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A unified framework for some inexact proximal point algorithms

NUMERICAL FUNCTIONAL ANALYSIS AND OPTIMIZATION 2001年第7-8期22卷 1013-1035页

作者： Solodov, MV Svaiter, BF Inst Matematica Pura & Aplicada BR-22460320 Rio De Janeiro Brazil

We present a unified framework for the design and convergence analysis of a class of algorithms based on approximate solution of proximal point subproblems. Our development further enhances the constructive approximation approach of the recently proposed hybrid projection-proximal and extragradient-proximal methods. Specifically, we introduce an even more flexible error tolerance criterion, as well as provide a unified view of these two algorithms. Our general method possesses global convergence and local (super)linear rate of convergence under standard assumptions, while using a constructive approximation criterion suitable for a number of specific implementations. For example, we show that close to a regular solution of a monotone system of semismooth equations, two Newton iterations are sufficient to solve the proximal subproblem within the required error tolerance. Such systems of equations arise naturally when reformulating the nonlinear complementarity problem.

关键词： maximal monotone operator proximal point algorithm approximation criteria

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Approximate Customized proximal point algorithms for Separable Convex Optimization

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Journal of the Operations Research Society of China 2023年第2期11卷 383-408页

作者： Hong-Mei Chen Xing-Ju Cai Ling-Ling Xu School of Mathematical Sciences Nanjing Normal UniversityNanjing 210023JiangsuChina Jiangsu Key Laboratory for NSLSCS Nanjing Normal UniversityNanjing 210023JiangsuChina

proximal point algorithm(PPA)is a useful algorithm framework and has good convergence *** difficulty is that the subproblems usually only have iterative *** this paper,we propose an inexact customized PPA framework for twoblock separable convex optimization problem with linear *** design two types of inexact error criteria for the *** first one is absolutely summable error criterion,under which both subproblems can be solved *** one of the two subproblems is easily solved,we propose another novel error criterion which is easier to implement,namely relative error *** relative error criterion only involves one parameter,which is more *** establish the global convergence and sub-linear convergence rate in ergodic sense for the proposed *** numerical experiments on LASSO regression problems and total variation-based image denoising problem illustrate that our new algorithms outperform the corresponding exact algorithms.

关键词： Inexact criteria proximal point algorithm Alternating direction method of multipliers Separable convex programming

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Four parameter proximal point algorithms

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NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS 2011年第2期74卷 544-555页

作者： Boikanyo, O. A. Morosanu, G. Cent European Univ Dept Math & Its Applicat H-1051 Budapest Hungary

Several strong convergence results involving two distinct four parameter proximal point algorithms are proved under different sets of assumptions on these parameters and the general condition that the error sequence converges to zero in norm. Thus our results address the two important problems related to the proximal point algorithm - one being that of strong convergence (instead of weak convergence) and the other one being that of acceptable errors. One of the algorithms discussed was introduced by Yao and Noor (2008) [7] while the other one is new and it is a generalization of the regularization method initiated by Lehdili and Moudafi (1996) [9] and later developed by Xu (2006) [8]. The new algorithm is also ideal for estimating the convergence rate of a sequence that approximates minimum values of certain functionals. Although these algorithms are distinct, it turns out that for a particular case, they are equivalent. The results of this paper extend and generalize several existing ones in the literature. (C) 2010 Elsevier Ltd. All rights reserved.

关键词： proximal point algorithm Regularization method Monotone operator Weak convergence Strong convergence Minimizer

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On relaxation of some customized proximal point algorithms for convex minimization: from variational inequality perspective

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COMPUTATIONAL OPTIMIZATION AND APPLICATIONS 2019年第3期73卷 871-901页

作者： Ma, Feng High Tech Inst Xian Xian 710025 Shaanxi Peoples R China

The proximal point algorithm (PPA) is a fundamental method for convex programming. When applying the PPA to solve linearly constrained convex problems, we may prefer to choose an appropriate metric matrix to define the proximal regularization, so that the computational burden of the resulted PPA can be reduced, and sometimes even admit closed form or efficient solutions. This idea results in the so-called customized PPA (also known as preconditioned PPA), and it covers the linearized ALM, the primal-dual hybrid gradient algorithm, ADMM as special cases. Since each customized PPA owes special structures and has popular applications, it is interesting to ask wether we can design a simple relaxation strategy for these algorithms. In this paper we treat these customized PPA algorithms uniformly by a mixed variational inequality approach, and propose a new relaxation strategy for these customized PPA algorithms. Our idea is based on correcting the dual variables individually and does not rely on relaxing the primal variables. This is very different from previous works. From variational inequality perspective, we prove the global convergence and establish a worst-case convergence rate for these relaxed PPA algorithms. Finally, we demonstrate the performance improvements by some numerical results.

关键词： Convex minimization proximal point algorithm Relaxation Augmented Lagrangian method

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Two accelerated double inertial algorithms for variational inequalities on Hadamard manifolds

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COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION 2025年 145卷

作者： Tan, Bing Abass, Hammed Anuoluwapo Li, Songxiao Oyewole, Olawale Kazeem Southwest Univ Sch Math & Stat Chongqing 400715 Peoples R China Sefako Makgatho Hlth Sci Univ Dept Math & Appl Math POB 94 ZA-0204 Pretoria South Africa Shantou Univ Dept Math Shantou 515821 Peoples R China Tshwane Univ Technol Dept Math ZA-0007 Pretoria South Africa

Two modified double inertial proximal point algorithms are proposed for solving variational inequality problems with a pseudomonotone vector field in the settings of a Hadamard manifold. Weak convergence of the proposed methods is attained without the requirement of Lipschitz continuity conditions. The convergence efficiency of the proposed algorithms is improved with the help of the double inertial technique and the non-monotonic self-adaptive step size rule. We present a numerical experiment to demonstrate the effectiveness of the proposed algorithm compared to several existing ones. The results extend and generalize many recent methods in the literature.

关键词： Variational inequality problem Double inertial proximal point algorithm Uniformly continuous Hadamard manifold

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Customized proximal point algorithms for linearly constrained convex minimization and saddle-point problems: a unified approach

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COMPUTATIONAL OPTIMIZATION AND APPLICATIONS 2014年第1-2期59卷 135-161页

作者： Gu, Guoyong He, Bingsheng Yuan, Xiaoming Nanjing Univ Dept Math Nanjing 210093 Jiangsu Peoples R China Nanjing Univ Int Ctr Management Sci & Engn Nanjing 210093 Jiangsu Peoples R China Nanjing Univ Dept Math Nanjing 210093 Jiangsu Peoples R China Hong Kong Baptist Univ Dept Math Hong Kong Hong Kong Peoples R China

This paper focuses on some customized applications of the proximal point algorithm (PPA) to two classes of problems: the convex minimization problem with linear constraints and a generic or separable objective function, and a saddle-point problem. We treat these two classes of problems uniformly by a mixed variational inequality, and show how the application of PPA with customized metric proximal parameters can yield favorable algorithms which are able to make use of the models' structures effectively. Our customized PPA revisit turns out to unify some algorithms including some existing ones in the literature and some new ones to be proposed. From the PPA perspective, we establish the global convergence and a worst-case O(1/t) convergence rate for this series of algorithms in a unified way.

关键词： Convex minimization Saddle-point problem proximal point algorithm Convergence rate Customized algorithms Splitting algorithms

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A class of nonlinear proximal point algorithms for variational inequality problems

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INTERNATIONAL JOURNAL OF COMPUTER MATHEMATICS 2015年第7期92卷 1385-1401页

作者： He, Hongjin Cai, Xingju Han, Deren Hangzhou Dianzi Univ Sch Sci Dept Math Hangzhou 310018 Zhejiang Peoples R China Nanjing Normal Univ Sch Math Sci Jiangsu Key Lab NSLSCS Nanjing 210023 Jiangsu Peoples R China

This work is motivated by a recent work on an extended linear proximal point algorithm (PPA) [B.S. He, X.L. Fu, and Z.K. Jiang, proximal-point algorithm using a linear proximal term, J. Optim. Theory Appl. 141 (2009), pp. 299-319], which aims at relaxing the requirement of the linear proximal term of classical PPA. In this paper, we make further contributions along the line. First, we generalize the linear PPA-based contraction method by using a nonlinear proximal term instead of the linear one. A notable superiority over traditional PPA-like methods is that the nonlinear proximal term of the proposed method may not necessarily be a gradient of any functions. In addition, the nonlinearity of the proximal term makes the new method more flexible. To avoid solving a variational inequality subproblem exactly, we then propose an inexact version of the developed method, which may be more computationally attractive in terms of requiring lower computational cost. Finally, we gainfully employ our new methods to solve linearly constrained convex minimization and variational inequality problems.

关键词： 90C25 90C30 65K10 convex minimization problem proximal point algorithm variational inequality problem nonlinear proximal term

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LINEAR RATES OF ASYMPTOTIC REGULARITY FOR HALPERN-TYPE ITERATIONS

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MATHEMATICS OF COMPUTATION 2025年第353期94卷 1323-1333页

作者： Cheval, Horatiu Leustean, Laurentiu Univ Bucharest Fac Math & Comp Sci LOS Acad 14 Bucharest Romania Romanian Acad Sim Stoilow Inst Math POB 1-764 Bucharest Romania Inst Log & Data Sci Popa Tatu 18 Bucharest Romania

In this note we apply a lemma due to Sabach and Shtern to compute linear rates of asymptotic regularity for Halpern-type nonlinear iterations studied in optimization and nonlinear analysis.

关键词： Rates of asymptotic regularity Halpern iteration proximal point algorithm

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On finite convergence of proximal point algorithms for variational inequalities

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JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 2005年第1期312卷 148-158页

作者： Xiu, NH Zhang, JZ Beijing Jiaotong Univ Dept Appl Math Beijing 100044 Peoples R China City Univ Hong Kong Dept Math Kowloon Hong Kong Peoples R China

In this paper, we first characterize finite convergence of an arbitrary iterative algorithm for solving the variational inequality problem (VIP), where the finite convergence means that the algorithm can find an exact solution of the problem in a finite number of iterations. By using this result, we obtain that the well-known proximal point algorithm possesses finite convergence if the solution set of VIP is weakly sharp. As an extension, we show finite convergence of the inertial proximal method for solving the general variational inequality problem under the condition of weak g-sharpness. (c) 2005 Elsevier Inc. All rights reserved.

关键词： variational inequalities proximal point algorithm inertial proximal method finite convergence weak sharpness

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Strong Convergence of Regularized New proximal point algorithms

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 2019年第3期181卷 864-882页

作者： Rouhani, Behzad Djafari Moradi, Sirous Univ Texas El Paso Dept Math Sci El Paso TX 79968 USA Arak Univ Fac Sci Dept Math Arak *** Iran

We consider the regularization of two proximal point algorithms (PPA) with errors for a maximal monotone operator in a real Hilbert space, previously studied, respectively, by Xu, and by Boikanyo and Morosanu, where they assumed the zero set of the operator to be nonempty. We provide a counterexample showing an error in Xu's theorem, and then we prove its correct extended version by giving a necessary and sufficient condition for the zero set of the operator to be nonempty and showing the strong convergence of the regularized scheme to a zero of the operator. This will give a first affirmative answer to the open question raised by Boikanyo and Morosanu concerning the design of a PPA, where the error sequence tends to zero and a parameter sequence remains bounded. Then, we investigate the second PPA with various new conditions on the parameter sequences and prove similar theorems as above, providing also a second affirmative answer to the open question of Boikanyo and Morosanu. Finally, we present some applications of our new convergence results to optimization and variational inequalities.

关键词： Maximal monotone operator proximal point algorithm Resolvent operator Metric projection Hilbert space 47J25 47H05 47H09 90C29 90C90

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