-optimal solution of (50) with (ρ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' µ) = (ρk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' µk) and limk→∞ µk = ∞,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' it follows from Lemma 9 with (ρ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' µ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' ǫ) = (ρk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' µk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' ǫk) and (xǫ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' yǫ) = (xk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' yk) that for all sufficiently large k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' one has f(xk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' yk) ≤ f ∗ + Lfω �� µ−1 k ( ˜f ∗ hi − ˜flow) � + 2ǫk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' ˜f(xk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' yk) ≤ ˜f ∗(xk) + ρ−1 k � f ∗ − flow + Lfω �� µ−1 k ( ˜f ∗ hi − ˜flow) � + 2ǫk � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' ��[˜g(xk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' yk)]+ ��2 ≤ µ−1 k � ˜f ∗(xk) − ˜flow + ρ−1 k � f ∗ − flow + Lfω �� µ−1 k ( ˜f ∗ hi − ˜flow) � + 2ǫk �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' By the continuity of f, ˜f and ˜f ∗ (see Assumption 3(i) and Lemma 6(ii)), limk→∞(xk, yk) = (x∗, y∗), limk→∞(ρk, µk, ǫk) = (∞, ∞, 0), limθ↓0 ω(θ) = 0, and taking limits as k → ∞ on both sides of the above relations, we obtain that f(x∗, y∗) ≤ f ∗, ˜f(x∗, y∗) ≤ ˜f ∗(x∗) and [˜g(x∗, y∗)]+ = 0, which along with (42) and (43) imply that f(x∗, y∗) = f ∗ and y∗ ∈ Argminz{ ˜f(x∗, z)|˜g(x∗, z) ≤ 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Hence, (x∗, y∗) is an optimal solution of (42) as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' 23 We next prove Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Before proceeding, we establish several technical lemmas below, which will be used to prove Theorem 6 subsequently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Suppose that Assumptions 3 and 4 hold and that (xǫ, yǫ, zǫ) is an ǫ-stationary point of problem (50) for some ǫ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Let Dy, ˜g, ρ, µ, Lf, L ˜ f and G be given in (30), (42), (50) and Assumption 4, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Then we have ∥[˜g(xǫ, zǫ)]+∥ ≤ (2µG)−1Dy(ρ−1ǫ + L ˜ f), (101) ∥[˜g(xǫ, yǫ)]+∥ ≤ (2µG)−1Dy(ρ−1ǫ + ρ−1Lf + L ˜ f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' (102) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' We first prove (101).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Since (xǫ, yǫ, zǫ) is an ǫ-stationary point of (50), it follows from Definition 2 that dist(0, ∂zPρ,µ(xǫ, yǫ, zǫ)) ≤ ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Also, by (47) and (50), one has Pρ,µ(x, y, z) = f(x, y) + ρ( ˜f(x, y) + µ ∥[˜g(x, y)]+∥2) − ρ( ˜f(x, z) + µ ∥[˜g(x, z)]+∥2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' (103) Using these relations, we have dist � 0, ∂z ˜f(xǫ, zǫ) + 2µ l � i=1 [˜gi(xǫ, zǫ)]+∇z˜gi(xǫ, zǫ) � ≤ ρ−1ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Hence, there exists s ∈ ∂z ˜f(xǫ, zǫ) such that ���s + 2µ l � i=1 [˜gi(xǫ, zǫ)]+∇z˜gi(xǫ, zǫ) ��� ≤ ρ−1ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' (104) Let ˆzxǫ and G be given in Assumption 4(iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' It then follows that ˆzxǫ ∈ Y and −˜gi(xǫ, ˆzxǫ) ≥ G > 0 for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Notice that [˜gi(xǫ, zǫ)]+˜gi(xǫ, zǫ) ≥ 0 for all i and ∥zǫ − ˆzxǫ∥ ≤ Dy due to (30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Using these,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' (104),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' and the convexity of ˜f(xǫ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' ·) and ˜gi(xǫ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' ·) for all i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' we have ˜f(xǫ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' zǫ) − ˜f(xǫ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' ˆzxǫ) + 2µG l � i=1 [˜gi(xǫ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' zǫ)]+ ≤ ˜f(xǫ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' zǫ) − ˜f(xǫ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' ˆzxǫ) − 2µ l � i=1 [˜gi(xǫ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' zǫ)]+˜gi(xǫ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' ˆzxǫ) ≤ ˜f(xǫ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' zǫ) − ˜f(xǫ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' ˆzxǫ) + 2µ l � i=1 [˜gi(xǫ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' zǫ)]+(˜gi(xǫ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' zǫ) − ˜gi(xǫ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' ˆzxǫ)) ≤ ⟨s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' zǫ − ˆzxǫ⟩ + 2µ l � i=1 [˜gi(xǫ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' zǫ)]+⟨∇z˜gi(xǫ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' zǫ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' zǫ − ˆzxǫ⟩ = ⟨s + 2µ l � i=1 [˜g(xǫ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' zǫ)]+∇z˜gi(xǫ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' zǫ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' zǫ − ˆzxǫ⟩ ≤ ρ−1Dyǫ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' (105) where the first inequality is due to −˜gi(xǫ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' ˆzxǫ) ≥ G for all i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' the second inequality follows from [˜gi(xǫ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' zǫ)]+˜gi(xǫ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' zǫ) ≥ 0 for all i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' the third inequality is due to s ∈ ∂z ˜f(xǫ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' zǫ) and the convexity of ˜f(xǫ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' ·) and ˜gi(xǫ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' ·) for all i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' and the last inequality follows from (30) and (104).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' In view of (30), (105), and L ˜ f-Lipschitz continuity of ˜f(x, y) (see Assumption 4), one has ∥[˜g(xǫ, zǫ)]+∥ ≤ l � i=1 [˜gi(xǫ, zǫ)]+ (105) ≤ (2µG)−1(ρ−1Dyǫ + ˜f(xǫ, ˆzxǫ) − ˜f(xǫ, zǫ)) ≤ (2µG)−1(ρ−1Dyǫ + L ˜ f∥ˆzxǫ − zǫ∥) (30) ≤ (2µG)−1Dy(ρ−1ǫ + L ˜ f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Hence, (101) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' We next prove (102).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Since (xǫ, yǫ, zǫ) is an ǫ-stationary point of (50), it follows from Definition 2 that dist(0, ∂yPρ,µ(xǫ, yǫ, zǫ)) ≤ ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' This together with (103) implies that dist � 0, ∂yf(xǫ, yǫ) + ρ∂y ˜f(xǫ, yǫ) + 2ρµ∇y˜g(xǫ, yǫ)[˜g(xǫ, yǫ)]+ � ≤ ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Hence, there exists s ∈ ∂yf(xǫ, yǫ) and ˜s ∈ ∂y ˜f(xǫ, yǫ) such that ∥s + ρ˜s + 2ρµ∇y˜g(xǫ, yǫ)[˜g(xǫ, yǫ)]+∥ ≤ ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' (106) 24 Let ¯ A(xǫ, yǫ) = {i|˜gi(xǫ, yǫ) > 0, 1 ≤ i ≤ l}, ˆzxǫ and G be given in Assumption 4(iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' It then follows that ˆzxǫ ∈ Y and −˜gi(xǫ, ˆzxǫ) ≥ G > 0 for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Using these and the convexity of ˜gi(xǫ, ·) for all i, we have ⟨∇y˜g(xǫ, yǫ)[˜g(xǫ, yǫ)]+, yǫ − ˆzxǫ⟩ = � i∈ ¯ A(xǫ,yǫ) ⟨∇y˜gi(xǫ, yǫ), yǫ − ˆzxǫ⟩[gi(xǫ, yǫ)]+ ≥ � i∈ ¯ A(xǫ,yǫ) (˜gi(xǫ, yǫ) − ˜gi(xǫ, ˆzxǫ))[˜gi(xǫ, yǫ)]+ ≥ � i∈ ¯ A(xǫ,yǫ) G[˜gi(xǫ, yǫ)]+ = G l � i=1 [˜gi(xǫ, yǫ)]+ ≥ G ∥[˜g(xǫ, yǫ)]+∥ , (107) where the first inequality follows from the convexity of ˜g(xǫ, ·) and the second inequality is due to −˜gi(xǫ, ˆzxǫ) ≥ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' It then follows from this,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' (106) and (107) that Dyǫ ≥ ∥s + ρ˜s + 2ρµ∇y˜g(xǫ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' yǫ)[˜g(xǫ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' yǫ)]+∥ · ∥yǫ − ˆzxǫ∥ ≥ ⟨s + ρ˜s + 2ρµ∇y˜g(xǫ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' yǫ)[˜g(xǫ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' yǫ)]+,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' yǫ − ˆzxǫ⟩ = ⟨s + ρ˜s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' yǫ − ˆzxǫ⟩ + 2ρµ⟨∇y˜g(xǫ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' yǫ)[˜g(xǫ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' yǫ)]+,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' yǫ − ˆzxǫ⟩ (107) ≥ − (∥s∥ + ρ∥˜s∥) ∥yǫ − ˆzxǫ∥ + 2ρµG ∥[˜g(xǫ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' yǫ)]+∥ ≥ −(Lf + ρL ˜ f)Dy + 2ρµG ∥[˜g(xǫ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' yǫ)]+∥ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' (108) where the last inequality follows from ∥yǫ − ˆzxǫ∥ ≤ Dy and the fact that ∥s∥ ≤ Lf and ∥˜s∥ ≤ L ˜ f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' which are due to (30),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' s ∈ ∂yf(xǫ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' yǫ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' ˜s ∈ ∂y ˜f(xǫ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' yǫ) and Assumption 4(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' By (108), one can immediately see that (102) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Lemma 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Suppose that Assumptions 3 and 4 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Let f, ˜f, ˜g, Dy, flow, ˜f ∗ and Pρ,µ be given in (29), (30), (32), (43) and (50), Lf, L ˜ f and G be given in Assumptions 3 and 4, (xǫ, yǫ, zǫ) be an ǫ-stationary point of (50) for some ǫ > 0, and ˜λ = 2µ[˜g(xǫ, zǫ)]+, ˆλ = 2ρµ[˜g(xǫ, yǫ)]+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' (109) Then we have dist � ∂f(xǫ, yǫ) + ρ∂ ˜f(xǫ, yǫ) − ρ(∇x ˜f(xǫ, zǫ) + ∇x˜g(xǫ, zǫ)˜λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' 0) + ∇˜g(xǫ, yǫ)ˆλ � ≤ ǫ, (110) dist � 0, ρ(∂z ˜f(xǫ, zǫ) + ∇z˜g(xǫ, zǫ)˜λ) � ≤ ǫ, (111) ∥[˜g(xǫ, zǫ)]+∥ ≤ (2µG)−1Dy(ρ−1ǫ + L ˜ f), (112) |⟨˜λ, ˜g(xǫ, zǫ)⟩| ≤ (2µ)−1G−2D2 y(ρ−1ǫ + L ˜ f)2, (113) | ˜f(xǫ, yǫ) − ˜f ∗(xǫ)| ≤ max � ρ−1(max z Pρ,µ(xǫ, yǫ, z) − flow), (2µ)−1G−2D2 yL ˜ f(ρ−1ǫ + ρ−1Lf + L ˜ f) � , (114) ∥[˜g(xǫ, yǫ)]+∥ ≤ (2µG)−1Dy(ρ−1ǫ + ρ−1Lf + L ˜ f), (115) |⟨ˆλ, ˜g(xǫ, yǫ)⟩| ≤ (2µ)−1ρG−2D2 y(ρ−1ǫ + ρ−1Lf + L ˜ f)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' (116) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Since (xǫ, yǫ, zǫ) is an ǫ-stationary point of (50), it easily follows from (103), (109) and Definition 2 that (110) and (111) hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Also, it follows from (101) and (102) that (112) and (115) hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' In addition, in view of (109), (112) and (115), one has |⟨˜λ, ˜g(xǫ, zǫ)⟩| (109) = 2µ ∥[˜g(xǫ, zǫ)]+∥2 (112) ≤ (2µ)−1G−2D2 y(ρ−1ǫ + L ˜ f)2, |⟨ˆλ, ˜g(xǫ, yǫ)⟩| (109) = 2ρµ ∥[˜g(xǫ, yǫ)]∥+∥2 (115) ≤ (2µ)−1ρG−2D2 y(ρ−1ǫ + ρ−1Lf + L ˜ f)2, and hence (113) and (116) hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Also, observe from the definition of Pρ,µ in (50) that �Pµ(xǫ, yǫ) − min z �Pµ(xǫ, z) = ρ−1(max z Pρ,µ(xǫ, yǫ, z) − f(xǫ, yǫ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' 25 Using this, (32), (47) and (93), we obtain that ˜f(xǫ, yǫ) + µ ∥[˜g(xǫ, yǫ)]+∥2 (47) = �Pµ(xǫ, yǫ) = min z �Pµ(xǫ, z) + ρ−1(max z Pρ,µ(xǫ, yǫ, z) − f(xǫ, yǫ)) (32)(93) ≤ ˜f ∗(xǫ) + ρ−1(max z Pρ,µ(xǫ, yǫ, z) − flow).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' (117) On the other hand, let λ∗ ∈ Rl + be an optimal Lagrangian multiplier of problem (43) with x = xǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' It then follows from Lemma 6(i) that ∥λ∗∥ ≤ G−1L ˜ fDy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Using these and (115), we have ˜f ∗(xǫ) = min y � ˜f(xǫ, y) + ⟨λ∗, ˜g(xǫ, y)⟩ � ≤ ˜f(xǫ, yǫ) + ⟨λ∗, ˜g(xǫ, yǫ)⟩ ≤ ˜f(xǫ, yǫ) + ∥λ∗∥∥[˜g(xǫ, yǫ)]+∥ ≤ ˜f(xǫ, yǫ) + (2µ)−1G−2D2 yL ˜ f(ρ−1ǫ + ρ−1Lf + L ˜ f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' By this and (117), one can see that (114) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' We are now ready to prove Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Proof of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Observe from (51) that problem (50) can be viewed as min x,y max z {Pρ,µ(x, y, z) = h(x, y, z) + p(x, y) − q(z)} , where h(x, y, z) = f1(x, y) + ρ ˜f1(x, y) + ρµ ∥[˜g(x, y)]+∥2 − ρ ˜f1(x, z) − ρµ ∥[˜g(x, z)]+∥2, p(x, y) = f2(x) + ρ ˜f2(y) and q(z) = ρ ˜f2(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Hence, problem (50) is in the form of (6) with H = Pρ,µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' By Assumption 3, (45), (46), ρ = ε−1 and µ = ε−2, one can see that h is �L-smooth on its domain, where �L is given in (61).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Also, notice from Algorithm 6 that ǫ0 = ε5/2 ≤ ε/2 = ǫ/2 due to ε ∈ (0, 1/4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Consequently, Algorithm 2 can be suitably applied to problem (50) with ρ = ε−1 and µ = ε−2 for finding an ǫ-stationary point (xǫ, yǫ, zǫ) of it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' In addition, notice from Algorithm 6 that �Pµ(x0, y0) ≤ miny �Pµ(x0, y) + ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Using this, (50) and ρ = ε−1, we obtain max z Pρ,µ(x0, y0, z) (50) = f(x0, y0) + ρ( �Pµ(x0, y0) − min z �Pµ(x0, z)) ≤ f(x0, y0) + ρε = f(x0, y0) + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' (118) By this and (28) with H = Pρ,µ, ǫ = ε, ǫ0 = ε5/2, ˆx0 = (x0, y0), Dq = Dy and L∇h = �L, one has Pρ,µ(xǫ, yǫ, zǫ) ≤ max z Pρ,µ(x0, y0, z) + εDy/4 + 2ε5(�L−1 + 4D2 y�Lε−2) (118) ≤ 1 + f(x0, y0) + εDy/4 + 2ε5(�L−1 + 4D2 y�Lε−2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' It then follows from this and Lemma 11 with ǫ = ε, ρ = ε−1 and µ = ε−2 that (xǫ, yǫ, zǫ) satisfies the relations (62)-(68).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' We next show that at most � N evaluations of ∇f1, ∇ ˜f1, ∇˜g and proximal operator of f2 and ˜f2 are respectively performed in Algorithm 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Indeed,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' by (31),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' (32),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' (45),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' (47) and (50),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' one has min x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content='y max z Pρ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content='µ(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' z) (50) = min x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content='y {f(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' y) + ρ( �Pµ(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' y) − min z �Pµ(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' z))} ≥ min (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content='y)∈X ×Y f(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' y) (32) = flow,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' (119) min{Pρ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content='µ(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' z)|(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' z) ∈ X × Y × Y} (50) = min{f(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' y) + ρ( �Pµ(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' y) − �Pµ(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' z))|(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' z) ∈ X × Y × Y} (47) = min{f(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' y) + ρ( ˜f(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' y) + µ∥[˜g(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' y)]+∥2 − ˜f(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' z) − µ∥[˜g(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' z)]+∥2)|(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' z) ∈ X × Y × Y} ≥ flow + ρ( ˜flow − ˜fhi) − ρµ˜g2 hi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' (120) where the last inequality follows from (31),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' (32) and (45).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' In addition, let (x∗, y∗) be an optimal solution of (42).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' It then follows that f(x∗, y∗) = f ∗ and [˜g(x∗, y∗)]+ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' By these, (31), (47) and (50), one has min x,y max z Pρ,µ(x, y, z) ≤ max z Pρ,µ(x∗, y∗, z) (50) = f(x∗, y∗) + ρ � �Pµ(x∗, y∗) − min z �Pµ(x∗, z) � (47) = f(x∗, y∗) + ρ( ˜f(x∗, y∗) + µ∥[˜g(x∗, y∗)]+∥2 − min z { ˜f(x∗, z) + µ∥[˜g(x∗, z)]+∥2}) (31) ≤ f ∗ + ρ( ˜fhi − ˜flow).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' (121) 26 For convenience of the rest proof, let H = Pρ,µ, H∗ = min x,y max z Pρ,µ(x, y, z), Hlow = min{Pρ,µ(x, y, z)|(x, y, z) ∈ X × Y × Y}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' (122) In view of these, (118), (119), (120), (121), ρ = ε−1 and µ = ε−2, we obtain that max z H(x0, y0, z) (118) ≤ f(x0, y0) + 1, flow (119) ≤ H∗ (121) ≤ f ∗ + ρ( ˜fhi − ˜flow) = f ∗ + ε−1( ˜fhi − ˜flow), Hlow (120) ≥ flow + ρ( ˜flow − ˜fhi) − ρµ˜g2 hi = flow + ε−1( ˜flow − ˜fhi) − ε−3˜g2 hi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Using these and Theorem 2 with ǫ = ε, ˆx0 = (x0, y0), Dp = � D2x + D2y, Dq = Dy, ǫ0 = ε5/2, L∇h = �L, α = ˜α, δ = ˜δ, and H, H∗, Hlow given in (122), we can conclude that Algorithm 6 performs at most � N evaluations of ∇f1, ∇ ˜f1, ∇˜g and proximal operator of f2 and ˜f2 for finding an approximate solution (xǫ, yǫ) of problem (42) satisfying (62)-(68).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' 6 Concluding remarks For the sake of simplicity, first-order penalty methods are proposed only for problem (3) in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' It would be interesting to extend them to problem (1) by using a standard technique (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=', see [39]) for handling the constraint g(x, y) ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' In addition, a single subproblem with static penalty and tolerance parameters is solved in our methods (Algorithms 4 and 6), which may be conservative in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' To make the methods possibly practically more efficient, it would be natural to modify them by solving a sequence of subproblems with dynamic penalty and tolerance parameters instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' These along with numerical experiments will be left for the future research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' References [1] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Allende and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Still.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Solving bilevel programs with the KKT-approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Mathematical pro- gramming, 138(1):309–332, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' [2] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Bard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Practical bilevel optimization: algorithms and applications, volume 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Springer Science & Business Media, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' [3] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Bennett, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Kunapuli, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Hu, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Pang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Bilevel optimization and machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' In IEEE World Congress on Computational Intelligence, pages 25–47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Springer, 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' [4] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Bertinetto, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Henriques, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Torr, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Vedaldi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Meta-learning with differentiable closed-form solvers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' In International Conference on Learning Representations, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' [5] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Chen and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Rockafellar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Convergence rates in forward–backward splitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' SIAM Journal on Optimization, 7(2):421–444, 1997.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' [6] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Chen, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Sun, and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Yin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' A single-timescale stochastic bilevel optimization method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' arXiv preprint arXiv:2102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content='04671, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' [7] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Clarke.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Optimization and nonsmooth analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' SIAM, 1990.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' [8] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Colson, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Marcotte, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Savard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' An overview of bilevel optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Annals of operations research, 153(1):235–256, 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' [9] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Crockett, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Fessler, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Bilevel methods for image reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Foundations and Trends® in Signal Processing, 15(2-3):121–289, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' [10] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Dempe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Foundations of bilevel programming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Springer Science & Business Media, 2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' [11] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Dempe, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Kalashnikov, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' P´erez-Vald´es, and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Kalashnykova.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Bilevel programming prob- lems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Energy Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Springer, Berlin, 10:978–3, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' [12] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Dempe and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Zemkoho.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Bilevel optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' In Springer optimization and its applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' 161.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Springer, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' 27 [13] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Dempe and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Zemkoho.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' The bilevel programming problem: reformulations, constraint qual- ifications and optimality conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Mathematical Programming, 138(1):447–473, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' [14] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Dontchev and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Rockafellar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Implicit functions and solution mappings, volume 543.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Springer, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' [15] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Feurer and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Hutter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Hyperparameter optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' In Automated machine learning, pages 3–33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Springer, Cham, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' [16] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Franceschi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Donini, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Frasconi, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Pontil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Forward and reverse gradient-based hy- perparameter optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' In International Conference on Machine Learning, pages 1165–1173, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' [17] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Franceschi, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Frasconi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Salzo, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Grazzi, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Pontil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Bilevel programming for hyperpa- rameter optimization and meta-learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' In International Conference on Machine Learning, pages 1568–1577, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' [18] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Guo and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Yang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Randomized stochastic variance-reduced methods for stochastic bilevel opti- mization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' arXiv e-prints, pages arXiv–2105, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' [19] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Hansen, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Jaumard, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Savard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' New branch-and-bound rules for linear bilevel programming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' SIAM Journal on scientific and Statistical Computing, 13(5):1194–1217, 1992.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' [20] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Hong, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content='-T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Wai, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Wang, and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Yang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' A two-timescale framework for bilevel optimization: Complexity analysis and application to actor-critic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' arXiv preprint arXiv:2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content='05170, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' [21] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Hu, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Xiao, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Liu, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Toh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' An improved unconstrained approach for bilevel optimiza- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' arXiv preprint arXiv:2208.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content='00732, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' [22] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Ishizuka and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Aiyoshi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Double penalty method for bilevel optimization problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Annals of Operations Research, 34(1):73–88, 1992.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' [23] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Ji, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Lee, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Liang, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Poor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Convergence of meta-learning with task-specific adapta- tion over partial parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Advances in Neural Information Processing Systems, 33:11490–11500, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' [24] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Ji, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Yang, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Liang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Bilevel optimization: Nonasymptotic analysis and faster algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' arXiv preprint arXiv:2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content='07962, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' [25] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Kaplan and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Tichatschke.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Proximal point methods and nonconvex optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Journal of global Optimization, 13(4):389–406, 1998.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' [26] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Khanduri, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Zeng, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Hong, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content='-T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Wai, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Wang, and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Yang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' A near-optimal algorithm for stochastic bilevel optimization via double-momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Advances in Neural Information Processing Systems, 34, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' [27] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Konda and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Tsitsiklis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Actor-critic algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Advances in neural information processing systems, 12, 1999.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' [28] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Kong and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Monteiro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' An accelerated inexact proximal point method for solving nonconvex- concave min-max problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' SIAM Journal on Optimization, 31(4):2558–2585, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' [29] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Kovalev and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Gasnikov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' The first optimal algorithm for smooth and strongly-convex-strongly- concave minimax optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' arXiv preprint arXiv:2205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content='05653, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' [30] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Liu, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Simonyan, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Yang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Darts: Differentiable architecture search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' In International Conference on Learning Representations, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' [31] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Liu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Gao, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Zhang, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Meng, and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Lin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Investigating bi-level optimization for learning and vision from a unified perspective: A survey and beyond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' IEEE Transactions on Pattern Analysis and Machine Intelligence, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' [32] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Lopez-Paz and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Ranzato.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Gradient episodic memory for continual learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Advances in neural information processing systems, 30, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' 28 [33] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content='-Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Luo, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Pang, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Ralph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Mathematical programs with equilibrium constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Cam- bridge University Press, 1996.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' [34] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Ma, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Yao, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Ye, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Zhang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Combined approach with second-order optimality conditions for bilevel programming problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' arXiv preprint arXiv:2108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content='00179, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' [35] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Maclaurin, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Duvenaud, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Adams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Gradient-based hyperparameter optimization through reversible learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' In International conference on machine learning, pages 2113–2122, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' [36] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Madry, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Makelov, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Schmidt, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Tsipras, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Vladu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Towards deep learning models resistant to adversarial attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' In International Conference on Learning Representations, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' [37] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Mirrlees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' The theory of moral hazard and unobservable behaviour: Part I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' The Review of Economic Studies, 66(1):3–21, 1999.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' [38] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Nesterov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Gradient methods for minimizing composite functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Mathematical programming, 140(1):125–161, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' [39] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Nocedal and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Wright.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Numerical optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Springer, 1999.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' [40] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Outrata, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Kocvara, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Zowe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Nonsmooth approach to optimization problems with equilibrium constraints: theory, applications and numerical results, volume 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Springer Science & Business Media, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' [41] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Pedregosa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Hyperparameter optimization with approximate gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' In International conference on machine learning, pages 737–746, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' [42] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Rajeswaran, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Finn, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Kakade, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Levine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Meta-learning with implicit gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Advances in neural information processing systems, 32, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' [43] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Shi, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Lu, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Zhang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' An extended Kuhn–Tucker approach for linear bilevel programming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Applied Mathematics and Computation, 162(1):51–63, 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' [44] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Shimizu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Ishizuka, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Bard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Nondifferentiable and two-level mathematical programming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Springer Science & Business Media, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' [45] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Sinha, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Malo, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Deb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' A review on bilevel optimization: from classical to evolutionary approaches and applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' IEEE Transactions on Evolutionary Computation, 22(2):276–295, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' [46] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Szegedy, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Zaremba, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Sutskever, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Bruna, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Erhan, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Goodfellow, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Fergus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Intriguing properties of neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' arXiv preprint arXiv:1312.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content='6199, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' [47] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Vicente and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Calamai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Bilevel and multilevel programming: A bibliography review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Journal of Global optimization, 5(3):291–306, 1994.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' [48] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Von Stackelberg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Market structure and equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Springer Science & Business Media, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' [49] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Ward and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Borwein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Nonsmooth calculus in finite dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' SIAM Journal on control and optimization, 25(5):1312–1340, 1987.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' [50] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Ye.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Constraint qualifications and optimality conditions in bilevel optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' In Bilevel Optimization, pages 227–251.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Springer, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' [51] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Ye, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Yuan, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Zeng, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Zhang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Difference of convex algorithms for bilevel programs with applications in hyperparameter selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' Mathematical Programming, pages 1–34, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'} +page_content=' 29' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQfv_6i/content/2301.01716v1.pdf'}