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 nu...
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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.
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 o...
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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.
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...
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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 to the solution set. (C) 2003 Elsevier Science B.V. All rights reserved.
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 strongl...
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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.
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 prob...
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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.
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 ...
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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].
We present a unified framework for the design and convergence analysis of a class of algorithms based on approximate solution of proximalpoint subproblems. Our development further enhances the constructive approximat...
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We present a unified framework for the design and convergence analysis of a class of algorithms based on approximate solution of proximalpoint 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.
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 ...
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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.
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 modifica...
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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.
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 t...
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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.
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