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Bregman monotone optimization algorithms

SIAM JOURNAL ON CONTROL AND OPTIMIZATION 2003年第2期42卷 596-636页

作者： Bauschke, HH Borwein, JM Combettes, PL Univ Guelph Dept Math & Stat Guelph ON N1G 2W1 Canada Simon Fraser Univ Ctr Expt & Construct Math Burnaby BC V5A 1S6 Canada Univ Paris 06 Lab Jacques Louis Lions F-75005 Paris France

A broad class of optimization algorithms based on Bregman distances in Banach spaces is unified around the notion of Bregman monotonicity. A systematic investigation of this notion leads to a simplified analysis of numerous algorithms and to the development of a new class of parallel block-iterative surrogate Bregman projection schemes. Another key contribution is the introduction of a class of operators that is shown to be intrinsically tied to the notion of Bregman monotonicity and to include the operators commonly found in Bregman optimization methods. Special emphasis is placed on the viability of the algorithms and the importance of Legendre functions in this regard. Various applications are discussed.

关键词： Banach space block-iterative method Bregman distance Bregman monotone Bregman projection B-class operator convex feasibility problem essentially smooth function essentially strict convex function Fejer monotone Legendre function monotone operator proximal mapping proximal point algorithm resolvent subgradient projection

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Solving monotone inclusions with linear multi-step methods

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MATHEMATICAL PROGRAMMING 2003年第3期96卷 469-487页

作者： Pennanen, T Svaiter, BF Helsinki Sch Econ Dept Management Sci Helsinki 00101 Finland Inst Matematica Pura & Aplicada BR-22460320 Rio De Janeiro Brazil

In this paper anew class of proximal-like algorithms for solving monotone inclusions of the form T(x) There Exists 0 is derived. It is obtained by applying linear multi-step methods (LMM) of numerical integration in order to solve the differential inclusion (x) over dot (t) is an element of -T(x(t)), which can be viewed as a generalization of the steepest decent method for a convex function. It is proved that under suitable conditions on the parameters of the LMM, the generated sequence converges weakly to a point in the solution set T-1 (0). The LMM is very similar to the classical proximal point algorithm in that both are based on approximately evaluating the resolvants of T. Consequently, LMM can be used to derive multi-step versions of many of the optimization methods based on the classical proximal point algorithm. The convergence analysis allows errors in the computation of the iterates, and two different error criteria are analyzed, namely, the classical scheme with summable errors, and a recently proposed more constructive criterion.

关键词： proximal point algorithm monotone operator numerical integration strong stability relative error criterion

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Convergence of a splitting inertial proximal method for monotone operators

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JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 2003年第2期155卷 447-454页

作者： Moudafi, A Oliny, M Univ Antilles Guyane DSI GRIMAAG Schoelcher 97200 Martinique France

A forward-backward inertial procedure for solving the problem of finding a zero of the sum of two maximal monotone operators is proposed and its convergence is established under a cocoercivity condition with respect t... 详细信息

关键词： monotone operators enlargements proximal point algorithm cocoercivity splitting algorithm projection convergence

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Strong convergence of a proximal-type algorithm in a Banach space

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SIAM JOURNAL ON OPTIMIZATION 2003年第3期13卷 938-945页

作者： Kamimura, SJ Takahashi, W Hitotsubashi Univ Grad Sch Int Corp STrategy Chiyoda Ku Tokyo 1018439 Japan Tokyo Inst Technol Dept Math & Comp Sci Meguro Ku Tokyo 1528552 Japan

In this paper, we study strong convergence of the proximal point algorithm. It is known that the proximal point algorithm converges weakly to a solution of a maximal monotone operator, but it fails to converge strongly. Then, in [Math. Program., 87 (2000), pp. 189 202], Solodov and Svaiter introduced the new proximal-type algorithm to generate a strongly convergent sequence and established a convergence property for it in Hilbert spaces. Our purpose is to extend Solodov and Svaiter's result to more general Banach spaces. Using this, we consider the problem of finding a minimizer of a convex function.

关键词： proximal point algorithm maximal monotone operator Banach space strong convergence

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The proximal point algorithm with genuine superlinear convergence for the monotone complementarity problem

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SIAM JOURNAL ON OPTIMIZATION 2000年第2期11卷 364-379页

作者： Yamashita, N Fukushima, M Kyoto Univ Grad Sch Informat Dept Appl Math & Phys Kyoto 6068501 Japan

In this paper, we consider a proximal point algorithm ( PPA) for solving monotone nonlinear complementarity problems (NCP). PPA generates a sequence by solving subproblems that are regularizations of the original problem. It is known that PPA has global and superlinear convergence properties under appropriate criteria for approximate solutions of subproblems. However, it is not always easy to solve subproblems or to check those criteria. In this paper, we adopt the generalized Newton method proposed by De Luca, Facchinei, and Kanzow to solve subproblems and adopt some NCP functions to check the criteria. Then we show that the PPA converges globally provided that the solution set of the problem is nonempty. Moreover, without assuming the local uniqueness of the solution, we show that the rate of convergence is superlinear in a genuine sense, provided that the limit point satis es the strict complementarity condition.

关键词： nonlinear complementarity problem proximal point algorithm genuine superlinear convergence

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Convergence of prox-regularization methods for generalized fractional programming

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RAIRO-OPERATIONS RESEARCH 2002年第1期36卷 73-94页

作者： Roubi, A Fac Sci & Tech Dept Math & Informat Settat Morocco

We analyze the convergence of the prox-regularization algorithms introduced in [1], to solve generalized fractional programs, without assuming that the optimal solutions set of the considered problem is nonempty, and since the objective functions are variable with respect to the iterations in the auxiliary problems generated by Dinkelbach-type algorithms DT1 and DT2, we consider that the regularizing parameter is also variable. On the other hand we study the convergence when the iterates are only eta(k)-minimizers of the auxiliary problems. This situation is more general than the one considered in [1]. We also give some results concerning the rate of convergence of these algorithms, and show that it is linear and some times superlinear for some classes of functions. Illustrations by numerical examples are given in [1].

关键词： generalized fractional programs Dinkelbach-type algorithms proximal point algorithm rate of convergence

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A unified framework for some inexact proximal point algorithms

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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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Decision Support for Location Problems in Town Planning

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International Transactions in Operational Research 2002年第3期9卷 261-278页

作者： Patz, Renate Spitzner, Jan Tammer, Christiane University of Applied Science Department of Technology Transfer Merseburg D-06217 Germany Department of Mathematics Martin-Luther University Germany

Urban development and town planning need an adequate decision-making process. European cities, in particular, are compact. Urban elements and functions are in a constant state of change. Moreover, the large number of historic buildings and areas means a sensitive and responsible approach must be taken. The aim of this paper is to consider special location problems in town planning. We formulate multi-criteria location problems, derive optimality conditions and present a geometric algorithm and an interactive procedure including a proximal point algorithm for solving multi-criteria location problems. In this paper, we use location theory as a possible method to help determine the location of a children’s playground in a newly-built district of Halle, Germany. International Federation of Operational Research Societies 2002.

关键词： Duality assertions Geometrical algorithm Location problem Multi-criteria decision-making proximal point algorithm Town planning

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A weak-to-strong convergence principle for Fejer-monotone methods in Hilbert spaces

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MATHEMATICS OF OPERATIONS RESEARCH 2001年第2期26卷 248-264页

作者： Bauschke, HH Combettes, PL Okanagan Univ Coll Dept Math & Stat Kelowna BC V1V 1V7 Canada Univ Paris 06 Anal Numer Lab F-75005 Paris France

We consider a wide class of iterative methods arising in numerical mathematics and optimization that are known to converge only weakly. Exploiting an idea originally proposed by Haugazeau, we present a simple modification of these methods that makes them strongly convergent without additional assumptions. Several applications are discussed.

关键词： convex feasibility Fejer-monotonicity firmly nonexpansive mapping fixed point Haugazeau maximal monotone operator projection proximal point algorithm resolvent subgradient algorithm

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Weak and strong convergence of approximating fixed points and applications

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NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS 2001年第7期47卷 4981-4993页

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

A discussion on weak and strong approximating fixed points was presented. The nonlinear ergodic theorems of Baillon's type were analyzed. Weak and strong convergence theorems of Mann's type and Halpern's type in Banach spaces were studied. Iterative methods for approximation of common fixed points for families of nonexpansive mappings were established. The feasibility problem by convex combinations of nonexpansive retractions and convex minimization problem of finding a minimizer of a convex function were considered using the results.

关键词： fixed point nonexpansive mapping nonlinear ergodic theorem fasibility problem proximal point algorithm

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