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CONVERGENCE RATE OF inexact proximal point algorithmS FOR OPERATOR WITH HOLDER METRIC SUBREGULARITY

SIAM JOURNAL ON OPTIMIZATION 2023年第3期33卷 1996-2020页

作者： Wang, Jinhua Li, Chong Ng, K. F. Hangzhou Normal Univ Dept Math Hangzhou 311121 Peoples R China Zhejiang Univ Sch Math Sci Hangzhou Qianjiang Distinguished Expert Hangzhou 310058 Peoples R China Chinese Univ Hong Kong Dept Math Hong Kong Peoples R China Chinese Univ Hong Kong IMS Hong Kong Peoples R China

We study the issue of strong convergence of inexact proximal point algorithms (introduced by Rockafellar in [SIAM J. Control Optim., 14 (1976), pp. 877-898]) for maximal monotone operators on Hilbert spaces. A unified global/local strong convergence of inexact proximal point algorithms is established under the Holder metrically subregular condition. Furthermore, quantitative estimates on the convergence rate of inexact proximal point algorithms are also provided. Applying to the special case of the classical (exact) proximal point algorithm, our results improve the corresponding ones in [G. Li and B. S. Mordukhovich, SIAM J. Optim., 22 (2012), pp. 1655-1684]. Finally, as applications, global/local strong convergence and estimates on the convergence rate of inexact proximal point algorithms for optimization problems are presented.

关键词： inexact proximal point algorithm convergence rate Holder metric subregularity

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An inexact proximal point algorithm for maximal monotone vector fields on Hadamard manifolds

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OPERATIONS RESEARCH LETTERS 2013年第6期41卷 586-591页

作者： Tang, Guo-ji Huang, Nan-jing Guangxi Univ Nationalities Sch Sci Nanning 530006 Guangxi Peoples R China Sichuan Univ Dept Math Chengdu 610064 Sichuan Peoples R China

In this paper, an inexact proximal point algorithm concerned with the singularity of maximal monotone vector fields is introduced and studied on Hadamard manifolds, in which a relative error tolerance with squared summable error factors is considered. It is proved that the sequence generated by the proposed method is convergent to a solution of the problem. Moreover, an application to the optimization problem on Hadamard manifolds is given. The main results presented in this paper generalize and improve some corresponding known results given in the literature. (C) 2013 Elsevier By. All rights reserved.

关键词： inexact proximal point algorithm Hadamard manifold Maximal monotone vector field

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Accelerated differential inclusion for convex optimization

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OPTIMIZATION 2023年第5期72卷 1139-1170页

作者： Luo, Hao Peking Univ Sch Math Sci Beijing Peoples R China

This paper introduces a second-order differential inclusion for unconstrained convex optimization. In continuous level, solution existence in proper sense is obtained and exponential decay of a novel Lyapunov function along with the solution trajectory is derived as well. Then in discrete level, based on numerical discretizations of the continuous model, two inexact proximal point algorithms are proposed, and some new convergence rates are established via a discrete Lyapunov function.

关键词： Convex optimization inexact proximal point algorithm differential inclusion existence Lyapunov function

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Linear conditioning, weak sharpness and finite convergence for equilibrium problems

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JOURNAL OF GLOBAL OPTIMIZATION 2020年第2期77卷 405-424页

作者： Luong Van Nguyen Ansari, Qamrul Hasan Qin, Xiaolong Hong Duc Univ Dept Nat Sci Thanh Hoa Vietnam Aligarh Muslim Univ Dept Math Aligarh 202002 Uttar Pradesh India King Fahd Univ Petr & Minerals Dept Math & Stat Dhahran Saudi Arabia Hangzhou Normal Univ Dept Math Hangzhou Zhejiang Peoples R China

The present paper first provides sufficient conditions and characterizations for linearly conditioned bifunction associated with an equilibrium problem. It then introduces the notion of weak sharp solution for equilibrium problems which is analogous to the linear conditioning notion. This new notion generalizes and unifies the notion of weak sharp minima for optimization problems as well as the notion of weak sharp solutions for variational inequality problems. Some sufficient conditions and characterizations of weak sharpness are also presented. Finally, we study the finite convergence property of sequences generated by some algorithms for solving equilibrium problems under linear conditioning and weak shapness assumptions. An upper bound of the number of iterations by which the sequence generated by proximal point algorithm converges to a solution of equilibrium problems is also given.

关键词： Equilibrium problems Linear conditioning Weak sharpness Finite convergence inexact proximal point algorithm

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On the Convergence Rate of a proximal point algorithm for Vector Function on Hadamard Manifolds

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Journal of the Operations Research Society of China 2017年第3期5卷 405-417页

作者： Feng-Mei Tang Ping-Liang Huang College of Science Shanghai UniversityShanghai 200444China

The proximal point algorithm has many interesting applications,such as signal recovery,signal processing and *** recent years,the proximal point method has been extended to Riemannian *** main advantages of these extensions are that nonconvex problems in classic sense may become geodesic convex by introducing an appropriate Riemannian metric,constrained optimization problems may be seen as unconstrained *** this paper,we propose an inexact proximal point algorithm for geodesic convex vector function on Hadamard *** the assumption that the objective function is coercive,the sequence generated by this algorithm converges to a Pareto critical *** the objective function is coercive and strictly geodesic convex,the sequence generated by this algorithm converges to a Pareto optimal ***,under the weaker growth condition,we prove that the inexact proximal point algorithm has linear/superlinear convergence rate.

关键词： inexact proximal point algorithm Hadamard manifolds Convergence rate Pareto critical point Pareto optimal point

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Rate of convergence for proximal point algorithms on Hadamard manifolds

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OPERATIONS RESEARCH LETTERS 2014年第6-7期42卷 383-387页

作者： Tang, Guo-ji Huang, Nan-jing Guangxi Univ Nationalities Sch Sci Nanning 530006 Guangxi Peoples R China Sichuan Univ Dept Math Chengdu 610064 Sichuan Peoples R China

In this paper, an estimate of convergence rate concerned with an inexact proximal point algorithm for the singularity of maximal monotone vector fields on Hadamard manifolds is discussed. We introduce a weaker growth condition, which is an extension of that of Luque from Euclidean spaces to Hadamard manifolds. Under the growth condition, we prove that the inexact proximal point algorithm has linear/superlinear convergence rate. The main results presented in this paper generalize and improve some corresponding known results. (C) 2014 Elsevier B.V. All rights reserved.

关键词： inexact proximal point algorithm Hadamard manifold Convergence rate Maximal monotone vector field

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A partial inexact alternating direction method for structured variational inequalities

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OPTIMIZATION 2014年第7期63卷 1043-1055页

作者： Peng, Zheng Zhu, Wenxing Fuzhou Univ Coll Math & Comp Sci Fuzhou 350108 Peoples R China Fuzhou Univ Ctr Discrete Math & Theoret Comp Sci Fuzhou 350108 Peoples R China

In this article, a new method is proposed for solving a class of structured variational inequalities (SVIs). The proposed method is referred to as the partial inexact proximal alternating direction (piPAD) method. In the method, two subproblems are solved independently. One is handled by an inexact proximal point method and the other is solved directly. This feature is the major difference between the proposed method and some existing alternating direction-like methods. The convergence of the piPAD method is proved. Two examples of the modern convex optimization problem arising from engineering and information sciences, which can be reformulated into the encountered SVIs, are presented to demonstrate the applicability of the piPAD method. Also, some preliminary numerical results are reported to validate the feasibility and efficiency of the piPAD method.

关键词： structured variational inequalities separable convex optimization alternating direction method inexact proximal point algorithm

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Convergence analysis of a relaxed extragradient-proximal point algorithm application to variational inequalities

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OPTIMIZATION 2005年第2期54卷 191-213页

作者： Hue, TT Strodiot, JJ Fac Univ Notre Dame Paix Unite Optimisat & Controle Dept Math B-5000 Namur Belgium Hue Univ Dept Math Fac Sci Hue Vietnam

In this paper we present a relaxed version of an extragradient-proximal point algorithm recently proposed by Solodov and Svaiter for finding a zero of a maximal monotone operator defined on a Hilbert space. The aim is to introduce a family of parameters in order to accelerate the rate of convergence of this algorithm. First we study the convergence and rate of convergence of the relaxed algorithms and then we apply them to the generalized variational inequality problem. For this problem, the operator is a sum of two operators: the first one is single-valued, monotone and continuous and the second one is the subdifferential of a nonsmooth lower semi-continuous proper convex function phi. To make the subproblems easier to solve, we consider, as in the bundle methods, piecewise linear convex approximations of phi. We explain how to construct these approximations and how the subproblems fall within the framework of our relaxed extragradient-proximal point algorithm. We prove the convergence of the resulting algorithm without assuming a Dunn property on the single-valued operator. Finally, we report some numerical experiences to illustrate the behavior of our implementable algorithm for different values of the relaxation factor. A comparison with another algorithm is also given.

关键词： inexact proximal point algorithm enlargement of a maximal monotone operator extragradient method variational inequalities forward-backward bundle method quadratic programming problems

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An approximate proximal-extragradient type method for monotone variational inequalities

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JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 2004年第2期300卷 362-374页

作者： He, BS Yang, ZH Yuan, XM Nanjing Univ Dept Math Nanjing 210093 Peoples R China City Univ Hong Kong Dept Math Hong Kong Hong Kong Peoples R China

proximal point algorithms (PPA) are attractive methods for monotone variational inequalities. The approximate versions of PPA are more applicable in practice. A modified approximate proximal point algorithm (APPA) presented by Solodov and Svaiter [Math. Programming, Ser. B 88 (2000) 371-389] relaxes the inexactness criterion significantly. This paper presents an extended version of Solodov-Svaiter's APPA. Building the direction from current iterate to the new iterate obtained by Solodov-Svaiter's APPA, the proposed method improves the profit at each iteration by choosing the optimal step length along this direction. In addition, the inexactness restriction is relaxed further. Numerical example indicates the improvement of the proposed method. (C) 2004 Elsevier Inc. All rights reserved.

关键词： monotone variational inequality inexact proximal point algorithm

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关于单调变分不等式的不精确邻近点算法的收敛性分析

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高等学校计算数学学报 2003年第4期25卷 336-343页

作者：王治华南京大学数学系南京 210093

1 引言设Ω是R;的一个非空闭凸子集,F（u）是一个从R;到自身的一个单调连续映射,变分不等式问题是寻找一个向量u;∈Ω,使得（1.1）我们用VI（Ω,F）来表示（1.1）式.变分不等式问题包括非线性互补问题（当Ω=R;）和

关键词： monotone variational inequalities proximal point algorithm inexact proximal point algorithm accelerative method.

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