diff --git "a/9tE3T4oBgHgl3EQfSQmN/content/tmp_files/load_file.txt" "b/9tE3T4oBgHgl3EQfSQmN/content/tmp_files/load_file.txt"
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+page_content='Adaptive proximal algorithms for convex optimization  under local Lipschitz continuity of the gradient*  Puya Latafat†  Andreas Themelis‡  Lorenzo Stella§  Panagiotis Patrinos†  Abstract  Backtracking linesearch is the de facto approach for minimizing continuously differentiable  functions with locally Lipschitz gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' In recent years, it has been shown that in the convex  setting it is possible to avoid linesearch altogether, and to allow the stepsize to adapt based on  a local smoothness estimate without any backtracks or evaluations of the function value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' In this  work we propose an adaptive proximal gradient method, adaPGM, that uses novel tight estimates  of the local smoothness modulus which leads to less conservative stepsize updates and that can  additionally cope with nonsmooth terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' This idea is extended to the primal-dual setting where  an adaptive three term primal-dual algorithm, adaPDM, is proposed which can be viewed as an  extension of the PDHG method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' Moreover, in this setting the fully adaptive adaPDM+ method is  proposed that avoids evaluating the linear operator norm by invoking a backtracking procedure,  that, remarkably, does not require extra gradient evaluations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' Numerical simulations demonstrate  the effectiveness of the proposed algorithm compared to the state of the art.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='  Keywords.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' Convex minimization · proximal gradient method · primal-dual algorithms · locally Lip-  schitz gradient · linesearch-free adaptive stepsizes  AMS subject classifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' 65K05 · 90C06 · 90C25 · 90C30 · 90C47  Contents  1  Introduction  2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='1  Contributions .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
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+page_content='  15  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
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+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='  17  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='1  Dual support vector machine problem .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='  18  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='2  Least absolute deviation regression and square-root lasso .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='  18  5  Conclusions  19  Appendix  21  A Convergence analysis for Section 3  21  Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
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+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='  22  References  25  1  Introduction  Backtracking linesearch is one of the most successful ideas in smooth optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' It is well known  that gradient descent with linesearch converges under mild differentiability assumptions [8, §1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='  Even in the Lipschitz differentiable case such techniques can lead to significant speedups compared  to using a constant stepsize dictated by a global Lipschitz modulus due to their ability to adapt to the  local geometry of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' In this work we explore an alternative approach which can cope with  nonsmooth formulations, does not require any backtracking procedures, does not require function  value evaluations, and yet, similar to linesearch based methods only requires local Lipschitz differ-  entiablity of the smooth term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' The key property that allows for this improvement is the assumption  of convexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='  Our work is inspired by [49] where a gradient descent algorithm xk+1 = xk − γk+1∇f(xk) is  studied with stepsize updated at each iteration by the rule  γk+1 = min  �  γk  �  1 +  γk  γk−1 ,  ∥xk − xk−1∥  2∥∇f(xk) − ∇f(xk−1)∥  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='1)  The idea of using an estimate for the Lipschitz modulus has also been explored in the setting of  variational inequalities [71, 65, 12, 70, 10], but often at the cost of enforcing the stepsize sequence  to be nonincreasing, which can lead to slow convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' Allowing the stepsize to increase is a  crucial feature of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='1) which our proposed methods maintain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' It is worth noting that in [48] another  adaptive scheme, aGRAAL, was proposed for hemivariational inequalities which also allows for  2  increasing stepsizes (see also [1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' We also mention recent works for smooth minimization such  as [64, 52] whose adaptive rule is designed to guarantee worst-case rates, and [3] that exploits a  continuous-time viewpoint to develop adaptive algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='  In [49] it was observed that the line of proof therein does not provide any route for generalization  to the composite proximal setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' In this work, additionally to showing that this is in fact possible  we will actually provide larger stepsizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' As better detailed in the discussion before Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='1, the  improvement is partially due to tighter estimates of the geometry of f in which both Lipschitz and  (inverse) cocoercivity estimates are taken into account, namely  Lk � ⟨∇f(xk−1) − ∇f(xk), xk−1 − xk⟩  ∥xk−1 − xk∥2  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='2a)  and  Ck �  ∥∇f(xk−1) − ∇f(xk)∥2  ⟨∇f(xk−1) − ∇f(xk), xk−1 − xk⟩  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='2b)  for a pair of points xk−1, xk ∈ �n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' Note that Lk and Ck are the inverse of the Barzilai-Borwein  stepsize choices [5], which have been considered in the setting of gradient descent [58, 24], however  their convergence results is limited to the quadratic setting (see also [63, 15] for extensions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' Our  proposed adaptive proximal gradient scheme adaPGM (Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='1) combines the two estimates  and involves the following update rule  γk+1 = min  �������γk  �  1 +  γk  γk−1 ,  γk  2  ��γkLk(γkCk − 1)�  +  �������  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='3)  on the stepsize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' Note that whenever γkCk ≤ 1 the update reduces to γk+1 = γk  �  1 +  γk  γk−1 , effectively  strictly increasing the stepsize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' Regardless, this update is easily seen to be less conservative than  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='1), the one prescribed in [49, Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' 1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' see Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='4 for the details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='  In the second part of the paper this idea is extended to the primal-dual setting to address general  problems of the form  minimize  x∈�n  ϕ(x) � f(x) + g(x) + h(Ax),  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='4)  where A is a linear mapping, g and h are (possibly nonsmooth) extended-real-valued convex func-  tions, and f is convex and typically assumed to be Lipschitz differentiable (this is relaxed to local  Lipschitz continuity of the gradient here, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' Assumption II).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='  In the past decade primal-dual algorithms have gained a lot of popularity in areas ranging from  machine learning and signal processing to control [20, 62, 37, 36, 39, 40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' Their popularity is primar-  ily due to their ability to achieve full splitting on composite problems of the form (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' Moreover,  inherent properties of first order operations facilitates block-coordinate and distributed variants, see  for instance [9, 44, 42, 39, 30, 46, 45, 23, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='  There is a large body of literature on primal-dual algorithms see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=', [16, 29, 67, 22, 13, 11,  21, 40, 69].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' Despite employing different techniques in their convergence analysis, the majority of  existing methods rely on establishing a Fejér-type inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' In fact, most can be viewed as in-  telligent applications of a monotone splitting technique such as forward-backward splitting (FBS),  Douglas-Rachford splitting (DRS) and forward-backward-forward splitting (FBFS) for solving the  associated primal-dual optimality conditions, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=', [35, 67, 22, 13, 11, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' More recently, the  introduction of new splitting techniques such as AFBA [41, 43], NOFOB [31], forward-Douglas-  Rachford-forward [60], forward-backward-half forward [14], forward-reflected-backward [51], has  3  led to new primal-dual algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' We remark also that when A = id in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='4), one can directly  solve the problem without any lifting by using the three-term splitting [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' There exists also an  adaptive/linesearch variant of this algorithm, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=', [56], which requires potentially costly extra  gradient evaluations during the backtracking procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='  Unlike the above described correspondence with splitting techniques, our proposed method can-  not be viewed as an instance of any splitting technique for solving general monotone inclusions,  in that it relies heavily on the knowledge that the operators involved are subdifferentials of convex  functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' Although the proposed idea is extendable to other primal-dual methods such as those in  [41], our focus here is on an adaptive variant of PDHG [16, 67, 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' An interesting algorithm in this  line of work was proposed in [66] which however cannot handle the third nonsmooth term g in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='  In adaPDM (Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='1) we provide a different stepsize rule that not only can handle (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='4) but  also inherits the same idea of using tighter estimates Ck, Lk as in the case of the proximal gradient  method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='  A second consideration for primal-dual methods is that in the usual (nonadaptive) setting the  primal-dual stepsizes γ, σ should typically satisfy a condition of the form γσ∥A∥2 ≤ 1 − γLf  2  (see  for instance [22, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='1], [41, Prop 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' In practical applications the norm of the linear operator  may be costly to compute and can lead to smaller stepsize, and thus slower convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' Recently,  in the setting where f ≡ 0, a linesearch procedure was proposed in [50] for selecting the stepsizes  based on an estimate of the norm of the linear operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' A linesearch extension of [48] in the primal-  dual setting was also proposed in [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' We propose a different linesearch procedure that naturally  integrates our adaptive primal-dual algorithm, see adaPDM+ (Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='2), to handle the more  general problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='4) without any extra gradient evaluations during backtracks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='1  Contributions  The main contributions of the paper are summarized below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' This paper proposes a nonmonotone adaptive stepsize rule for the proximal gradient method that  departs from the usual linesearch technique and adapts to the local geometry of the smooth func-  tion by combining local estimates of cocoercivity and Lipschitz moduli of the differentiable term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='  Compared to [49], even when restricted to the case of gradient descent, it allows for less restrictive  stepsizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' Through this observation, convergence of the aforementioned work in the proximal case  follows immediately as a by-product of our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' This idea is extended to the primal-dual setting where an adaptive three term splitting for com-  posite minimization problems is developed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' The proposed algorithm can be viewed as an extension  of the V˜u-Condat algorithm [22, 67], itself being an extension of the PDHG algorithm [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' As a final contribution, a fully adaptive variant of the primal-dual method is presented in the sense  that it no longer requires evaluating the norm of the linear operator A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' Remarkably, the proposed  linesearch does not require any extra gradient evaluations and can be implemented efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='2  Organization  We conclude this section by introducing the adopted notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' The proposed adaptive proximal gra-  dient method adaPGM is formally studied in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' The underlying idea is then extended to the  primal-dual setting in Section 3, where adaPDM is presented that can handle one additional nons-  mooth term composed with a linear operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' The issue of estimating the norm of the linear operator  4  is resolved through the introduction of a linesearch procedure in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' The convergence re-  sults for both variants of the primal-dual algorithm are presented in a unified fashion in Section  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='3 with some of the proofs deferred to the appendix Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' Numerical simulations for the  proposed algorithms are presented in Section 4, and Section 5 concludes the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='3  Notation  The set of real and extended-real numbers are � � (−∞, ∞) and � � � ∪ {∞}, while the positive  and strictly positive reals are �+ � [0, ∞) and �++ � (0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' We use the notation [x]+ = max{0, x}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='  With id we indicate the identity function defined on a suitable space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' We denote by ⟨·, ·⟩ and ∥·∥ the  standard Euclidean inner product and the induced norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' Given a set E ⊆ �n, with int E, relint E and  bdry E we respectively denote its interior, relative interior, and boundary, and for a sequence (xk)k∈�  we write (xk)k∈� ⊆ E to indicate that xk ∈ E for all k ∈ �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' The indicator function of E is denoted by  δE, namely δE(x) = 0 if x ∈ E and ∞ otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='  A sequence (wk)k∈� is said to be quasi-Fejér monotone with respect to V ⊆ �n if for all v ∈ V,  there exists a summable nonnegative sequence (εk = εk(v))k∈� such that for all k ∈ �  ∥wk+1 − v∥2 ≤ ∥wk − v∥2 + εk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='  The domain and epigraph of an extended-real-valued function h : �n → � are the sets dom h �  {x ∈ �n | h(x) < ∞} and epi h � {(x, c) ∈ �n × � | h(x) ≤ c}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' Function h is said to be proper if  dom h � ∅, and lower semicontinuous (lsc) if epi h is a closed subset of �n+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' We say that h is  level bounded if its c-sublevel set lev≤c h � {x ∈ �n | h(x) ≤ c} is bounded for all c ∈ �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' The conju-  gate of h, is defined by h∗(y) � supx∈�n {⟨y, x⟩ − h(x)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='  2  Adaptive proximal gradient method  The proximal gradient method (PGM) is the natural extension of gradient descent for constrained  and nonsmooth problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' It addresses nonsmooth minimization problems by splitting them into sum  of two terms as follows  minimize  x∈�n  ϕ(x) � f(x) + g(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='1)  Throughout this section the following underlying assumptions are imposed on problem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='  Assumption I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' The following hold in problem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='1):  (i) f : �n → � is convex and has locally Lipschitz continuous gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='  (ii) g : �n → � is proper, lsc, and convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='  (iii) A solution exists: arg min ϕ � ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='  In addition to the gradient of the differentiable term, the fundamental oracle of PGM is the  proximal mapping defined as  proxτg(x) � arg min  w∈�n  �  g(w) + 1  2τ∥w − x∥2�  ,  where τ > 0 is a given stepsize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' In the convex setting the proximal map is single valued and in  fact Lipschitz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' It is well known that for many applications of interest such as constrained or regular-  ized problems the nonsmooth term admits closed form proximal operator (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=', projection on sets,  5  Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='1 Adaptive proximal gradient method (adaPGM)  Require  starting point x−1 ∈ �n and stepsizes γ0 ≥ γ−1 > 0  Initialize x0 = proxγ0g(x−1 − γ0∇f(x−1))  Repeat for k = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' until convergence  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='1:  Set ∆k � γkLk(γkCk − 1), where Lk and Ck are as in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='2)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='2:  Define the stepsize as  γk+1 = γk min  � �  1 +  γk  γk−1 ,  1  2 √[∆k]+  �  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='3:  xk+1 = proxγk+1g(xk − γk+1∇f(xk))  shrinkage operator, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' The most common variant of PGM involves constant stepsize that is upper  bounded by 2/L f, where L f is the global Lipschitz constant of ∇f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' A common strategy in practice is  to estimate such modulus via backtracking linesearch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='  Diverging from the linesearch technique we propose adaPGM that adaptively selects the step-  sizes based on the estimates Ck, Lk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' Note that an adaptive gradient method with g ≡ 0 was proposed  in [49] with a different update rule, see (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='1), where it was observed that the adopted line of proof  does not seem to provide any route for generalization to account for a nonsmooth term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' This appears  to be fundamentally due to the fact that the analysis therein revolves around the fact that consecutive  iterates are a multiple of the gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' In contrast, we circumvent this by combining the subgradient  inequality for the nonsmooth term g at three different pairs of points, namely (x⋆, xk+1), (xk+1, xk),  and (xk−1, xk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' In addition, the combined use of the quantities Lk and Ck as in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='2) allows for esti-  mating, along with that of the gradient, the local Lipschitz constant of the forward operator id−γ∇f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='  This appears to be fundamental for recovering the update rule (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='1) of [49], and in fact leads to the  less conservative update rule of adaPGM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='1  Local estimates of Lipschitz and cocoercivity moduli  Throughout, we will make use of the following shorthand for the forward operator with stepsize γk:  Hk � id − γk∇f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='  As we are about to show, the combined adoption of the estimates Lk and Ck provides an estimate of  the Lipschitz modulus of the forward operator Hk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' Suppose that Assumption I holds, and let xk−1, xk ∈ �n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' Then, with Lk and Ck as in  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='2) the following hold:  (i) ∥∇f(xk−1) − ∇f(xk)∥2 = CkLk∥xk−1 − xk∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='  (ii) ∥Hk(xk−1) − Hk(xk)∥2 = �1 − γkLk(2 − γkCk)�∥xk−1 − xk∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='  (iii) Lk ≤ ∥∇f(xk−1)−∇f(xk)∥  ∥xk−1−xk∥  ≤ Ck ≤ Lf,V, where L f,V is a Lipschitz modulus for ∇f on an open convex  set V containing xk−1 and xk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='  6  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' The first assertion is of trivial verification, and similarly the third one follows from the  Cauchy-Schwarz inequality and the Baillon-Haddad theorem [4, Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' 10], see also [6, Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='  To conclude, observe that  ∥Hk(xk−1) − Hk(xk)∥2 = ∥xk−1 − xk∥2 + γ2  k∥∇f(xk−1) − ∇f(xk)∥2  − 2γk⟨∇f(xk−1) − ∇f(xk), xk−1 − xk⟩,  from which assertion 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='1(ii) follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='  Our convergence analysis will rely on first establishing boundedness of the generated sequence,  thereby entailing the existence of a Lipschitz constant Lf,V > 0 for ∇f on a bounded convex set  V that contains the iterates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' It will then follow from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='1(iii) that both Ck and Lk are upper  bounded by this quantity, which in turn will be used to show that this modulus provides a lower  bound for the stepsize separating it from zero, and convergence will be established by means of  Fejér-monotonicity arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' Before that, we show how the combined use of Lk and Ck can be  employed to estimate the progress of the iterates generated by PGM with arbitrary stepsizes, not  necessarily dictated by the update rule of adaPGM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' Suppose Assumption I holds, and consider a sequence (xk)k∈� generated by PGM  iterations xk+1 = proxγk+1g(xk − γk+1∇f(xk)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' Then, for any x⋆ ∈ arg min ϕ  1  2∥xk+1 − x⋆∥2 + γk+1(1 + ρk+1)Pk + 1−εk+1ρk+1  2  ∥xk − xk+1∥2  ≤ 1  2∥xk − x⋆∥2 + ρk+1γk+1Pk−1 + ρk+1  � 1−γkLk(2−γkCk)  2εk+1  − ρk+1(1 − γkLk)  �  ∥xk−1 − xk∥2  holds for any k ≥ 1 and εk > 0, where Lk and Ck are as in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='2), ρk �  γk  γk−1 , and Pk � ϕ(xk) − min ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' The subgradient characterization  Hk+1(xk)−xk+1  γk+1  = xk−xk+1  γk+1  − ∇f(xk) ∈ ∂g(xk+1)  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='2)  of xk+1 = proxγk+1g(xk − γk+1∇f(xk)) implies  0 ≤ g(x⋆) − g(xk+1) + ⟨∇f(xk), x⋆ − xk+1⟩ −  1  γk+1 ⟨xk − xk+1, x⋆ − xk+1⟩  = g(x⋆) − g(xk+1) + ⟨∇f(xk), x⋆ − xk+1⟩  (A)  +  1  2γk+1 ∥xk − x⋆∥2 −  1  2γk+1 ∥xk+1 − x⋆∥2 −  1  2γk+1 ∥xk − xk+1∥2  holds for any solution x⋆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' We next proceed to upper bound the term (A) as  (A) = ⟨∇f(xk), x⋆ − xk⟩ + ⟨∇f(xk), xk − xk+1⟩  = ⟨∇f(xk), x⋆ − xk⟩ + 1  γk ⟨Hk(xk−1) − xk, xk+1 − xk⟩ + 1  γk ⟨Hk(xk−1) − Hk(xk), xk − xk+1⟩  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='2)  ≤  f(x⋆) − f(xk)  +  g(xk+1) − g(xk)  + 1  γk ⟨Hk(xk−1) − Hk(xk), xk − xk+1⟩  (B)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='  Next, we bound the term (B) by εk+1-Young’s inequality as  (B) ≤ εk+1  2γk ∥xk − xk+1∥2 +  1  2εk+1γk ∥Hk(xk−1) − Hk(xk)∥2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='1(ii)  = εk+1  2γk ∥xk − xk+1∥2 + 1−γkLk(2−γkCk)  2εk+1γk  ∥xk−1 − xk∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='3)  7  Let ϕ⋆ � min ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' The three inequalities combined give  0 ≤ ϕ⋆ − ϕ(xk) +  1  2γk+1 ∥xk − x⋆∥2 −  1  2γk+1 ∥xk+1 − x⋆∥2  +  � εk+1  2γk −  1  2γk+1  �  ∥xk − xk+1∥2 + 1−γkLk(2−γkCk)  2εk+1γk  ∥xk−1 − xk∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='  Using again the subgradient (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='2) (since ∂ϕ = ∇f + ∂g) one has  vk � xk−1−xk  γk  − (∇f(xk−1) − ∇f(xk)) ∈ ∂ϕ(xk),  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='4)  hence  0 ≤ γk+1  γk  �  ϕ(xk−1) − ϕ(xk) − ⟨vk, xk−1 − xk⟩  �  = γk+1  γk  �  ϕ(xk−1) − ϕ(xk) − 1  γk ∥xk − xk−1∥2 + ⟨∇f(xk−1) − ∇f(xk), xk−1 − xk⟩  �  = γk+1  γk  �  ϕ(xk−1) − ϕ(xk) − 1−γkLk  γk  ∥xk − xk−1∥2�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='5)  The proof now follows by summing the last two inequalities and multiplying by γk+1, observing that  ϕ⋆ − ϕ(xk) + γk+1  γk (ϕ(xk−1) − ϕ(xk)) = γk+1  γk Pk−1 − (1 + γk+1  γk )Pk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='2  Convergence results  While adaPGM can be seen as a special case of the more general primal-dual adaPDM, the conver-  gence results of adaPGM is obtained under fewer restrictions (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' For this reason, we  provide a dedicated proof for the adaptive proximal gradient algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' Suppose that Assumption I holds, and consider the iterates generated by adaPGM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='  Then, for any x⋆ ∈ arg min ϕ and with Uk = Uk(x⋆) defined as  Uk � 1  2∥xk − x⋆∥2 + 1  4∥xk − xk−1∥2 + γk(1 + ρk)Pk−1,  the following hold:  (i) Uk+1 ≤ Uk − �  ≥0  1  4 − ρ2  k+1∆k  �∥xk−1 − xk∥2 − γk  �  ≥0  1 + ρk − ρ2  k+1  �Pk−1 for all k ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='  (ii) (The sequence (xk)k∈� is bounded and) γk >  1  2Lf,V holds for every k ≥ k0 �  �  2 log2  1  γ0L f,V  �  +,  where L f,V is a Lipschitz modulus for ∇f on a compact convex set V containing (xk)k∈�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='  (iii) The sequence (xk)k∈� converges to a solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' The claim remains true if the stepsizes are chosen  in such a way that (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='3) holds with “≤”, as long as (γk)k∈� is bounded away from zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' In what follows, let αk � γkLk and βk � γkCk, so that ∆k = αk(βk − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='  ♠ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='3(i) Defining  ˜Uk � 1  2∥xk − x⋆∥2 + 1−εkρk  2  ∥xk − xk−1∥2 + γk(1 + ρk)Pk−1,  the inequality in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='2 reads  ˜Uk+1 ≤ ˜Uk − γk  �  1 + ρk − ρ2  k+1  �  Pk−1 −  � 1−εkρk  2  + ρ2  k+1(1 − αk) − ρk+1  1−αk(2−βk)  2εk+1  �  ∥xk−1 − xk∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='  8  With εk �  1  2ρk , one has ˜Uk = Uk and the inequality simplifies to  Uk+1 ≤ Uk − γk  �  1 + ρk − ρ2  k+1  �  Pk−1 −  � 1  4 + ρ2  k+1αk(1 − βk)  �  ∥xk−1 − xk∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='  If βk ≤ 1, (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=', ∆k ≤ 0), one directly obtains that Uk+1 ≤ Uk − 1  4∥xk − xk−1∥2 −γk  �  1 + ρk − ρ2  k+1  �  Pk−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='  Otherwise, the update rule for γk implies that the coefficient in curly brackets is greater or equal than  zero, and the claim follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='  ♠ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='3(ii) The proven inequality implies that (xk)k∈� is quasi-Fejér monotone of type II with respect  to the set of solutions (in the sense of [19, Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='1]) and in particular is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' As such, V and  Lf,V as in the statement exist, and consequently Ck ≤ Lf,V holds for every k ≥ 1 by cocoercivity  of ∇f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' Suppose that γk ≤ 1/L f,V for k = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' , K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' Then, for these k one has that γkCk ≤ γkLf,V ≤ 1,  hence [∆k]+ = 0 and the update rule step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='2 then implies that ρ2  k+1 = 1+ρk > 1, which inductively  gives ρ2  k ≥ 2 for all k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' , K (since ρ0 ≥ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' We then have  γ2  K = ρ2  Kγ2  K−1 ≥ 2γ2  K−1 ≥ · · · ≥ 2Kγ2  0  showing that in at most k0 iterations one stepsize exceeds 1/L f,V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' Because of the update rule step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='2,  note that if γk ≤ 1/Lf,V (hence γkCk ≤ 1) then γk > γk−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' The lower bound on the stepsizes will thus  follow once we show that γk−1 > 1/L f,V and γk < 1/L f,V implies that γk >  1  2L f,V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' In this case, since  γk < γk−1 necessarily βk−1 > 1 and  (γkL f,V)2 ≥ (γkCk−1)2 =  1  4αk−1(βk−1−1)β2  k−1 ≥ 1  4  βk−1  βk−1−1 = 1  4  �  1 +  1  βk−1−1  �  > 1  4,  where the second inequality owes to the fact that βk−1 ≥ αk−1, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='1(iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='  ♠ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='3(iii) We begin by observing that, as is apparent from its proof, assertion 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='3(i) remains valid if  the identity in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='3) is replaced by an inequality “≤”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' Either way, a telescoping argument yields that  γk(1 + ρk − ρ2  k+1)Pk−1 → 0 as k → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' Since (xk)k∈� is quasi-Fejér monotone (of Type II), by virtue  of [19, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='11(i)–(iv)] it suffices to show that there exists a subsequence of (xk)k∈� converging to  a solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' Equivalently, it suffices to show that lim infk→∞ Pk = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' If lim supk→∞(1 + ρk − ρ2  k+1) > 0,  then a telescoping argument in the descent condition of assertion 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='3(i) yields the claim, since γk  is bounded away from zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' Alternatively, since 1 + ρk − ρ2  k+1 ≥ 0, necessarily 1 + ρk − ρ2  k+1 → 0,  from which it easily follows that lim infk→∞ ρk > 1, and that therefore γk → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' In this case, since  γk(1 + ρk)Pk−1 ≤ Uk this directly proves that Pk → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='  As a consequence of the above result, it can be easily seen that [49, Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' 1] can in fact cope  with nonsmooth problems of the form (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' This will be apparent once we show that the proposed  stepsize update rule is less conservative than that of [49, Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='4 (Comparison with [49, Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' 1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' With γk+1 as in step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='2, note that  γk+1 ≥ min  �  γk  �  1 + ρk,  ∥xk−xk−1∥  2∥∇f(xk)−∇f(xk−1)∥  �  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='6)  holds for every k, and that the right-hand side corresponds to the stepsize update of [49, Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='  Indeed, the inequality easily follows by observing that  ∥xk−xk−1∥  2∥∇f(xk)−∇f(xk−1)∥ = 1  2  �  1  LkCk ≤ 1  2  �  γkCk  LkCk[γkCk−1]+ =  γk  2 √[∆k]+ ,  where the first identity owes to Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='1(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' As a consequence, in addition to accommodating  proximal terms, adaPGM also comes with a less conservative stepsize update rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='  9  Inequality (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='6) cannot be reiterated inductively, and in particular there is no guarantee that,  iteration-wise, the entire sequence of stepsizes produced by adaPDM is larger than that generated  by [49, Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' 1], even if the algorithms are started with same initial conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' This is nevertheless  enough to infer convergence of [49, Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' 1] applied to composite minimization problems as in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='  Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='5 (Proximal extension of [49, Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' 1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' Suppose that Assumption I holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' Then, PGM  iterations xk+1 = proxγk+1g(xk − γk+1∇f(xk)) with stepsize rule  γk+1 = min  �  γk  �  1 + ρk,  ∥xk−xk−1∥  2∥∇f(xk)−∇f(xk−1)∥  �  as in [49, Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' 1] converge to a solution of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' The validity of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='3(iii) guarantees that the generated sequence remains bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='  As also observed in [49, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' 1], this guarantees that γk ≥  1  2L f,V (up to possibly excluding initial  iterates), where L f,V is a Lipschitz constant of ∇f on a bounded open convex set V that contains all  the iterates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' The proof follows by invoking Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='3(iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='  3  Adaptive three-term primal-dual methods  In this section the idea of adaptively estimating the local geometry of f will be extended to composite  problems of the form (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='4), which we rewrite here for the reader’s convenience  minimize  x∈�n  ϕ(x) � f(x) + g(x) + h(Ax).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='  Although inevitably the analysis in this setting is more complicated, the key idea of using the es-  timates Ck, Lk in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='2) remains the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' We will first propose an adaptive algorithm under the as-  sumption that the norm of the linear operator A is known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' This assumpion will then be lifted through  a certain linesearch procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='  Throughout, the following standing assumptions are imposed on problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='  Assumption II (Requirements for the primal-dual setting).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' The following hold in problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='4):  (i) f : �n → � is continuously differentiable with locally Lipschitz continuous gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='  (ii) h : �m → � and g : �n → � are proper convex and lsc, and A : �n → �m is a linear  mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='  (iii) A solution exists: arg min ϕ � ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='  (iv) The problem is strictly feasible: there exists x ∈ relint dom g such that Ax ∈ relint dom h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='  A well-established approach for addressing (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='4) is to lift the problem into the primal-dual space  and solve the associated convex-concave saddle point problem  minimize  x∈�n  maximize  y∈�m  L(x, y) � f(x) + g(x) + ⟨Ax, y⟩ − h∗(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='  This lifting is the key to splitting the composed term h ◦ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' Moreover, in doing so the primal and  the dual solutions can be obtained simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' A pair z⋆ = (x⋆, y⋆) will be referred to as a  primal-dual solution if the following primal-dual optimality condition holds  0 ∈ −Ax⋆ + ∂h∗(y⋆),  0 ∈ A⊤y⋆ + ∇f(x⋆) + ∂g(x⋆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='1)  10  The set of all such points would be denoted by S⋆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' Under the constraint qualification of Assump-  tion II(iv), the set of solutions for the dual problem is nonempty, and thus so is S⋆, and the duality  gap is zero, see [59, Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='1] and [6, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' Moreover, the pair (x⋆, y⋆) is a primal-dual  solution if and only if x⋆ is a primal and y⋆ is a dual solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' Since (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='4) is a convex problem, the  primal-dual solution pairs are equivalently characterized by the saddle point inequality  L(x⋆, y) ≤ L(x⋆, y⋆) ≤ L(x, y⋆)  ∀(x, y) ∈ �n × �m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='2)  In our analysis we will measure deviation from L(x⋆, y⋆) along the primal and dual sequences using  shorthand notations  Pk � L(xk, y⋆) − L(x⋆, y⋆) = (f + g)(xk) − ( f + g)(x⋆) + ⟨xk − x⋆, A⊤y⋆⟩  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='3)  and  Qk � L(x⋆, y⋆) − L(x⋆, yk) = h∗(yk) − h∗(y⋆) + ⟨Ax⋆, y⋆ − yk⟩  which are both positive due to the saddle point inequality (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='1  Algorithmic overview  The proposed algorithm is presented in adaPDM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' It can be viewed as an adaptive variant of the  the algorithm proposed in [22, 67], which itself is an extension of the PDHG method [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' In com-  parison to the aforementioned algorithms with constant stepsizes, here, a varying and potentially  increasing stepsize rule is proposed based on the estimates Ck, Lk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' Moreover, in PDHG the dual up-  date combines two consequent primal updates as 2Axk − Axk−1 (in our notation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' In [41, Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' 3] it  was shown that many primal-dual algorithms can be unified by modifying the dual update to use  terms of the form θAxk + (1 − θ)Axk−1 followed by a correction step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' While depending on the appli-  cation this can lead to parallel implementations and potentially larger stepsizes compared to PDHG  (with θ = 2), the improvement in speed is limited by the use of global estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' Differently from the  aforementioned works, in adaPDM the mixing constant θ is selected adaptively as 1 + γk+1/γk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' When  viewed as an extension of the proximal gradient method adaPGM this idea appears natural.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' In fact,  this was proposed in [66] as a primal-dual extension of [49] where superior convergence rate com-  pared to constant stepsize variants was observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' Our proposed update rule differs substantially and  inherits the tighter estimates in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='2) while permitting for the third nonsmooth term g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='1 (comparison to the proximal gradient method).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' When h ≡ 0 (and A = 0), problem  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='4) reduces to problem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='1), and  (i) One has that adaPDM with δ = 0 reduces to adaPGM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' In fact, in this case apparently yk ≡ 0  and xk+1 = proxγk+1g(xk − γk+1∇f(xk)), and since ξk ≡ 0 the stepsize update on γk reduces to  γk+1 = γk min  �������  �  1 +  γk  γk−1 ,  1  �  2�  |∆k|+∆k�  ������� = γk min  � �  1 +  γk  γk−1 ,  1  2 √[∆k]+  �  ,  which is precisely the stepsize update rule in adaPGM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='  (ii) Pk in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='3) reduces to the one in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' However, while for adaPGM it was sufficient to  show lim inf Pk = 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=', that any limit point ˆx satisfies ϕ(ˆx) = min ϕ, in the primal-dual setting  an argument through the cost function cannot be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' In fact, as it will become evident in the  11  Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='1 Adaptive primal-dual method (adaPDM)  Require  primal/dual (square inverse) stepsize ratio t > 0  stepsize parameters δ ≥ 0, c > 1 + δ  initial primal-dual pair (x−1,y0) ∈ �n × �m and stepsizes γ−1 ≤ γ0 ≤ 1/2ct∥A∥  Initialize x0 = proxγ0g  �x−1 − γ0∇f(x−1) − γ0A⊤y0�  Repeat for k = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' until convergence  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='1:  Set ∆k � γkLk(γkCk − 1) and ξk � t2γ2  k∥A∥2, where Lk and Ck are as in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='2)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='2:  Define the stepsizes as  ���������������  γk+1 = min  �������γk  �  1 +  γk  γk−1 ,  1  2ct∥A∥, γk  �  1−4ξk(1+δ)2  2(1+δ)  �√  ∆2  k+ξk(1−4ξk(1+δ)2)+∆k  �  �������  σk+1 = t2γk+1  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='3:  yk+1 = proxσk+1h∗  �  yk + σk+1  ��1 + γk+1  γk  �Axk − γk+1  γk Axk−1��  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='4:  xk+1 = proxγk+1g  �xk − γk+1∇f(xk) − γk+1A⊤yk+1�  proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='2(iv), although any limit point (ˆx, ˆy) of (xk, yk)k∈� generated by either one  of Algorithms 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='1 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='2 satisfies  L(ˆx, y⋆) = L(x⋆, ˆy) = L(x⋆, y⋆),  ∀(x⋆, y⋆) ∈ S⋆,  convergence can only be guaranteed by establishing sufficient descent in terms of the residual  ∥zk+1 − zk∥ (enforced in the algorithms through introduction of the parameter δ > 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' It is only  if f +g is strictly convex that δ = 0 can be permitted, as will be shown in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='2(iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='  Notice also that Algorithms 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='1 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='2 are not symmetric with respect to the primal and dual  variables and a different algorithm can be obtained by applying the algorithm to the dual problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='2  Linesearch variant without linear operator norm  In this subsection we introduce a fully adaptive primal-dual algorithm that removes any dependence  on the norm of the linear operator A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' As was also the case for the other two algorithms, adaPDM+  follows the convention of indexing with k all the variables that depend on quantities defined up  to iteration k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' This choice highlights the nested dependency of γk+1 and ηk+1 at step 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='4, which  appears to be solvable only by means of a linesearch procedure (see also the discussion before The-  orem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='  It should be noted that backtracking linesearch has been employed in combination with PDHG  in various forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' In [33, 32] it is used to potentially increase the speed of convergence by balancing  the primal and dual residuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' In [50] an adaptive linesearch algorithm is presented for PDHG and  the idea is extended to the composite form (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='4), see [50, Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' In addition to a different stepsize  update rule, a major difference here is that the backtracks involved do not require evaluations of the  gradient ∇f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' We also remark that adaPDM+ provides a practical way of initializing the stepsize, and  that it reduces to adaPDM if ηk is taken as ∥A∥ for all k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='  12  Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='2 Adaptive primal-dual method with linesearch (adaPDM+)  Require  primal/dual (square inverse) stepsize ratio t > 0  stepsize parameters δ ≥ 0 and c > 1 + δ  initial primal-dual pair (x−1, y0) ∈ �n × �m, estimate η0 of ∥A∥,  stepsizes γ−1 ≤ γ0 ∈ (0, 1/2tcη0], and backtracking parameters r, R > 1  Initialize x0 = proxγ0g  �  x−1 − γ0∇f(x−1) − γ0A⊤y0�  Repeat for k = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' until convergence  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='1:  Set ∆k � γkLk(γkCk − 1) and ¯��k � t2γ2  kη2  k(1 + δ)2, with Lk and Ck as in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='2)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='2:  Choose an estimate 0 < ˆηk+1 ≤ R max {1, ηk} of ∥A∥ (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=', ˆηk+1 = ηk)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='3:  while true do  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='4:  Define the stepsizes as  ���������������  γk+1 = min  �������γk  �  1 +  γk  γk−1 ,  1  2ctˆηk+1 , γk  �  1−4¯ξk  2(1+δ)  �√  ∆2  k+(tˆηk+1γk)2(1−4¯ξk)+∆k  �  �������  σk+1 = t2γk+1  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='5:  yk+1 = proxσk+1h∗  �  yk + σk+1  ��1 + γk+1  γk  �Axk − γk+1  γk Axk−1��  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='6:  Estimate ∥A∥ via  ηk+1 = ∥A⊤(yk+1 − yk)∥  ∥yk+1 − yk∥  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='7:  if γk+1 ≤ min  �������  1  2ctηk+1 , γk  �  (1−4¯ξk)  2(1+δ)  �√  ∆2  k+(tηk+1γk)2(1−4¯ξk)+∆k  �  ������� then break, else ˆηk+1 ← rˆηk+1  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='8:  xk+1 = proxγk+1g  �  xk − γk+1∇f(xk) − γk+1A⊤yk+1�  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='3  Convergence results  The convergence analysis will once again revolve around showing descent on a suitable merit func-  tion, this time defined as  Uk � 1  2∥xk − x⋆∥2 + 1−4ξk(1+δ)  4  ∥xk − xk−1∥2 +  1  2t2 ∥yk − y⋆∥2 + γk(1 + ρk)Pk−1,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='4)  where (x⋆, y⋆) ∈ S⋆ is any primal-dual optimal pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' The next theorem will allow us to study the  convergence of Algorithms 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='1 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='2 under a unified analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' It should be noted that, unless  ∥A∥ is known and ηk+1 is chosen greater or equal than that quantity (as it happens in adaPDM),  the following theorem does not furnish an implementable stepsize update rule, owing to the implicit  dependency between the stepsize γk+1 and the norm estimate ηk+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' The linesearch strategy prescribed  by adaPDM+ is designed to circumvent this issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' Suppose that Assumption II holds, and let t > 0, δ ≥ 0, and c > 1 + δ be fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='  Consider a sequence (xk, yk)k≥1 generated by  ���������  yk+1 = proxσk+1h∗  �  yk + σk+1  ��1 + γk+1  γk  �Axk − γk+1  γk Axk−1��  xk+1 = proxγk+1g  �  xk − γk+1∇f(xk) − γk+1A⊤yk+1�  ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='5)  13  starting from a triplet (x−1, x0, y0) ∈ �n × �n × �m and with initial primal stepsizes γ0 ≥ γ−1 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='  Denote ξk � tηkγk and ∆k � γkLk(γkCk − 1) with Lk and Ck as in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='2), η0 ≤  1  2ctγ0 and ηk+1 any such  that ∥A⊤(yk−yk+1)∥  ∥yk−yk+1∥  ≤ ηk+1 ≤ ηmax for some ηmax < ∞, k ∈ �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' Suppose that the sequences of stepsizes  comply with the rules  γk+1 = min  �������������  γk  �  1 +  γk  γk−1 ,  1  2ctηk+1  , γk  �  �  �  �  1 − 4ξk(1 + δ)2  2(1 + δ)  � �  ∆2  k + (tηk+1γk)2(1 − 4ξk(1 + δ)2) + ∆k  �  �������������  and σk = t2γk, k ∈ �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' Then, for any primal-dual solution (x⋆, y⋆) of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='4) and with Uk as in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='4),  the following hold:  (i) Uk+1 ≤ Uk −  δ  2t2(1+δ)∥yk − yk−1∥2 −  ≥δ/4(1+δ)  1−4ξk−4ρ2  k+1(∆k+ξk+1)  4  ∥xk − xk+1∥2 − γk(  ≥0  1 + ρk − ρ2  k+1)Pk−1  − γk+1Qk+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='  (ii) (The sequence (xk, yk)k∈� is bounded and) γk ≥ ˆγ > 0 for all k ≥ 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=', the stepsize sequence  is bounded away from zero (see (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='6) for the value of ˆγ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='  (iii) If δ > 0, the sequence (xk, yk)k∈� converges to a primal-dual solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='  (iv) If f + g is strictly convex, the set arg min ϕ = �x⋆� is a singleton, and the sequence (xk)k∈�  converges to x⋆ even when δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='  We remark that in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='2(iv) convergence of (yk)k∈� to a dual solution could be established  using a similar argument under the assumption that h∗ is strictly convex, but this is omitted here as  our interest lies in solving the primal problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' The sequential convergence results in the next  theorem for adaPDM follows from the more general Theorems 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='2(iii) and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='2(iv), specialized to  ηk ≡ ηmax = ∥A∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' For adaPDM+ the assertions follow from Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='2 this time with ηmax =  max {η0, r∥A∥, R∥A∥, rR}, as this furnishes an upper bound on both (ηk)k∈� and (ˆηk)k∈�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' Indeed, that  ηk+1 ≤ ηmax is a trivial consequence of the inequality ηk+1 ≤ ∥A∥, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' step 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' Similarly, observe  that whenever ˆηk+1 ≥ ∥A∥ the backtracking necessarily ends, for ˆηk+1 ≥ ηk+1 in this case and the  bound required at step 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='7 turns out to be looser than that already ensured at step 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' The claimed  bound then follows from the fact that ˆηk+1 is initialized at some value not larger than R max {1, ηk} ≤  max {R, R∥A∥}, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' step 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='2, and that it is augmented by a factor r at every failed backtracking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='  (Having R max {1, ηk} as opposed to Rηk is a technical requirement to address the possibility of  ηk = 0 when A⊤ is not injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=')  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='3 (convergence of Algorithms 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='1 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' Suppose that Assumption II holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' Then,  all the assertions Theorems 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='2(i) to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='2(iv) remain valid for both of the sequences generated by  Algorithms 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='1 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='  4  Numerical simulations  In this section the performance of the proposed algorithms is evaluated through a series of simula-  tions on standard problems on both synthetic data as well as datasets from the LIBSVM library [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='  All the algorithms are implemented in the Julia programming language and are available online1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='  1https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='com/pylat/adaptive-proximal-algorithms  14  When applicable, the following algorithms are included in the comparisons:  PGM  Proximal gradient method with constant stepsize 1/L f  PGM-ls  Proximal gradient method with backtracking [61, LS1], [26, Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' 3]  Nesterov-ls  Nesterov’s acceleration of PGM-ls [54], [7, §10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='7]  aGraal  The golden ratio algorithm [48]  PDHG  The algorithm of [16]  VC  The algorithm proposed in [67, 22]  MP-ls  The linesearch method of Malitsky and Pock [50, Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' 4]  adaPGM  Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='1  adaPGM-MM  Proximal extension of Malitsky and Mishchenko [49, Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' 1]2  adaPDM  Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='1  adaPDM+  Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='2  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='1  Adaptive proximal gradient  We compare the performance of adaPGM (Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='1) on three practical problems that can be  cast as (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' In the figures, the distance of the cost from the minimum cost is plotted against the  number of the gradient evaluations, which in all problems correspond to the most costly operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='  For adaPGM we always set γ0 = γ−1 equal to some estimate of the Lipschitz modulus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' It should  be noted that the algorithm is not sensitive to the choice of initial stepsize and the improvement  gained by more involved initial estimates is negligable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' For this reason, for globally Lipschitz-  smooth problems we set the initial step equal to 1/L f where Lf denotes the Lipschitz constant of  ∇f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' For cubic regularization we initialized with γ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' For backtracking linesearch variants we used  standard parameter choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' For aGraal, we set the algorithm parameters as suggested in [48, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='  5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' The inital point x0 was set to zero for all algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' The results of all the simulations are reported  in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='1  Logistic regression  We consider ℓ1-regularized logistic regression problem  minimize  x∈�n+1  1  m  �m  i=1  �yi log(si) + (1 − yi) log(1 − si)� + λ∥x∥1,  where λ > 0 is the regularization parameter, m, n are the number of samples and features, the pair  ai ∈ �n+1 denotes the i-th sample (up to absorbing the bias terms), yi ∈ {−1, 1} is the associated  label, and si = (1 + exp(−a⊤  i x))−1 is the logistic sigmoid function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' In all the simulations λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='01  was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='2  Cubic regularization  The subproblem in the cubic Newton method [55] involves minimizing  minimize  x∈�n+1  1  2⟨x, Qx⟩ + ⟨x, q⟩ + M  6 ∥x∥3,  where Q ∈ �n+1×n+1, q ∈ �n+1, and M > 0 is some regularization parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' In the simulations,  the Hessian and gradient Q, q are generated for the logsitic loss problem evalated at zero on the  2See Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='  15  0  200  400  600  800  1,000  10−11  10−8  10−5  10−2  # of calls to ∇f  ϕ(xk) − min ϕ  mushroom dataset (m = 8124, n = 112, density=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='19)  0  200  400  600  800  1,000  10−13  10−10  10−7  10−4  10−1  # of calls to ∇f  a5a dataset (m = 6414, n = 123, density=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='11)  aGraal  Nesterov-ls  PGM-ls  PGM  adaPGM-MM  adaPGM  0  200  400  600  800  1,000  10−12  10−9  10−6  10−3  100  # of calls to ∇f  phishing dataset (m = 11055, n = 68, density=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='44)  (a) Logistic regression problem of §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='1  0  20  40  60  80  100  10−15  10−11  10−7  10−3  # of calls to ∇f  ϕ(xk) − min ϕ  mushroom dataset (m = 8124, n = 112, density=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='19)  0  20  40  60  80  100  10−15  10−12  10−9  10−6  10−3  100  # of calls to ∇f  a5a dataset (m = 6414, n = 123, density=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='11)  0  20  40  60  80  100  10−15  10−12  10−9  10−6  10−3  # of calls to ∇f  phishing dataset (m = 11055, n = 68, density=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='44)  (b) Cubic regularization problem of §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='2  0  500  1,000  1,500  2,000  10−15  10−11  10−7  10−3  101  # of calls to ∇f  ϕ(xk) − min ϕ  m = 100, n = 300, n⋆ = 30  0  500  1,000  1,500  2,000  10−15  10−11  10−7  10−3  101  # of calls to ∇f  m = 500, n = 1000, n⋆ = 100  0  500  1,000  1,500  2,000  10−15  10−11  10−7  10−3  101  # of calls to ∇f  m = 4000, n = 1000, n⋆ = 100  (c) Lasso problem of §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='3  Figure 1: Simulations for problem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='1)  16  mushroom, a5a, and phishing datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' Moreover, M = 1 is used in the plots noting that the behavior  of the algorithms for different values is similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='3  Regularized least squares  Consider the lasso problem  minimize  x∈�n  1  2∥Ax − b∥2 + ∥x∥1,  where matrix A ∈ �m×n and vector b ∈ �n are generated based on the procedure described in [53,  §6], and n⋆ denotes the number of nonzero elements of the solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' In all simulations, the parameter  controlling the magnitude of the primal solution was set equal to one (ρ = 1 in the notation of [53,  §6]), but the general behavior of the algorithms is similar for alternative values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='2  Adaptive primal-dual algorithms  We now compare the performance of adaPDM (Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='1) and its operator norm–free extension  adaPDM+ (Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='2) on problems fitting into formulation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='  Optimality criterion  In this case, an optimality criterion based on the primal-dual optimality conditions is employed that  uses quantities provided by all methods being compared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' Specifically, denoting  T(x, y) �  �  ∂h∗(y) − Ax, ∂g(x) + ∇f(x) + A⊤y  �  the operator associated with the primal-dual optimality conditions (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='1), for any primal-dual se-  quence (zk)k∈� = (xk, yk)k∈� generated by any of the algorithms included in the simulation we asso-  ciate a sequence vk = (vk  1, vk  2) ∈ Tzk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' Noting that ∥vk∥ = 0 implies optimality of the pair (xk, yk) (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='1)), more generally the quantity  ∥vk∥ ≥ dist(0, Tzk)  serves as a measure of optimality providing a fair comparison between different methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' In all of  the included algorithms this quantity is readily available based on the optimality condition of the  primal and dual updates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' In the case of adaPDM+ (as well as adaPDM, the V˜u-Condat and PDHG  algorithms, all being a special case of adaPDM+), it follows from steps 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='5 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='8 that  vk+1  1  = σ−1  k+1(yk − yk+1) + γk+1  γk  �  Axk − Axk−1�  +  �  Axk − Axk+1�  ∈ ∂h∗(yk+1) − Axk+1  vk+1  2  = γ−1  k+1  �  xk − xk+1�  + ∇f(xk+1) − ∇f(xk) ∈ ∂(g + f)(xk+1) + A⊤yk+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='  In the case of MP-ls [50, Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' 4], this quantity is similarly obtained according to the optimality  conditions of the proximal updates at steps 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='a therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='  Stepsize selection  The stepsize parameters for Algorithms 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='1 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='2 were set as follows δ = 10−8, c = (1+10−3)(1+δ),  γ0 = γ−1 = 1/(2ctη0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' In the presented simulations for comparison purposes we used η0 = ∥A∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' How-  ever, it was observed that the algorithms are not sensitive to the initial choices of η0, γ0 and there is  17  little need for tuning these parameters and any reasonable estimate results in very similar trajecto-  ries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' We observed that the stepsize ratio t requires some tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' In our preliminary simulations for  Algorithms 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='1 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='2, on the problems presented in here, we ran a grid search for t ∈ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='01, 100]  and observed that often t = 1 performed well enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' For the V˜u-Condat and PDHG algorithms, we  used the heuristic stepsize selection rule suggested in [38, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='53)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' In the case of MP-ls, in [50,  §5] it was suggested to set t equal to the ratio obtained by tuning the constant stepsize regime of the  V˜u-Condat algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' In our experiments we used this heuristic but often found better performance  in the range t ∈ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='01, 100].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' Therefore, the plots for MP-ls are tuned for best performance by a grid  search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='1  Dual support vector machine problem  The support vector machine (SVM) problem consists of  minimize  x∈�n  C  N  �  n=1  max  �  0, 1 − bi(a⊤  i x + x0)  �  + 1  2∥x∥2,  where the pair (ai, bi) represent the i-th data and C > 0 is some positive constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' Computationally,  a popular approach is to instead consider the dual SVM problem [34, §12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='1]  minimize  α1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=',αN  1  2∥ �N  i=1 αibiai∥2 − �N  i=1 αi  subject to 0 ≤ αi ≤ C,  i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' , N  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='1)  �N  i=1 αibi = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='  In the simulations we used C = 1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' The problem is cast in the standard form (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='4) by letting f  represent the quadratic cost, g the box constraints, h = δ{0} and A = [b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' , bN] ∈ �1×N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' Notice that  the dual vectors yk are scalars in this case, and in particular one has ∥A⊤(yk+1 − yk)∥ = ∥A∥∥yk+1 − yk∥  for any k, where ∥A∥ is a vector norm which is negligible to compute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' Apparently, in this case (upto  dicarding inital iterates) the linesearch variant adaPDM+ produces the same iterates of adaPDM  with no computational advantage, and is thus omitted from the plots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' The results of this simulation  are reported in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='2  Least absolute deviation regression and square-root lasso  As a final application we consider regularized regression, consisting of  minimize  x∈�n  ∥Ax − b∥p + λ∥x∥1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='2)  Whenever p = 2, this problem is referred to as square-root lasso, and for p = 1 it is known as the  least absolute deviation (LAD) regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' Both problems are cast as (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='4) by letting f ≡ 0, g = ∥·∥1,  h = ∥ · − b∥p, and A = A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' For the regression data A ∈ �m×n, b ∈ �m, three different datasets from  the LIBSVM library were used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' The results are reported in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='  Having f = 0, adaPDM reduces to PDHG with worse (constant) stepsizes and is thus omitted  from the comparisons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' Instead, as is apparent from Figure 3, the linesearch variant adaPDM+ (as  well as MP-ls, for similar reasons) excels by adaptively employing tighter bounds of the linear  operator norm along one-dimensional subspaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' We ran the simulations for λ ∈ {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='1, 1, 10}, but  only report them for λ = 10 remarking that the behavior is very similar for all values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='  18  0  1,000  2,000  3,000  4,000  5,000  10−5  10−3  10−1  101  103  ∥v∥  heart dataset (m = 270, n = 13, density= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='96)  0  1,000  2,000  3,000  4,000  5,000  10−1  100  101  102  103  svmguide3 dataset (m = 1243, n = 22, density=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='8)  VC  MP-ls  adaPDM  0  1,000  2,000  3,000  4,000  5,000  10−1  100  101  102  103  104  mushroom dataset (m = 8124, n = 112, density=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='19)  0  1,000  2,000  3,000  4,000  5,000  10−5  10−3  10−1  101  # of calls to ∇f  ∥v∥  0  1,000  2,000  3,000  4,000  5,000  10−3  10−2  10−1  100  101  102  103  # of calls to ∇f  0  1,000  2,000  3,000  4,000  5,000  10−1  100  101  102  103  # of calls to ∇f  Figure 2: Simulations for the dual SVM problem (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' First row: C = 1, second row: C = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' The x-axis  reports gradient evaluations, which is the most expensive operation (since A ∈ �1×N, calls to A and A⊤  are negligible).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' As explained, in this case adaPDM+ is indistinguishable from adaPDM and thus omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='  adaPDM and MP-ls are tuned for best performance by a grid search for t ∈ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='1, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='  5  Conclusions  In this paper we studied convex composite problems involving the sum of a locally Lipschitz dif-  ferentiable term and two nonsmooth terms, one of which composed with a linear operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' The  primal-dual algorithm adaPDM was proposed that updates the stepsizes adaptively using already  available information without any backtracking procedures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' The stepsize update is based on a novel  rule that combines estimates of local cocoercivity and Lipschitz continuity of the gradient of the  differentiable function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' When the linear operator is zero we obtain adaPGM that not only extends  the adaptive gradient descent algorithm of [49], but also involves a less conservative stepsize update  rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' More generally, we further waive the computation of the norm of the linear operator (required  by the stepsize update rule) through a linesearch procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' The resulting algorithm, adaPDM+, can  be implemented efficiently given that the backtracks do not require additional gradient evaluations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='  Future research directions involve extending the presented line of proof to the setting of varia-  tions inequalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' In particular, projection-based splittings such as [41, 31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' Other potentials direc-  tions include nonmonotone extensions for problems satisfying the so-called weak Minty assump-  tions [28, 57], as well as block coordinate and stochastic variants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' It would also be interesting to  explore extensions of the ideas presented in this work to other types of adaptive settings such as  those in [47, 27, 72, 68].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='  19  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='8  1  104  10−2  10−1  100  101  102  103  ∥v∥  housing dataset  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='8  1  104  10−2  10−1  100  101  102  103  104  abalone dataset  PDHG  MP-ls  adaPDM+  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='8  1  104  10−2  10−1  100  101  102  103  104  105  cpusmall dataset  0  500  1,000  1,500  2,000  2,500  3,000  10−5  10−3  10−1  101  103  # of calls to A and A⊤  ∥v∥  0  500  1,000  1,500  2,000  2,500  3,000  10−5  10−3  10−1  101  103  # of calls to A and A⊤  0  500  1,000  1,500  2,000  2,500  3,000  10−5  10−3  10−1  101  103  105  # of calls to A and A⊤  Figure 3: Simulations for problem (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='2) of Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='2 with λ = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' First row: regularized least-absolute  deviation (p = 1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' second row: square-root lasso (p = 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' adaPDM+ and MP-ls are tuned for best  performance by a grid search for t ∈ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='01, 100]  50  60  70  80  90  100  0  20  40  60  # of iterations  γk  aGraal  Nesterov  PGM-ls  PGM  adaPGM-MM  adaPGM  50  60  70  80  90  100  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='5  1  10−3  # of iterations  MP-ls  adaPDM  50  60  70  80  90  100  2 · 10−2  4 · 10−2  6 · 10−2  8 · 10−2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='12  # of iterations  MP-ls  adaPDM+  Figure 4: A demonstrative plot of stepsize magnitudes of the algorithms in a window of 50 iterations ex-  tracted from the simulations in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' Left: logistic regression (mushroom dataset);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' center: dual  SVM (mushroom dataset, C = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' right: square-root lasso (housing dataset, λ = 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' Primal-dual algo-  rithms are compared on problems where the ratio t2 = σk/γk coincides, so that the plots are representative  also for the dual stepsizes σk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' As is evident from the left plot, and as commented after Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='4, de-  spite the fact that the stepsize update rule of adaPGM is less conservative than that of adaPGM-MM the  comparison does not carry over iterationwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='  20  A  Convergence analysis for Section 3  We begin by establishing an inequality for the iterates in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='5) that would eventually under proper  stepsize choices imply quasi-Fejér monotonicity of the generated sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='  Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' Let (γk)k∈�, (σk)k∈� be sequences of strictly positive scalars, and starting from a triplet  (x−1, x0, y0) ∈ �n×�n×�m let (xk, yk)k∈� be recursively defined by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' Then, for any εk, τk, µk > 0  and ηk ≥ ∥A⊤(yk−yk−1)∥  ∥yk−yk−1∥  (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=', ηk = ∥A∥) it holds that  0 ≤ 1  2∥xk − x⋆∥2 − 1  2∥xk+1 − x⋆∥2 + εk+1ρk+1+µk+1ηk+1γk+1−1  2  ∥xk − xk+1∥2  + ρk+1  � 1−αk(2−βk)  2εk+1  + τk+1ηk+1γk+1  2  − ρk+1(1 − αk)  �  ∥xk−1 − xk∥2  +  γk+1  2σk+1 ∥yk − y⋆∥2 −  � γk+1  2σk+1 − ηk+1γk+1  2  � 1  µk+1 + ρk+1  τk+1  ��  ∥yk+1 − yk∥2 −  γk+1  2σk+1 ∥yk+1 − y⋆∥2  + γk+1ρk+1Pk−1 − γk+1(1 + ρk+1)Pk − γk+1  �L(x⋆, y⋆) − L(x⋆, yk+1)�,  where ρk+1 � γk+1  γk , αk � γkLk and βk � γkCk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' Let ¯yk+1 � yk + σk+1  γk+1  γk A(xk − xk−1), so that yk+1 = proxσk+1h∗(¯yk+1 + σk+1Axk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' The charac-  terization of yk and xk+1 as in the respective updates then reads  ¯yk+1−yk+1  σk+1  + Axk ∈ ∂h∗(yk+1)  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='1a)  and  Hk+1(xk)−xk+1  γk+1  − A⊤yk+1 = xk−xk+1  γk+1  − ∇f(xk) − A⊤yk+1 ∈ ∂g(xk+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='1b)  In particular, for any solution pair (x⋆, y⋆) ∈ S⋆  0  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='1b)  ≤ g(x⋆) − g(xk+1) + ⟨A⊤yk+1, x⋆ − xk+1⟩ +  (A′)  ⟨∇f(xk), x⋆ − xk+1⟩  +  1  2γk+1 ∥xk − x⋆∥2 −  1  2γk+1 ∥xk+1 − x⋆∥2 −  1  2γk+1 ∥xk − xk+1∥2  and  0  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='1a)  ≤ h∗(y⋆) − h∗(yk+1) − ⟨Axk, y⋆ − yk+1⟩ −  1  σk+1 ⟨¯yk+1 − yk+1, y⋆ − yk+1⟩  = h∗(y⋆) − h∗(yk+1) − ⟨Auk+1, y⋆ − yk+1⟩  +  1  2σk+1 ∥yk − y⋆∥2 −  1  2σk+1 ∥yk+1 − yk∥2 −  1  2σk+1 ∥yk+1 − y⋆∥2,  where uk+1 � (1 + ρk+1)xk − ρk+1xk−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' We next proceed to upper bound the term (A′) as  (A′) = ⟨∇f(xk),x⋆ − xk⟩+⟨∇f(xk),xk − xk+1⟩  = ⟨∇f(xk),x⋆ − xk⟩+ 1  γk ⟨Hk(xk−1)− xk,xk+1 − xk⟩+ 1  γk ⟨Hk(xk−1)−Hk(xk),xk − xk+1⟩  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='1b)  ≤  f(x⋆)− f(xk) +g(xk+1)−g(xk)+⟨A⊤yk,xk+1 − xk⟩+ 1  γk ⟨Hk(xk−1)−Hk(xk),xk − xk+1⟩  (B)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='  21  Let ϕ = f + g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' We bound the term (B) as done in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='3) and combine the three inequalities to obtain  0 ≤ ( f + g)(x⋆) − ( f + g)(xk) +  1  2γk+1 ∥xk − x⋆∥2 −  1  2γk+1 ∥xk+1 − x⋆∥2  +  � εk+1  2γk −  1  2γk+1  �  ∥xk − xk+1∥2 + 1−γkLk(2−γkCk)  2εk+1γk  ∥xk−1 − xk∥2  + h∗(y⋆) − h∗(yk+1) + ⟨A⊤yk, xk+1 − xk⟩ + ⟨A⊤yk+1, x⋆ − xk+1⟩ − ⟨Auk+1, y⋆ − yk+1⟩  +  1  2σk+1 ∥yk − y⋆∥2 −  1  2σk+1 ∥yk+1 − yk∥2 −  1  2σk+1 ∥yk+1 − y⋆∥2  Using again the subgradient (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='1), one has  vk � xk−1−xk  γk  − (∇f(xk−1) − ∇f(xk)) − A⊤yk ∈ ∂( f + g)(xk),  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='2)  hence  0 ≤ γk+1  γk  �  (f + g)(xk−1) − ( f + g)(xk) − ⟨vk, xk−1 − xk⟩  �  = γk+1  γk  �  (f + g)(xk−1) − ( f + g)(xk)  �  − γk+1  γk  1−γkLk  γk  ∥xk − xk−1∥2 + γk+1  γk ⟨A⊤yk, xk−1 − xk⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='3)  Sum the last two inequalities and use the identity ρk+1(xk − xk−1) = uk+1 − xk to obtain  0 ≤  1  2γk+1 ∥xk − x⋆∥2 −  1  2γk+1 ∥xk+1 − x⋆∥2 + ⟨Ax⋆,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' yk+1⟩ − ⟨Auk+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' y⋆⟩  +  � εk+1  2γk −  1  2γk+1  �  ∥xk − xk+1∥2 +  � 1−γkLk(2−γkCk)  2εk+1γk  − γk+1  γk  1−γkLk  γk  �  ∥xk−1 − xk∥2  +  1  2σk+1 ∥yk − y⋆∥2 −  1  2σk+1 ∥yk+1 − yk∥2 −  1  2σk+1 ∥yk+1 − y⋆∥2 +  �  h∗(y⋆) − h∗(yk+1)  �  + γk+1  γk  �  ( f + g)(xk−1) − ( f + g)(xk)  �  −  �  ( f + g)(xk) − ( f + g)(x⋆)  �  (C)  + ⟨A⊤(yk+1 − yk),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' uk+1 − xk+1⟩  (D)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='  To conclude, observe that  (C) = ρk+1Pk−1 − (1 + ρk+1)Pk + ⟨uk+1 − x⋆, A⊤y⋆⟩  and  (D) = γk+1  γk ⟨xk − xk−1, A⊤(yk+1 − yk)⟩ + ⟨xk − xk+1, A⊤(yk+1 − yk)⟩  ≤ γk+1  γk  τk+1ηk+1  2  ∥xk−1 − xk∥2 + µk+1ηk+1  2  ∥xk+1 − xk∥2 + ηk+1  2  � 1  µk+1 +  γk+1  γkτk+1  �  ∥yk+1 − yk∥2,  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='4)  where µk, τk > 0 are parameters related to the Fenchel-Young inequality, so that the proof follows  from the identity ⟨Ax⋆, yk+1 − y⋆⟩ = L(x⋆, yk+1) − L(x⋆, y⋆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='  Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='  ♠ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='2(i) Denoting  ˜Uk � 1  2∥xk − x⋆∥2 + 1−εkρk−µkηkγk  2  ∥xk − xk−1∥2 +  γk  2σk ∥yk − y⋆∥2 + γk(1 + ρk)Pk−1,  the inequality in Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='1 can be expressed as  ˜Uk+1 ≤ ˜Uk −  � γk  2σk −  γk+1  2σk+1  �  ∥yk − y⋆∥2 − γk(1 + ρk − ρ2  k+1)Pk−1  −  � 1−εkρk−µkηkγk  2  + ρk+1  �  ρk+1(1 − αk) − 1−αk(2−βk)  2εk+1  − τk+1ηk+1γk+1  2  ��  ∥xk−1 − xk∥2  −  � γk+1  2σk+1 − ηk+1γk+1  2  � 1  µk+1 + ρk+1  τk+1  ��  ∥yk+1 − yk∥2 − γk+1  �L(x⋆, y⋆) − L(x⋆, yk+1)�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='  22  Since σk = t2γk and ξk = t2γ2  kη2  k, by selecting εk �  1  2ρk and µk � 2(1 + δ)tξ  1/2  k one has that ˜Uk = Uk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='  Note also that the coefficient of ∥xk − xk−1∥ in Uk is strictly positive since the stepsize update ensures  ξk ≤ 1/c2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' Moreover, with this choice the above inequality becomes  Uk+1 ≤ Uk − γk(1 + ρk − ρ2  k+1)Pk−1 − γk+1  �L(x⋆,y⋆) − L(x⋆,yk+1)�  −  �  1−4ξk(1+δ)  4  + ρk+1  �  ρk+1αk(1 − βk) −  τk+1ξ  1/2  k+1  2t  ��  ∥xk−1 − xk∥2 − 1  2t  �  1+2δ  2t(1+δ) −  ξ  1/2  k+1ρk+1  τk+1  �  ∥yk+1 − yk∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='  We now set τk+1 � ρk+1µk+1 = 2(1 + δ)tξ  1/2  k+1ρk+1 so the inequality overall simplifies to the one of the  statement  Uk+1 ≤ Uk − γk(1 + ρk − ρ2  k+1)Pk−1 − γk+1  �L(x⋆, y⋆) − L(x⋆, yk+1)�  −  � 1−4ξk(1+δ)  4  + ρk+1(ρk+1αk(1 − βk) − ρk+1ξk+1(1 + δ))  �  ≥  δ  4(1+δ)  ∥xk−1 − xk∥2 −  δ  2t2(1+δ)∥yk+1 − yk∥2,  upto ensuring that the coefficient of Pk−1 is positive and that the inequality for the coefficient of  ∥xk − xk−1∥2 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' The former is of trivial verification, having ρk+1 ≤  �  1 + ρk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' It thus remains to  show that  1  4 −  δ  4(1+δ) − ξk(1 + δ) − ρ2  k+1(∆k − ξk+1(1 + δ)) ≥ 0  holds for every k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' By using the fact that ξk+1 = (tηk+1γk+1)2 = (tηk+1γk)2ρ2  k+1, this reduces to the  second-order inequality (in ρ2  k+1)  (tηk+1γk)2(1 + δ)ρ4  k+1 + ∆kρ2  k+1 − �  1  4(1+δ) − ξk(1 + δ)� ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='  Note that the bound γk ≤  1  2ctηk implies  1  4(1+δ) − ξk(1 + δ) ≥  c2−(1+δ)2  4c2(1+δ) > 0, thus ensuring that the  inequality always admits solutions for small enough ρ2  k+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' Namely, letting ¯ξk � ξk(1 + δ)2  ρ2  k+1 ≤  −∆k +  �  ∆2  k + (tηk+1γk)2(1 − 4¯ξk)  2(1 + δ)(tηk+1γk)2  =  1 − 4¯ξk  2(1 + δ)  �  ∆k +  �  ∆2  k + (tηk+1γk)2(1 − 4¯ξk)  �,  which is indeed guaranteed by one of the bounds on γk+1 = γkρk+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' Note that the second expression  removes the singularity in case ηk+1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='  ♠ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='2(ii) To prove the claim, we will show that whenever γk+1 < γk occurs, then necessarily γk+1  is lower bounded as in the statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' The proof will then follow from a trivial inductive argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='  Suppose that γk+1 < γk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' If γk+1 =  1  2ctηk+1 , then γk+1 ≥  1  2ctηmax .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' Otherwise, necessarily  γ2  k > γ2  k+1 =  γ2  k(1 − 4¯ξk)  2(1 + δ)  � �  ∆2  k + (tηk+1γk)2(1 − 4¯ξk) + ∆k  �  ≥  γ2  k ¯c  2(1 + δ)  � �  ∆2  k + (tηmaxγk)2¯c + ∆k  �,  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='5)  23  where ¯c � c2−(1+δ)2  c2  , and the second inequality uses the fact that 1−4¯ξk ≥ ¯c > 0 together with the fact  that (0, ∞) ∋ x �→  x  √  b2+a2x2 +b is increasing for any value of a, b ∈ �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' By comparing the outermost  terms of the chain of inequalities we then obtain that  �  ∆2  k + (tηmaxγk)2¯c + ∆k >  ¯c  2(1+δ)  ⇔  ∆k >  ¯c  4(1+δ) − (1 + δ)(tηmaxγk)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='  We now distinguish two cases:  Case 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' ∆k ≤ 0, and in particular γk >  √  ¯c  2(1+δ)tηmax .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' Then, since x �→  1  √  x2+b2 +x is increasing for  any value of b ∈ �, by setting ∆k ← 0 in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='5) we obtain  γ2  k+1 ≥ γk  √  ¯c  2(1 + δ)tηmax  ≥  ¯c  (2(1 + δ)tηmax)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='  Case 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' ∆k > 0 or, equivalently, γkCk > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' Denoting α � γkLf,V, one has that α ≥ γkLk and  α ≥ γkCk > 1, hence that ∆k ≤ α(α − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' Arguing as in the previous case, this time by setting  ∆k ← α(α − 1) in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='5), yields  γ2  k+1L2  f,V ≥  α2¯c  2(1 + δ)  � �  α2(α − 1)2 + (tηmaxγk)2¯c + α(α − 1)  �  ≥  α¯c  2(1 + δ)  � �  (α − 1)2 + (tηmax/L f,V)2¯c + α − 1  �  ≥ min  �  Lf,V  √  ¯c  2(1+δ)tηmax ,  ¯c  4(1+δ)  �  ,  where the last inequality owes to the fact that [1, ∞) ∋ x �→  x  √  (x−1)2+b2 +x−1 attains the infimum  at either 1 or ∞ for any b ∈ �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='  Putting all the cases together yields  γk ≥ min  �  γ0,  4√  ¯c  √  2(1+δ)tηmaxLf,V ,  √  ¯c  2 √(1+δ)L f,V ,  1  2ctηmax ,  √  ¯c  2(1+δ)tηmax  �  > 0  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='6)  (where we remind that ¯c = c2−(1+δ)2  c2  ), establishing the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='  ♠ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='2(iii) Since the sequence (xk, yk)k∈� is quasi-Fejér monotone, it suffices to show that there exists  a subsequence converging to a primal-dual solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' We start by showing that lim infk→∞ Pk = 0  which can be ensured arguing as in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='3(iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' If lim supk→∞(1 + ρk − ρ2  k+1) > 0, then a  telescoping argument in 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='2(i) yields the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' Alternatively, by the stepsize update rule one has that  1+ρk −ρ2  k+1 ≥ 0 and therefore 1+ρk −ρ2  k+1 → 0, from which it easily follows that lim infk→∞ ρk > 1  and that therefore γk → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' Consequently, since γk(1 + ρk)Pk−1 ≤ Uk ≤ U0, this directly proves that  Pk → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='  Let (ˆx, ˆy) denote a limit point of (xk, yk−1)k∈�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' This implies that  Pz⋆(ˆx) = L(ˆx, y⋆) − L(x⋆, y⋆) = 0,  24  and in particular ˆx ∈ arg min L(·, y⋆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' If f +g is strictly convex, then so is L(·, y⋆) and necessarily ˆx =  x⋆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' Alternatively, if δ > 0, telescoping the inequality in assertion 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='2(i) yields that both ∥xk+1 − xk∥  and ∥yk−1 − yk∥ vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' By the optimality condition for step 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='4 we have  1  γk+1 (xk+1 − xk) ∈ ∇f(xk) + A⊤yk+1 + ∂g(xk+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='  Passing to the limit, using Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='2(ii) and outer semicontinuity of ∂g yields 0 ∈ ∇f(ˆx)+∂g(ˆx)+  A⊤ˆy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' Similarly, for the dual variable,  1  σk+1 (yk+1 − yk) + γk+1  γk (xk − xk−1) ∈ ∂h∗(yk+1) − Axk+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='  A trivial induction argument reveals that γk+1  γk is upper bounded by max  � γ0  γ−1 , 1  2(1 +  √  5)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' Therefore,  passing to the limit along the same subsequence and recalling that σk+1 = t2γk+1 is lower bounded  yield 0 ∈ ∂h∗(ˆy) − Aˆx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' Along with the previous inclusion, primal-dual optimality of the limit pair  (ˆx, ˆy) is established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content='  References  [1] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' Alacaoglu, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' B¨ohm, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
+page_content=' Malitsky, Beyond the golden ratio for variational inequality  algorithms, arXiv preprint arXiv:2212.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
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+page_content='  [2] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
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+page_content='  [72] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
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+page_content=' Gu, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}
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+page_content='  29' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfSQmN/content/2301.04431v1.pdf'}