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A new inertial-type hybrid projection-proximal algorithm for monotone inclusions

APPLIED MATHEMATICS AND COMPUTATION 2010年第9期215卷 3149-3162页

作者： Mainge, Paul-Emile Merabet, Nora Univ Antilles Guyane DSI Lab CEREGMIA Campus Schoelcher Guyane 97233 Martinique France United Arab Emirates Univ Coll Sci Dept Math Sci Al Ain U Arab Emirates

This paper investigates an enhanced proximal algorithm with interesting practical features and convergence properties for solving non-smooth convex minimization problems, or approximating zeroes of maximal monotone operators, in Hilbert spaces. The considered algorithm involves a recent inertial-type extrapolation technique, the use of enlargement of operators and also a recently proposed hybrid strategy, which combines inexact computation of the proximal iteration with a projection. Compared to other existing related methods, the resulting algorithm inherits the good convergence properties of the inertial-type extrapolation and the relaxed projection strategy. It also inherits the relative error tolerance of the hybrid proximal-projection method. As a special result, an update of inexact Newton-proximal method is derived and global convergence results are established. (C) 2009 Elsevier Inc. All rights reserved.

关键词： Convex minimization Maximal monotone operator proximal point algorithm Inexact computations Inertial extrapolation Inexact Newton method

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A rapid algorithm for a class of linear complementarity problems

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APPLIED MATHEMATICS AND COMPUTATION 2007年第2期188卷 1647-1655页

作者： Xu, M. H. Luan, G. F. Jiangsu Polytech Univ Dept Informat Sci Changzhou 213164 Peoples R China Jiangsu Polytech Univ Dept Environm Sci Changzhou 213164 Peoples R China

In this paper, we propose a new rapid projection method for solving a class of linear complementarity problems based on matrix split technique and the idea of proximal point algorithm. The global convergence of the method is analyzed. Numerical experiments show that the new method compared with some existing methods has more efficiency and robustness in solving kinds of linear complementarity problems and can be applied very easily. Numerical experiments also show that the new method for those problems is almost not sensitive to the parameters used in this method. (c) 2006 Published by Elsevier Inc.

关键词： linear complementarity proximal point algorithm projection method variational inequalities

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Convergence Analysis of Primal-Dual algorithms for a Saddle-point Problem: From Contraction Perspective

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SIAM JOURNAL ON IMAGING SCIENCES 2012年第1期5卷 119-149页

作者： He, Bingsheng Yuan, Xiaoming Hong Kong Baptist Univ Dept Math Hong Kong Hong Kong Peoples R China Nanjing Univ Dept Math Nanjing 210093 Jiangsu Peoples R China Nanjing Univ Natl Key Lab Novel Software Technol Nanjing 210093 Jiangsu Peoples R China

Recently, some primal-dual algorithms have been proposed for solving a saddle-point problem, with particular applications in the area of total variation image restoration. This paper focuses on the convergence analysis of these primal-dual algorithms and shows that their involved parameters (including step sizes) can be significantly enlarged if some simple correction steps are supplemented. Some new primal-dual-based methods are thus proposed for solving the saddle-point problem. We show that these new methods are of the contraction type: the iterative sequences generated by these new methods are contractive with respect to the solution set of the saddle-point problem. The global convergence of these new methods thus can be obtained within the analytic framework of contraction-type methods. The novel study on these primal-dual algorithms from the perspective of contraction methods substantially simplifies existing convergence analysis. Finally, we show the efficiency of the new methods numerically.

关键词： saddle-point problem total variation image restoration primal-dual method contraction method proximal point algorithm

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A proximal method for solving nonlinear minmax location problems with perturbed minimal time functions via conjugate duality

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JOURNAL OF GLOBAL OPTIMIZATION 2019年第1期74卷 121-160页

作者： Grad, Sorin-Mihai Wilfer, Oleg Univ Vienna Fac Math Oskar Morgenstern Pl 1 A-1090 Vienna Austria Tech Univ Chemnitz Fac Math D-09107 Chemnitz Germany

We investigate via a conjugate duality approach general nonlinear minmax location problems formulated by means of an extended perturbed minimal time function, necessary and sufficient optimality conditions being delivered together with characterizations of the optimal solutions in some particular instances. A parallel splitting proximal point method is employed in order to numerically solve such problems and their duals. We present the computational results obtained in matlab on concrete examples, successfully comparing these, where possible, with earlier similar methods from the literature. Moreover, the dual employment of the proximal method turns out to deliver the optimal solution to the considered primal problem faster than the direct usage on the latter. Since our technique successfully solves location optimization problems with large data sets in high dimensions, we envision its future usage on big data problems arising in machine learning.

关键词： Gauge (Minkowski) function Minimal time function Minmax multifacility location problem Sylvester problem Apollonius problem proximal point algorithm Epigraphical projection Projection operator Machine learning

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FAST MULTIPLE-SPLITTING algorithmS FOR CONVEX OPTIMIZATION

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SIAM JOURNAL ON OPTIMIZATION 2012年第2期22卷 533-556页

作者： Goldfarb, Donald Ma, Shiqian Columbia Univ Dept Ind Engn & Operat Res New York NY 10027 USA Univ Minnesota Inst Math & Applicat Minneapolis MN 55455 USA

We present in this paper two different classes of general multiple-splitting algorithms for solving finite-dimensional convex optimization problems. Under the assumption that the function being minimized can be written as the sum of K convex functions, each of which has a Lipschitz continuous gradient, we prove that the number of iterations needed by the first class of algorithms to obtain an epsilon-optimal solution is O((K - 1)L/epsilon), where L is an upper bound on all of the Lipschitz constants. The algorithms in the second class are accelerated versions of those in the first class, where the complexity result is improved to O(root(K - 1)L/epsilon) while the computational effort required at each iteration is almost unchanged. To the best of our knowledge, the complexity results presented in this paper are the first ones of this type that have been given for splitting and alternating direction-type methods. Moreover, all algorithms proposed in this paper are parallelizable, which makes them particularly attractive for solving certain large-scale problems.

关键词： convex optimization variable splitting alternating direction augmented Lagrangian method alternating linearization method complexity theory decomposition smoothing techniques parallel computing proximal point algorithm optimal gradient method

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Multivalued regularized equilibrium problems

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JOURNAL OF GLOBAL OPTIMIZATION 2006年第3期35卷 483-492页

作者： Noor, Muhammad Aslam COMSATS Inst Informat Technol Dept Math Islamabad Pakistan

In this paper, we introduce a new class of equilibrium problems known as the multivalued regularized equilibrium problems. We use the auxiliary principle technique to suggest some iterative methods for solving multivalued regularized equilibrium problems. The convergence of the proposed methods is studied under some mild conditions. As special cases, we obtain a number of known and new results for solving various classes of regularized equilibrium problems and related optimization problems.

关键词： auxiliary principle convergence equilibrium problems proximal point algorithm variational inequalities

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Finding a zero of the sum of two maximal monotone operators

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 1997年第2期94卷 425-448页

作者： Moudafi, A Thera, M UNIV CADI AYYAD MARRAKECHMOROCCO UNIV LIMOGES URA CNRS 1586LACOLIMOGESFRANCE

In this paper, the equivalence between variational inclusions and a generalized type of Weiner-Hopf equation is established. This equivalence is then used to suggest and analyze iterative methods in order to find a zero of the sum of two maximal monotone operators. Special attention is given to the case where one of the operators is Lipschitz continuous and either is strongly monotone or satisfies the Dunn property. Moreover, when the problem has a nonempty solution set, a fixed-point procedure is proposed and its convergence is established provided that the Brezis-Crandall-Pazy condition holds true. More precisely, it is shown that this allows reaching the element of minimal norm of the solution set.

关键词： maximal monotone operators variational inequalities Weiner-Hopf equation Dunn property Yosida approximate regularization fixed-point methods proximal point algorithm Brezis-Crandall-Pazy condition

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Common solution for a finite family of minimization problem and fixed point problem for a pair of demicontractive mappings in Hadamard spaces

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REVISTA DE LA REAL ACADEMIA DE CIENCIAS EXACTAS FISICAS Y NATURALES SERIE A-MATEMATICAS 2020年第2期114卷 1-12页

作者： Chang, Shih-sen Wang, L. Wang, X. R. Zhao, L. C. China Med Univ Ctr Gen Educ Taichung 40402 Taiwan Yunnan Univ Finance & Econ Kunming 650221 Yunnan Peoples R China Yibin Univ Dept Math Yibin 644007 Sichuan Peoples R China

In this paper, an iterative algorithm to approximate a common solution of a finite family of minimization problem and fixed point problem for a pair of demicontractive mappings in Hadamard spaces are proposed. Under suitable conditions, some Delta-convergence and strong convergence theorems of the sequence generated by the algorithm to an element in the intersection of the set of solutions of such kind of minimization problem and fixed point problem in Hardmard space are proved. Our results complement and extend some recent results in literature.

关键词： Convex minimization problem CAT(0) space Hadamard space proximal point algorithm Non-spreading mapping Demi-contractive mapping

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A REGULARIZATION OF THE FRANK-WOLFE METHOD AND UNIFICATION OF CERTAIN NONLINEAR-PROGRAMMING METHODS

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MATHEMATICAL PROGRAMMING 1994年第3期65卷 331-345页

作者： MIGDALAS, A 1. Department of Mathematics Link?ping Institute of Technology S-581 83 Link?ping Sweden

The Frank-Wolfe linearization technique is a popular feasible direction algorithm for the solution of certain linearly constrained nonlinear problems. The popularity of this technique is due in part to its ability to exploit special constraint structures, such as network structures, and in part to the fact that it decomposes nonseparable problems over Cartesian product sets. However, the linearization which induces these advantages is also the source of the main disadvantages of the method: a sublinear rate of convergence and zigzagging behaviour. In order to avoid these disadvantages, a regularization penalty term is added to the objective of the direction generating subproblem. This results in a generic feasible direction method which also includes certain known nonlinear programming methods.

关键词： PSEUDOCONVEX PROGRAMMING DECOMPOSITION SCHEME FRANK WOLFE algorithm FEASIBLE DIRECTION METHODS proximal point algorithm GAP FUNCTION

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The primal Douglas-Rachford splitting algorithm for a class of monotone mappings with application to the traffic equilibrium problem

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MATHEMATICAL PROGRAMMING 1996年第1期72卷 1-15页

作者： Fukushima, M Graduate School of Information Science Nara Institute of Science and Technology Ikoma Nara Japan

We apply the Douglas-Rachford splitting algorithm to a class of multi-valued equations consisting of the sum of two monotone mappings. Compared with the dual application of the same algorithm, which is known as the alternating direction method of multipliers, the primal application yields algorithms that seem somewhat involved. However,the resulting algorithms may be applied effectively to problems with certain special structure. In particular we show that they can be used to derive decomposition algorithms for solving the variational inequality formulation of the traffic equilibrium problem.

关键词： splitting algorithms monotone mapping proximal point algorithm variational inequality problem traffic equilibrium

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