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Solving Lagrangian variational inequalities with applications to stochastic programming

MATHEMATICAL PROGRAMMING 2020年第2期181卷 435-451页

作者： Rockafellar, R. Tyrrell Sun, Jie Univ Washington Dept Math Box 354350 Seattle WA 98195 USA Curtin Univ Dept Math & Stat Bentley WA 6102 Australia

Lagrangian variational inequalities feature both primal and dual elements in expressing first-order conditions for optimality in a wide variety of settings where "multipliers" in a very general sense need to be brought in. Their stochastic version relates to problems of stochastic programming and covers not only classical formats with inequality constraints but also composite models with nonsmooth objectives. The progressive hedging algorithm, as a means of solving stochastic programming problems, has however focused so far only on optimality conditions that correspond to variational inequalities in primal variables alone. Here that limitation is removed by appealing to a recent extension of progressive hedging to multistage stochastic variational inequalities in general.

关键词： Stochastic variational inequality problems Stochastic programming problems Lagrangian variational inequalities Lagrange multipliers Progressive hedging algorithm proximal point algorithm Composite optimization

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EXISTENCE AND APPROXIMATION OF FIXED pointS OF FIRMLY NONEXPANSIVE-TYPE MAPPINGS IN BANACH SPACES

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SIAM JOURNAL ON OPTIMIZATION 2008年第2期19卷 824-835页

作者： Kohsaka, Fumiaki Takahashi, Wataru Tokyo Denki Univ Dept Informat Environm Chiba 2701382 Japan Tokyo Inst Technol Dept Math & Comp Sci Meguro Ku Tokyo 1528552 Japan

A class of nonlinear operators in Banach spaces is proposed. We call each operator in this class a firmly nonexpansive-type mapping. This class contains the classes of firmly nonexpansive mappings in Hilbert spaces and resolvents of maximal monotone operators in Banach spaces. We study the existence and approximation of fixed points of firmly nonexpansive-type mappings in Banach spaces.

关键词： Banach space convex optimization firmly nonexpansive mapping firmly nonexpansive-type mapping fixed point theorem monotone operator nonexpansive mapping proximal point algorithm

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On solving the minimization problem and the fixed-point problem for nonexpansive mappings in CAT(0) spaces

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OPTIMIZATION METHODS & SOFTWARE 2017年第1期32卷 182-192页

作者： Suparatulatorn, Raweerote Cholamjiak, Prasit Suantai, Suthep Univ Phayao Sch Sci Phayao 56000 Thailand Chiang Mai Univ Fac Sci Dept Math Chiang Mai 50200 Thailand

We propose a new modified proximal point algorithm combined with Halpern's iteration process for nonexpansive mappings in the framework of CAT(0) spaces. We establish a strong convergence theorem under some mild c... 详细信息

关键词： convex minimization problem fixed-point problem CAT(0) space proximal point algorithm nonexpansive mapping

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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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A Homotopy Alternating Direction Method of Multipliers for Linearly Constrained Separable Convex Optimization

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Journal of the Operations Research Society of China 2017年第2期5卷 271-290页

作者： Jiao Yang Yi-Qing Dai Zheng Peng Jie-Peng Zhuang Wen-Xing Zhu Hunan Vocational Institute of Safety Technology Changsha 410018China College of Mathematics and Computer Science Fuzhou UniversityFuzhou 350108China

Linearly constrained separable convex minimization problems have been raised widely in many real-world *** this paper,we propose a homotopy-based alternating direction method of multipliers for solving this kind of *** proposed method owns some advantages of the classical proximal alternating direction method of multipliers and homotopy *** some suitable condi-tions,we prove global convergence and the worst-case O(k/1)convergence rate in a nonergodic *** numerical results indicate effectiveness and efficiency of the proposed method compared with some state-of-the-art methods.

关键词： Separable convex optimization Alternating direction method of multipliers proximal point algorithm Homotopy method

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A partial PPA block-wise ADMM for multi-block linearly constrained separable convex optimization

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OPTIMIZATION 2021年第3期70卷 631-657页

作者： Shen, Yuan Zhang, Xingying Zhang, Xiayang Nanjing Univ Finance & Econ Sch Appl Math Nanjing Peoples R China Nanjing Inst Technol Dept Math & Phys Nanjing Peoples R China

The alternating direction method of multipliers (ADMM) is a classical effective method for solving two-block convex optimization subject to linear constraints. However, its convergence may not be guaranteed for multiple-block case without additional assumptions. One remedy might be the block-wise ADMM (BADMM), in which the variables are regrouped into two groups firstly and then the augmented Lagrangian function is minimized w.r.t. each block variable by the following scheme: using a Gauss-Seidel fashion to update the variables between each group, while using a Jacobi fashion to update the variables within each group. In order to derive its convergence property, a special proximal term is added to each subproblem. In this paper, we propose a new partial PPA block-wise ADMM where we only need to add proximal terms to the subproblems in the first group. At the end of each iteration, an extension step on all variables is performed with a fixed step size. As the subproblems in the second group are unmodified, the resulting sequence might yield better quality as well as potentially faster convergence speed. Preliminary experimental results show that the new algorithm is empirically effective on solving both synthetic and real problems when it is compared with several very efficient ADMM-based algorithms.

关键词： Convex optimization augmented Lagrangian alternating direction method of multipliers multi-block proximal point algorithm

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THE PROX-TIKHONOV-LIKE FORWARD-BACKWARD METHOD AND APPLICATIONS

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TAIWANESE JOURNAL OF MATHEMATICS 2015年第2期19卷 481-503页

作者： Sahu, D. R. Ansari, Q. H. Yao, J. C. Banaras Hindu Univ Dept Math Varanasi 221005 Uttar Pradesh India Natl Sun Yat Sen Univ Dept Appl Math Kaohsiung 804 Taiwan Aligarh Muslim Univ Dept Math Aligarh 202002 Uttar Pradesh India King Fahd Univ Petr & Minerals Dept Math & Stat Dhahran 31261 Saudi Arabia Kaohsiung Med Univ Ctr Fundamental Sci Kaohsiung 807 Taiwan King Abdulaziz Univ Dept Math Jeddah 21589 Saudi Arabia

It is known, by Rockafellar [SIAM J. Control Optim., 14 (1976), 877898], that the proximal point algorithm (PPA) converges weakly to a zero of a maximal monotone operator in a Hilbert space, but it fails to converge strongly. Lehdili and Moudafi [Optimization, 37(1996), 239-252] introduced the new prox-Tikhonov regularization method for PPA to generate a strongly convergent sequence and established a convergence property for it by using the technique of variational distance in the same space setting. In this paper, the prox-Tikhonov regularization method for the proximal point algorithm of finding a zero for an accretive operator in the framework of Banach space is proposed. Conditions which guarantee the strong convergence of this algorithm to a particular element of the solution set is provided. An inexact variant of this method with non-summable error sequence is also discussed.

关键词： Accretive operator Maximal monotone operator Metric projection mapping proximal point algorithm Regularization method Resolvent identity Strong convergence Uniformly Gateaux differentiable norm

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Approximating solutions of maximal monotone operators in Hilbert spaces

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JOURNAL OF APPROXIMATION THEORY 2000年第2期106卷 226-240页

作者： Kamimura, S Takahashi, W Tokyo Inst Technol Dept Math & Comp Sci Meguro Ku Tokyo 1528552 Japan

Let H he a real Hilbert space and let T:H --> 2(H) he a maximal monotone operator. In this paper, we first introduce two algorithms of approximating solutions of maximal monotone operators. One of them is to generate a strongly convergent sequence with limit v epsilon T(-1)0. The other is to discuss the weak convergence of the proximal point algorithm. Next, using these results, we consider the problem of finding a minimizer of a convex function. Our methods are motivated by Halpern's iteration and Mann's iteration. (C) 2000 Academic Press.

关键词： maximal monotone operator resolvent proximal point algorithm iteration strong convergence weak convergence

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Convergence of Dissipative-Like Dynamics and algorithms Governed by Set-Valued Nonexpansive Mappings

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BULLETIN OF THE MALAYSIAN MATHEMATICAL SCIENCES SOCIETY 2021年第2期44卷 1101-1121页

作者： Khatibzadeh, Hadi Piranfar, Mohsen Rahimi Rooin, Jamal Univ Zanjan Dept Math POB 45195-313 Zanjan Iran Inst Adv Studies Basic Sci Dept Math POB 45195-1159 Zanjan Iran

In this paper, we consider a differential inclusion governed by a set-valued nonexpansive mapping and study the asymptotic behavior (weak and strong convergence) of its solutions with various assumptions on this mapping. Then for a set-valued nonexpansive mapping, we define the corresponding resolvent (proximal) operator as a set-valued mapping and study some of its elementary properties. Subsequently, we apply the resolvent operator to state the implicit discretization of the differential inclusion and study the asymptotic behavior of its solutions which yields similar convergence results as in the continuous case. This provides an algorithm for approximating a fixed point of a set-valued nonexpansive mapping which extends the classical proximal point algorithm. An application to variational inequalities and a numerical comparison with another iterative method for approximating a fixed point of set-valued nonexpansive mappings are also presented.

关键词： Set-valued nonexpansive mapping Differential inclusion Asymptotic behavior Resolvent proximal point algorithm

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Equilibrium problems on Riemannian manifolds with applications

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JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 2019年第2期473卷 866-891页

作者： Wang, Xiangmei Lopez, Genaro Li, Chong Yao, Jen-Chih Guizhou Univ Coll Math & Stat Guiyang 550025 Guizhou Peoples R China Univ Seville Dept Math Anal E-41080 Seville Spain Zhejiang Univ Sch Math Sci Hangzhou 310027 Zhejiang Peoples R China China Med Univ China Med Univ Hosp Res Ctr Interneural Comp Taichung 40402 Taiwan

We study equilibrium problem on general Riemannian manifolds, focusing on existence of solutions and the convexity properties of the solution set. Our approach consists in relating the equilibrium problem to a suitable variational inequality problem on Riemannian manifolds, and is completely different from previous ones on this topic in the literature. We apply our results to the mixed variational inequality and the Nash equilibrium problem. Moreover, we formulate and analyze the convergence of the proximal point algorithm for the equilibrium problem. In particular, correct proofs are provided for the results claimed in J. Math. Anal. Appl. 388 (2012) 61-77 (i.e., Theorems 3.5 and 4.9 there) regarding the existence of the mixed variational inequality and the domain of the resolvent for the equilibrium problem on Hadamard manifolds. (C) 2019 Elsevier Inc. All rights reserved.

关键词： Riemannian manifold Equilibrium problem Variational inequality problem proximal point algorithm

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