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Finite Convergence of the proximal point algorithm for Variational Inequality Problems

SET-VALUED AND VARIATIONAL ANALYSIS 2013年第2期21卷 297-309页

作者： Matsushita, Shin-ya Xu, Li Akita Prefectural Univ Dept Elect & Informat Syst Akita Japan

In this paper, we establish sufficient conditions for guaranteeing finite termination of an arbitrary algorithm for solving a variational inequality problem in a Banach space. Applying these conditions, it shows that ... 详细信息

关键词： Variational inequality problem Weak sharp minima Finite termination proximal point algorithm Banach space Metric projection Generalized projection

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Two modified proximal point algorithms in geodesic spaces with curvature bounded above

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RENDICONTI DEL CIRCOLO MATEMATICO DI PALERMO 2019年第1期68卷 83-104页

作者： Kimura, Yasunori Kohsaka, Fumiaki Toho Univ Dept Informat Sci Funabashi Chiba 2748510 Japan Tokai Univ Dept Math Sci Hiratsuka Kanagawa 2591292 Japan

We obtain existence and convergence theorems for two variants of the proximal point algorithm involving proper lower semicontinuous convex functions in complete geodesic spaces with curvature bounded above.

关键词： CAT(1) space Convex function Fixed point Geodesic space Minimizer proximal point algorithm Resolvent

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A multi-parameter parallel ADMM for multi-block linearly constrained separable convex optimization

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APPLIED NUMERICAL MATHEMATICS 2022年 171卷 369-388页

作者： Shen, Yuan Gao, Qianming Yin, Xue Nanjing Univ Finance & Econ Sch Appl Math Nanjing 210023 Peoples R China

The alternating direction method of multipliers (ADMM) has been proved to be effective for solving two-block convex minimization model subject to linear constraints. However, the convergence of multiple-block convex minimization model with linear constraints may not be guaranteed without additional assumptions. Recently, some parallel multi-block ADMM algorithms which solve the subproblems in a parallel way have been proposed. This paper is a further study on this method with the purpose of improving the parallel multi-block ADMM algorithm by introducing more parameters. We propose two multi-parameter parallel ADMM algorithms with proximal point terms attached to all subproblems. Comparing with some popular parallel ADMM-based algorithms, the parameter conditions of the new algorithms are relaxed. Experiments on both real and synthetic problems are conducted to justify the effectiveness of the proposed algorithms compared to several efficient ADMM-based algorithms for multi-block problems. (C) 2021 IMACS. Published by Elsevier B.V. All rights reserved.

关键词： Multi-block Convex optimization Parallel computing proximal point algorithm Alternating direction method of multipliers

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Conical averagedness and convergence analysis of fixed point algorithms

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JOURNAL OF GLOBAL OPTIMIZATION 2022年第2期82卷 351-373页

作者： Bartz, Sedi Dao, Minh N. Phan, Hung M. Univ Massachusetts Dept Math Sci Kennedy Coll Sci Lowell MA 01854 USA Federat Univ Australia Sch Engn Informat Technol & Phys Sci Ballarat Vic 3353 Australia

We study a conical extension of averaged nonexpansive operators and the role it plays in convergence analysis of fixed point algorithms. Various properties of conically averaged operators are systematically investigated, in particular, the stability under relaxations, convex combinations and compositions. We derive conical averagedness properties of resolvents of generalized monotone operators. These properties are then utilized in order to analyze the convergence of the proximal point algorithm, the forward-backward algorithm, and the adaptive Douglas-Rachford algorithm. Our study unifies, improves and casts new light on recent studies of these topics.

关键词： Adaptive Douglas-Rachford algorithm Cocoercivity Conically averaged operator Forward-backward algorithm proximal point algorithm Strong monotonicity Weak monotonicity

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Revisiting jointly firmly nonexpansive families of mappings

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OPTIMIZATION 2022年第13期71卷 3819-3834页

作者： Sipos, Andrei Univ Bucharest Dept Comp Sci Fac Math & Comp Sci Res Ctr Log Optimizat & Secur LOS Bucharest Romania Romanian Acad Simion Stoilow Inst Math Bucharest Romania

Recently, the author, together with L. Leustean and A. Nicolae, introduced the notion of jointly firmly nonexpansive families of mappings in order to investigate in an abstract manner the convergence of proximal methods. Here, we further the study of this concept, by giving a characterization in terms of the classical resolvent identity, by improving on the rate of convergence previously obtained for the uniform case, and by giving a treatment of the asymptotic behaviour at infinity of such families.

关键词： Convergence of resolvents proximal point algorithm CAT(0) spaces jointly firmly nonexpansive families proof mining

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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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On Regularized Square-root Regression Problems: Distributionally Robust Interpretation and Fast Computations

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JOURNAL OF MACHINE LEARNING RESEARCH 2022年第1期23卷 1-39页

作者： Chu, Hong T. M. Toh, Kim-Chuan Zhang, Yangjing Natl Univ Singapore Dept Math Singapore 119076 Singapore Natl Univ Singapore Inst Operat Res & Analyt Singapore 119076 Singapore Chinese Acad Sci Acad Math & Syst Sci Inst Appl Math Beijing 100190 Peoples R China

Square-root (loss) regularized models have recently become popular in linear regression due to their nice statistical properties. Moreover, some of these models can be interpreted as the distributionally robust optimization counterparts of the traditional least-squares regularized models. In this paper, we give a unified proof to show that any square-root regularized model whose penalty function being the sum of a simple norm and a seminorm can be interpreted as the distributionally robust optimization (DRO) formulation of the corresponding least-squares problem. In particular, the optimal transport cost in the DRO formulation is given by a certain dual form of the penalty. To solve the resulting square -root regularized model whose loss function and penalty function are both nonsmooth, we design a proximal point dual semismooth Newton algorithm and demonstrate its efficiency when the penalty is the sparse group Lasso penalty or the fused Lasso penalty. Extensive experiments demonstrate that our algorithm is highly efficient for solving the square-root sparse group Lasso problems and the square-root fused Lasso problems.

关键词： square-root regularized model distributionally robust optimization proximal point algorithm semismooth Newton group Lasso

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The numerical reckoning of modified proximal point methods for minimization problems in non-positive curvature metric spaces

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INTERNATIONAL JOURNAL OF COMPUTER MATHEMATICS 2020年第1-2期97卷 245-262页

作者： Thounthong, Phatiphat Pakkaranang, Nuttapol Cho, Yeol Je Kumam, Wiyada Kumam, Poom KMUTNB Fac Tech Educ Renewable Energy Res Ctr Bangkok Thailand KMUTNB Fac Tech Educ Dept Teacher Training Elect Engn Bangkok Thailand KMUTT Fixed Point Res Lab Dept Math Fac Sci Room SCL 802Sci Lab Bldg126 Pracha Uthit Rd Bangkok 10140 Thailand KMUTT Theoret & Computat Sci Ctr TaCS Fac Sci Sci Lab Bldg126 Pracha Uthit Rd Bangkok 10140 Thailand Gyeongsang Natl Univ Dept Math Educ Jinju South Korea Univ Elect Sci & Technol China Sch Math Sci Chengdu Sichuan Peoples R China Rajamangala Univ Technol Thanyaburi RMUTT Fac Sci & Technol Dept Math & Comp Sci Program Appl Stat Pathum Thani Thailand China Med Univ China Med Univ Hosp Dept Med Res Taichung Taiwan

In this paper, we introduce a new modified proximal point algorithm for nonexpansive mappings in non-positive curvature metric spaces and also we prove the sequence generated by the proposed algorithms converges to a common solution between minimization problem and fixed point problem. Moreover, we give some numerical examples to illustrate our main results, that is, our algorithm is more efficient than the algorithm of Cholamjiak et al. and others.

关键词： CAT(0) spaces proximal point algorithm iterative method nonexapnsive mappings convex minimization problem

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Modified proximal symmetric ADMMs for multi-block separable convex optimization with linear constraints

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ANALYSIS AND APPLICATIONS 2022年第3期20卷 401-428页

作者： Shen, Yuan Zuo, Yannian Sun, Liming Zhang, Xiayang Nanjing Univ Finance & Econ Sch Appl Math Nanjing 210023 Peoples R China Nanjing Audit Univ Sch Stat & Data Sci Nanjing 211815 Peoples R China Nanjing Inst Technol Dept Math & Phys Nanjing 211167 Peoples R China

We consider the linearly constrained separable convex optimization problem whose objective function is separable with respect to m blocks of variables. A bunch of methods have been proposed and extensively studied in the past decade. Specifically, a modified strictly contractive Peaceman-Rachford splitting method (SC-PRCM) [S. H. Jiang and M. Li, A modified strictly contractive Peaceman-Rachford splitting method for multi-block separable convex programming, J. Ind. Manag. Optim. 14(1) (2018) 397-412] has been well studied in the literature for the special case of m = 3. Based on the modified SC-PRCM, we present modified proximal symmetric ADMMs (MPSADMMs) to solve the multi-block problem. In MPSADMMs, all subproblems but the first one are attached with a simple proximal term, and the multipliers are updated twice. At the end of each iteration, the output is corrected via a simple correction step. Without stringent assumptions, we establish the global convergence result and the O(1/t) convergence rate in the ergodic sense for the new algorithms. Preliminary numerical results show that our proposed algorithms are effective for solving the linearly constrained quadratic programming and the robust principal component analysis problems.

关键词： Separable convex optimization multiple blocks symmetric alternating direction method of multipliers proximal point algorithm

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New iterative algorithms for solving a class of split common solution problems and their applications

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JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 2024年 440卷

作者： Reich, Simeon Tuyen, Truong Minh Trang, Nguyen Thi Technion Israel Inst Technol Dept Math IL-32000 Haifa Israel Thai Nguyen Univ Sci Dept Math & Informat Thai Nguyen Vietnam Hanoi Univ Nat Resources & Environm Basic Sci Fac Hanoi Vietnam

The purpose of this paper is to introduce new iterative algorithms for approximating a solution to a class of monotone operator equations. More precisely, we study the split common solution problem with multiple output sets for monotone operator equations in Hilbert spaces. In order to solve this problem, we propose three new algorithms and establish strong convergence theorems for them.

关键词： Iterative algorithm algorithm Hilbert space Metric projection proximal point algorithm

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