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Approximation semigroups for resolvent maps

BANACH JOURNAL OF MATHEMATICAL ANALYSIS 2024年第2期18卷 1-17页

作者： Choi, Byoung Jin Ji, Un Cig Lim, Yongdo Palfia, Miklos Jeju Natl Univ Dept Math Educ Jeju 63243 South Korea Chungbuk Natl Univ Inst Ind & Appl Math Dept Math Cheongju 28644 South Korea Sungkyunkwan Univ Dept Math Suwon 16419 South Korea Corvinus Univ Budapest Dept Math Fovam Ter 8 H-1093 Budapest Hungary Univ Szeged Bolya Inst Interdisciplinary Excellence Ctr Arad vertanuk tere 1 H-6720 Szeged Hungary

In this paper, we extend the results for approximation semigroups for general resolvent maps including various resolvents of maps on a general convex geodesic metric space. For our study, we introduce the notion of (general) resolvent maps which is a generalization of the resolvent maps in Lawson (J Lie Theory 33, 361-376, 2023) and then we prove several useful properties for the resolvent map and construct the approximation semigroups for resolvent maps. We also study the convergence of a proximal point like algorithm for the general resolvent map.

关键词： Geodesic metric space Approximation semigroup Resolvent map Karcher mean proximal point algorithm

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New error bounds and their applications to convergence analysis of iterative algorithms

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MATHEMATICAL PROGRAMMING 2000年第2期88卷 341-355页

作者： Luo, ZQ McMaster Univ Dept Elect & Comp Engn Hamilton ON L8S 4K1 Canada

We present two new error bounds for optimization problems over a convex set whose objective function f is either semianalytic or gamma-strictly convex, with gamma greater than or equal to 1. We then apply these error bounds to analyze the rate of convergence of a wide class of iterative descent algorithms for the aforementioned optimization problem. Our analysis shows that the function sequence {f(x(k))} converges at least at the sublinear rate of k(-epsilon) for some positive constant epsilon, where k is the iteration index. Moreover, the distances from the iterate sequence {x(k)} to the set of stationary points of the optimization problem converge to zero at least sublinearly.

关键词： Variational inequality problem proximal point algorithm Convex minimization Linear convergence Systems Interior point algorithm Convexity Greater than or equal to Objective function Iterative algorithms Bound Converge

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An self-adaptive LQP method for constrained variational inequalities

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APPLIED MATHEMATICS AND COMPUTATION 2007年第2期189卷 1586-1600页

作者： Fu, Xiao-Ling Bnouhachem, Abdellah Nanjing Univ Dept Math Nanjing 210093 Peoples R China Nanjing Univ Sch Management Sci & Engn Nanjing 210093 Peoples R China

In this paper, we present a logarithmic-quadratic proximal (LQP) type prediction-correction methods for solving constrained variational inequalities VI(S,f), where S is a convex set with linear constraints. The computational load in each iteration is quite tiny. However, the number of iterations is significantly dependent on a parameter which balances the primal and dual variables. We then propose a self-adaptive prediction-correction method that adjusts the scalar parameter automatically. Under certain conditions, the global convergence of the proposed method is established. In order to demonstrate the efficiency of the proposed method, we provide numerical results for a convex nonlinear programming and traffic equilibrium problems. (c) 2006 Elsevier Inc. All rights reserved.

关键词： proximal point algorithm variational inequality prediction-correction traffic equilibrium problems

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A partially parallel splitting method for multiple-block separable convex programming with applications to robust PCA

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COMPUTATIONAL OPTIMIZATION AND APPLICATIONS 2016年第1期63卷 273-303页

作者： Hou, Liusheng He, Hongjin Yang, Junfeng Nanjing Xiaozhuang Univ Sch Math & Informat Technol Key Lab Trust Cloud Comp & Big Data Anal Nanjing 211171 Jiangsu Peoples R China Hangzhou Dianzi Univ Sch Sci Dept Math Hangzhou 310018 Zhejiang Peoples R China Nanjing Univ Dept Math Nanjing 210093 Jiangsu Peoples R China

We consider a multiple-block separable convex programming problem, where the objective function is the sum of m individual convex functions without overlapping variables, and the constraints are linear, aside from side constraints. Based on the combination of the classical Gauss-Seidel and the Jacobian decompositions of the augmented Lagrangian function, we propose a partially parallel splitting method, which differs from existing augmented Lagrangian based splitting methods in the sense that such an approach simplifies the iterative scheme significantly by removing the potentially expensive correction step. Furthermore, a relaxation step, whose computational cost is negligible, can be incorporated into the proposed method to improve its practical performance. Theoretically, we establish global convergence of the new method in the framework of proximal point algorithm and worst-case nonasymptotic O(1/t) convergence rate results in both ergodic and nonergodic senses, where t counts the iteration. The efficiency of the proposed method is further demonstrated through numerical results on robust PCA, i.e., factorizing from incomplete information of an unknown matrix into its low-rank and sparse components, with both synthetic and real data of extracting the background from a corrupted surveillance video.

关键词： Augmented Lagrangian method Multiple-block Convex programming Partially parallel splitting method proximal point algorithm

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An alternative extrapolation scheme of PDHGM for saddle point problem with nonlinear function

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COMPUTATIONAL OPTIMIZATION AND APPLICATIONS 2023年第1期85卷 263-291页

作者： Gao, Ying Zhang, Wenxing Univ Elect Sci & Technol China Sch Math Sci Chengdu Peoples R China

Primal-dual hybrid gradient (PDHG) method is a canonical and popular prototype for solving saddle point problem (SPP). However, the nonlinear coupling term in SPP excludes the application of PDHG on far-reaching real-world problems. In this paper, following the seminal work by Valkonen (Inverse Problems 30, 2014), we devise a variant iterative scheme for solving SPP with nonlinear function by exerting an alternative extrapolation procedure. The novel iterative scheme falls exactly into the proximal point algorithmic framework without any residuals, which indicates that the associated inclusion problem is "nearer" to the KKT mapping induced by SPP. Under the metrically regular assumption on KKT mapping, we simplify the local convergence of the proposed method on contractive perspective. Numerical simulations on a PDE-constrained nonlinear inverse problem demonstrate the compelling performance of the proposed method.

关键词： Primal-dual hybrid gradient (modified) proximal point algorithm Monotone inclusion Extrapolation Linearization Metric regularity Contraction

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Approximating a Solution of the Inclusion Problem for an Infinite Family of Monotone Operators in Hadamard Spaces and Its Applications

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NUMERICAL FUNCTIONAL ANALYSIS AND OPTIMIZATION 2022年第4期43卷 412-429页

作者： Ranjbar, Sajad Higher Educ Ctr Eghlid Dept Math Eghlid Iran

In this paper, a solution to the inclusion problem for an infinite family of monotone operators in Hadamard spaces is approximated. Strong convergence and Delta-convergence to a common zero of an infinite family of monotone operators are established. To support these results, some applications in convex optimization and fixed point theory are also presented.

关键词： proximal point algorithm inclusion problem monotone operator Hadamard space convex optimization fixed point theory

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proximal alternating directions method for structured variational inequalities

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 2007年第1期134卷 107-117页

作者： Xu, M. H. Jiangsu Polytech Univ Dept Informat Sci Changzhou Jiangsu Peoples R China Nanjing Univ Dept Math Nanjing 210008 Peoples R China

In the alternating directions method, the relaxation factor gamma is an element of (0, root 5+1/2) by Glowinski is useful in practical computations for structured variational inequalities. This paper points out that t... 详细信息

关键词： structured variational inequalities proximal point algorithm alternating directions method

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A note on Solodov and Tseng's methods for maximal monotone mappings

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JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 2010年第5期234卷 1522-1527页

作者： Zhao, Jinling Yang, Qingzhi Gao, Hongxiu Univ Sci & Technol Beijing Dept Math & Mech Beijing 100083 Peoples R China Nankai Univ Sch Math Tianjin 300071 Peoples R China Nankai Univ LPMC Tianjin 300071 Peoples R China Qingdao Technol Univ Sch Sci Qingdao 266520 Peoples R China

This paper considers the problem of finding a zero of the sum of a single-valued Lipschitz continuous mapping A and a maximal monotone mapping B in a closed convex set C. We first give some projection-type methods and extend a modified projection method proposed by Solodov and Tseng for the special case of B = N(c) to this problem, then we give a refinement of Tseng's method that replaces P(c) by P(ck). Finally, convergence of these methods is established. (C) 2010 Elsevier B.V. All rights reserved.

关键词： Maximal monotone proximal point algorithm Forward backward splitting method Orthogonal projection Relaxation

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A proximal parallel splitting method for minimizing sum of convex functions with linear constraints

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JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 2014年 256卷 36-51页

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

In this paper, we propose a proximal parallel decomposition algorithm for solving the optimization problems where the objective function is the sum of m separable functions (i.e., they have no crossed variables), and the constraint set is the intersection of Cartesian products of some simple sets and a linear manifold. The m subproblems are solved simultaneously per iterations, which are sum of the decomposed subproblems of the augmented Lagrange function and a quadratic term. Hence our algorithm is named as the 'proximal parallel splitting method'. We prove the global convergence of the proposed algorithm under some mild conditions that the underlying functions are convex and the solution set is nonempty. To make the subproblems easier, some linearized versions of the proposed algorithm are also presented, together with their global convergence analysis. Finally, some preliminary numerical results are reported to support the efficiency of the new algorithms. (C) 2013 Elsevier B.V. All rights reserved.

关键词： Parallel splitting method proximal point algorithm Separable structure Linearized algorithm Global convergence

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An efficient algorithm for Fantope-constrained sparse principal subspace estimation problem

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APPLIED MATHEMATICS AND COMPUTATION 2024年 475卷

作者： Liu, Yong-Jin Wan, Yuqi Lin, Lanyu Fuzhou Univ Ctr Appl Math Fujian Prov Sch Math & Stat Fuzhou 350108 Fujian Peoples R China Fuzhou Univ Sch Math & Stat Fuzhou 350108 Fujian Peoples R China

The Fantope-constrained sparse principal subspace estimation problem is initially proposed Vu et al. (Vu et al., 2013). This paper investigates a semismooth Newton based proximal point (P PASSN ) algorithm for solving the equivalent form of this problem, where a semismooth Newton (S SN ) method is utilized to optimize the inner problems involved in the P PASSN algorithm. Under standard conditions, the P PASSN algorithm is proven to achieve global convergence and asymptotic superlinear convergence rate. Computationally, we derive nontrivial expressions the Fantope projection and its generalized Jacobian, which are key ingredients for the P PASSN algorithm. Some numerical results on synthetic and real data sets are presented to illustrate the effectiveness of the proposed P PASSN algorithm for large-scale problems and superiority over the alternating direction method of multipliers (ADMM).

关键词： Fantope projection Semismooth Newton algorithm proximal point algorithm Generalized Jacobian

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