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A proximal point method for the variational inequality problem in Banach spaces

SIAM JOURNAL ON CONTROL AND OPTIMIZATION 2000年第5期39卷 1633-1649页

作者： Burachik, RS Scheimberg, S Univ Fed Rio de Janeiro COPPE IM BR-21945970 Rio De Janeiro Brazil

In this paper we prove well-definedness and weak convergence of the generalized proximal point method when applied to the variational inequality problem in reflexive Banach spaces. The proximal version we consider makes use of Bregman functions, whose original definition for finite dimensional spaces has here been properly extended to our more general framework.

关键词： maximal monotone operators proximal point algorithm Banach spaces convergence algorithmic scheme

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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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A logarithmic-quadratic proximal point scalarization method for multiobjective programming

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JOURNAL OF GLOBAL OPTIMIZATION 2011年第2期49卷 281-291页

作者： Gregorio, Ronaldo Oliveira, Paulo Roberto Univ Fed Rural Rio de Janeiro Technol & Languages Dept BR-26221010 Rio De Janeiro Brazil Univ Fed Rio de Janeiro Comp & Syst Engn Dept BR-21945970 Rio De Janeiro Brazil

We present a proximal point method to solve multiobjective programming problems based on the scalarization for maps. We build a family of convex scalar strict representations of a convex map F from R(n) to R(m) with respect to the lexicographic order on R(m) and we add a variant of the logarithmic-quadratic regularization of Auslender, where the unconstrained variables in the domain of F are introduced in the quadratic term. The nonegative variables employed in the scalarization are placed in the logarithmic term. We show that the central trajectory of the scalarized problem is bounded and converges to a weak pareto solution of the multiobjective optimization problem.

关键词： proximal point algorithm Scalar representations Multiobjective programming

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A Lyusternik-Graves theorem for the proximal point method

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COMPUTATIONAL OPTIMIZATION AND APPLICATIONS 2012年第3期52卷 785-803页

作者： Aragon Artacho, Francisco J. Gaydu, Michael Univ Alicante Dept Stat & Operat Res E-03080 Alicante Spain Univ Antilles Guyane LAMIA Dpt Math F-97159 Pointe A Pitre Guadeloupe France

We consider a generalized version of the proximal point algorithm for solving the perturbed inclusion yaT(x), where y is a perturbation element near 0 and T is a set-valued mapping acting from a Banach space X to a Banach space Y which is metrically regular around some point in its graph. We study the behavior of the convergent iterates generated by the algorithm and we prove that they inherit the regularity properties of T, and vice versa. We analyze the cases when the mapping T is metrically regular and strongly regular.

关键词： proximal point algorithm Generalized equations Perturbations Metric regularity Strong regularity

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A proximal point method for the sum of maximal monotone operators

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MATHEMATICAL METHODS IN THE APPLIED SCIENCES 2014年第17期37卷 2638-2650页

作者： Xiao, Hongying Zeng, Xueying China Three Gorges Univ Coll Sci Yichang 443002 Peoples R China Ocean Univ China Sch Math Sci Qingdao 266100 Peoples R China

In this paper, we concentrate on the maximal inclusion problem of locating the zeros of the sum of maximal monotone operators in the framework of proximal point method. Such problems arise widely in several applied mathematical fields such as signal and image processing. We define two new maximal monotone operators and characterize the solutions of the considered problem via the zeros of the new operators. The maximal monotonicity and resolvent of both of the defined operators are proved and calculated, respectively. The traditional proximal point algorithm can be therefore applied to the considered maximal inclusion problem, and the convergence is ensured. Furthermore, by exploring the relationship between the proposed method and the generalized forward-backward splitting algorithm, we point out that this algorithm is essentially the proximal point algorithm when the operator corresponding to the forward step is the zero operator. Copyright (c) 2013 John Wiley & Sons, Ltd.

关键词： proximal point algorithm maximal monotone operator generalized forward-backward splitting

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Projection and proximal point methods:: convergence results and counterexamples

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NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS 2004年第5期56卷 715-738页

作者： Bauschke, HH Matousková, E Reich, S Technion Israel Inst Technol Dept Math IL-32000 Haifa Israel Univ Guelph Dept Math & Stat Guelph ON N1G 2W1 Canada Acad Sci Czech Republ Inst Math CZ-11567 Prague Czech Republic

Recently, Hundal has constructed a hyperplane H, a cone K, and a starting point y(0) in l(2) such that the sequence of alternating projections ((PKPH)(n)y(0))nepsilonN converges weakly to some point in H boolean AND K, but not in norm. We show how this construction results in a counterexample to norm convergence for iterates of averaged projections;hence, we give an affirmative answer to a question raised by Reich two decades ago. Furthermore, new counterexamples to norm convergence for iterates of firmly nonexpansive maps (a la Genel and Lindenstrauss) and for the proximal point algorithm (a la Guler) are provided. We also present a counterexample, along with some weak and norm convergence results, for the new framework of string-averaging projection methods introduced by Censor et at. Extensions to Banach spaces and the situation for the Hilbert ball are discussed as well. (C) 2003 Elsevier Ltd. All rights reserved.

关键词： alternating projections averaged projections Hilbert space nonexpansive proximal point algorithm weak convergence

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A proximal point Analysis of the Preconditioned Alternating Direction Method of Multipliers

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 2017年第3期173卷 878-907页

作者： Bredies, Kristian Sun, Hongpeng Karl Franzens Univ Graz Inst Math & Sci Comp Heinrichstr 36 A-8010 Graz Austria Renmin Univ China Inst Math Sci 59 Zhongguancun St Beijing Peoples R China

We study preconditioned algorithms of alternating direction method of multipliers type for nonsmooth optimization problems. The alternating direction method of multipliers is a popular first-order method for general constrained optimization problems. However, one of its drawbacks is the need to solve implicit subproblems. In various applications, these subproblems are either easily solvable or linear, but nevertheless challenging. We derive a preconditioned version that allows for flexible and efficient preconditioning for these linear subproblems. The original and preconditioned version is written as a new kind of proximal point method for the primal problem, and the weak (strong) convergence in infinite (finite) dimensional Hilbert spaces is proved. Various efficient preconditioners with any number of inner iterations may be used in this preconditioned framework. Furthermore, connections between the preconditioned version and the recently introduced preconditioned Douglas-Rachford method for general nonsmooth problems involving quadratic-linear terms are established. The methods are applied to total variation denoising problems, and their benefits are shown in numerical experiments.

关键词： Alternating direction method of multipliers proximal point algorithm Weak convergence analysis Linear preconditioning techniques

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A proximal point Based Approach to Optimal Control of Affine Switched Systems

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DISCRETE EVENT DYNAMIC SYSTEMS-THEORY AND APPLICATIONS 2012年第1期22卷 61-81页

作者： Azhmyakov, Vadim Basin, Michael V. Raisch, Joerg CINVESTAV Dept Automat Control Mexico City 07360 DF Mexico Univ Nuevo Leon Dept Phys & Math Sci San Nicolas De Los Garza Mexico Tech Univ Berlin Fachgebiet Regelungssyst D-10587 Berlin Germany

This paper focuses on the proximal point regularization technique for a class of optimal control processes governed by affine switched systems. We consider switched control systems described by nonlinear ordinary differential equations which are affine in the input. The affine structure of the dynamical models under consideration makes it possible to establish some continuity/approximability properties and to specify these models as convex control systems. We show that, for some classes of cost functionals, the associated optimal control problem (OCP) corresponds to a conventional convex optimization problem in a suitable Hilbert space. The latter can be reliably solved using standard first-order optimization algorithms and consistent regularization schemes. In particular, we propose a conceptual numerical approach based on the gradient-type method and classic proximal point techniques.

关键词： Switched control systems Affine systems Optimal control proximal point algorithm

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On a generalized proximal point method for solving equilibrium problems in Banach spaces

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NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS 2012年第18期75卷 6456-6464页

作者： Burachik, Regina Kassay, Gabor Univ S Australia Sch Math & Stat Adelaide SA 5095 Australia Univ Babes Bolyai Fac Math & Comp Sci Cluj Napoca 400084 Romania

We introduce a regularized equilibrium problem in Banach spaces, involving generalized Received 14 April 2012 Bregman functions. For this regularized problem, we establish the existence and Accepted 20 July 2012 uniqueness of solutions. These regularizations yield a proximal-like method for solving equilibrium problems in Banach spaces. We prove that the proximal sequence is an asymptotically solving sequence when the dual space is uniformly convex. Moreover, we MSC: prove that all weak accumulation points are solutions if the equilibrium function is lower semicontinuous in its first variable. We prove, under additional assumptions, that the proximal sequence converges weakly to a solution. (C) 2012 Elsevier Ltd. All rights reserved.

关键词： Bregman function Equilibrium problem Normalized duality mapping proximal point algorithm Regularization method Totally convex function

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proximal point METHODS FOR MONOTONE OPERATORS IN BANACH SPACES

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TAIWANESE JOURNAL OF MATHEMATICS 2011年第1期15卷 259-281页

作者： Aoyama, Koji Kohsaka, Fumiaki Takahashi, Wataru Chiba Univ Dept Econ Chiba 2638522 Japan Oita Univ Dept Comp Sci & Intelligent Syst Oita 8701192 Japan Tokyo Inst Technol Dept Math & Comp Sci Meguro Ku Tokyo 1528552 Japan

Some fundamental properties of resolvents of monotone operators in Banach spaces are investigated. Using them, we study the asymptotic behavior of the sequences generated by two modifications of the proximal point alg... 详细信息

关键词： Convex function Firmly nonexpansive-type mapping Fixed point Minimization Monotone operator proximal point algorithm Uniformly convex Banach space Zero point

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