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Weak and strong convergence theorems for common zeros of accretive operators

JOURNAL OF INEQUALITIES AND APPLICATIONS 2014年第1期2014卷 1-10页

作者： Wang, Shenghua Li, Tian North China Elect Power Univ Dept Math & Phys Baoding 071003 Peoples R China

In this paper, we propose two proximal point algorithms for investigating common zeros of a family of accretive operators. Weak and strong convergence of the two algorithms are obtained in a Banach space.

关键词： accretive operator convergence resolvent proximal point algorithm zero point

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Common Zero for a Finite Family of Monotone Mappings in Hadamard Spaces with Applications

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MEDITERRANEAN JOURNAL OF MATHEMATICS 2018年第4期15卷 1-16页

作者： Chang, Shih-sen Yao, Jen-Chih Wen, Ching-Feng Yang, Li Qin, Li-Jian China Med Univ Ctr Gen Educ Taichung 40402 Taiwan Yibin Univ Coll Math Yibin 644007 Sichuan Peoples R China Kaohsiung Med Univ Ctr Fundamental Sci Kaohsiung 80708 Taiwan South West Univ Sci & Technol Sch Sci Mianyang 621010 Sichuan Peoples R China Kunming Univ Dept Math Kunming 650214 Yunnan Peoples R China

The purpose of this article is to propose a viscosity-type algorithms for solving the common zero for a finite family of monotone mappings in Hadamard spaces. Some applications to convex optimization problem in Hadama... 详细信息

关键词： Hadamard space CAT(0)space Monotone mapping proximal point algorithm Uniform convexity

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An efficient algorithm for globally minimizing a quadratic function under convex quadratic constraints

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MATHEMATICAL PROGRAMMING 2000年第3期87卷 401-426页

作者： An, LTH Natl Inst Appl Sci Rouen Math Lab Math Modelling & Appl Optimizat Grp F-76131 Mt St Aignan France

In this paper we investigate two approaches to minimizing a quadratic form subject to the intersection of finitely many ellipsoids. The first approach is the d.c. (difference of convex Functions) optimization algorithm (abbr. DCA) whose main tools are the proximal point algorithm and/or the projection subgradient method in convex minimization. The second is a branch-and-bound scheme using Lagrangian duality for bounding and ellipsoidal bisection in branching. The DCA was first introduced by Pham Dinh in 1986 for a general d.c. program and later developed by our various work is a local method but, from a good starting point, it provides often a global solution. This motivates us to combine the DCA and our branch and bound algorithm in order to obtain a good initial point for the DCA and to prove the globality of the DCA. In both approaches we attempt to use the ellipsoidal constrained quadratic programs as the main subproblems. The idea is based upon the fact that these programs can be efficiently solved by some available (polynomial and nonpolynomial time) algorithms, among them the DCA with restarting procedure recently proposed by Pham Dinh and Le Thi has been shown to be the most robust and fast For large-scale problems. Several numerical experiments with dimension up to 200 are given which show the effectiveness and the robustness of the DCA and the combined DCA-branch-and-bound algorithm.

关键词： d.c. optimization DCA branch and bound combined DCA-branch and bound projection subgradient method dual method Lagrangian duality proximal point algorithm

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On Finite Convergence of Iterative Methods for Variational Inequalities in Hilbert Spaces

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 2014年第3期161卷 701-715页

作者： Matsushita, Shin-ya Xu, Li Akita Prefectural Univ Dept Elect & Informat Syst Akita Japan

In a Hilbert space, we study the finite termination of iterative methods for solving a monotone variational inequality under a weak sharpness assumption. Most results to date require that the sequence generated by the method converges strongly to a solution. In this paper, we show that the proximal point algorithm for solving the variational inequality terminates at a solution in a finite number of iterations if the solution set is weakly sharp. Consequently, we derive finite convergence results for the gradient projection and extragradient methods. Our results show that the assumption of strong convergence of sequences can be removed in the Hilbert space case.

关键词： Variational inequality Weak sharp proximal point algorithm Metric projection Gradient projection method Extragradient method Hilbert space

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Some proximal algorithms for linearly constrained general variational inequalities

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OPTIMIZATION 2012年第5期61卷 505-524页

作者： Li, M. Yuan, X. M. Hong Kong Baptist Univ Dept Math Hong Kong Hong Kong Peoples R China Southeast Univ Sch Econ & Management Nanjing 210096 Jiangsu Peoples R China

Based on the classical proximal point algorithm (PPA), some PPA-based numerical algorithms for general variational inequalities (GVIs) have been developed recently. Inspired by these algorithms, in this article we propose some proximal algorithms for solving linearly constrained GVIs (LCGVIs). The resulted subproblems are regularized proximally, and they are allowed to be solved either exactly or approximately.

关键词： general variational inequality linear constraint proximal point algorithm inexact methods

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Uniformity and inexact version of a proximal method for metrically regular mappings

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JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 2007年第1期335卷 168-183页

作者： Aragon Artacho, Francisco J. Geoffroy, Michel H. Univ Antilles Guyane Dept Math Lab AOC F-97159 Pointe A Pitre Guadeloupe France Univ Alicante Dept Stat & Operat Res Alicante 03071 Spain

We study stability properties of a proximal point algorithm for solving the inclusion 0 is an element of T(x) when T is a set-valued mapping that is not necessarily monotone. More precisely we show that the convergence of our algorithm is uniform, in the sense that it is stable under small perturbations whenever the set-valued mapping T is metrically regular at a given solution. We present also an inexact proximal point method for strongly metrically subregular mappings and show that it is super-linearly convergent to a solution to the inclusion 0 is an element of T(x). (C) 2007 Elsevier Inc. All rights reserved.

关键词： proximal point algorithm set-valued mapping metric regularity strong subregularity strong regularity

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Stochastic incremental mirror descent algorithms with Nesterov smoothing

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NUMERICAL algorithmS 2024年第1期95卷 351-382页

作者： Bitterlich, Sandy Grad, Sorin-Mihai Tech Univ Chemnitz Fac Math D-09126 Chemnitz Germany Inst Polytech Paris Unite Math Appl ENSTA Paris F-91120 Palaiseau France

For minimizing a sum of finitely many proper, convex and lower semicontinuous functions over a nonempty closed convex set in an Euclidean space we propose a stochastic incremental mirror descent algorithm constructed by means of the Nesterov smoothing. Further, we modify the algorithm in order to minimize over a nonempty closed convex set in an Euclidean space a sum of finitely many proper, convex and lower semicontinuous functions composed with linear operators. Next, a stochastic incremental mirror descent Bregman-proximal scheme with Nesterov smoothing is proposed in order to minimize over a nonempty closed convex set in an Euclidean space a sum of finitely many proper, convex and lower semicontinuous functions and a prox-friendly proper, convex and lower semicontinuous function. Different to the previous contributions from the literature on mirror descent methods for minimizing sums of functions, we do not require these to be (Lipschitz) continuous or differentiable. Applications in Logistics, Tomography and Machine Learning modelled as optimization problems illustrate the theoretical achievements

关键词： Mirror descent method Stochastic algorithm Nesterov smoothing Incremental algorithm proximal point algorithm PET image reconstructions Location problem

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Multiclass, simultaneous route and departure time choice dynamic traffic assignment with an embedded spatial queuing model

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TRANSPORTMETRICA B-TRANSPORT DYNAMICS 2019年第1期7卷 124-146页

作者： Zhan, Xianyuan Ukkusuri, Satish, V Purdue Univ Lyles Sch Civil Engn W Lafayette IN 47907 USA

This paper develops a complementarity formulation for a multi-user class, simultaneous route and departure time choice dynamic user equilibrium (DUE) model. A path-based multiclass cell transmission model (mCTM) is embedded to propagate the traffic flow on the network. Heterogeneous user classes are incorporated in the new formulation and heterogeneity is based on different preferred arrival times, cost perception for travel time, early and late arrival penalties. Multiple model properties have been showed. The proposed model is solved as an equivalent non-monotone variational inequality (VI) problem defined on a product set. A modified proximal point algorithm is used to solve the proposed non-monotone VI problem. Numerical results show that the solution approach is able to find the equilibrium or close to equilibrium solutions. The new formulation and solution approach show the feasibility of solving the multiclass DUE problem for general traffic networks.

关键词： Dynamic traffic assignment transportation networks proximal point algorithm multi-user class spatial queue

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The asymptotic behavior of a class of nonlinear semigroups in Hadamard spaces

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JOURNAL OF FIXED point THEORY AND APPLICATIONS 2014年第1-2期16卷 189-202页

作者： Bacak, Miroslav Reich, Simeon Max Planck Inst D-04103 Leipzig Germany Technion Israel Inst Technol Dept Math IL-32000 Haifa Israel

We study a nonlinear semigroup associated with a nonexpansive mapping on an Hadamard space and establish its weak convergence to a fixed point. A discrete-time counterpart of such a semigroup, the proximal point algorithm, turns out to have the same asymptotic behavior. This complements several results in the literature-both classical and more recent ones. As an application, we obtain a new approach to heat flows in singular spaces for discrete as well as continuous times.

关键词： Asymptotic behavior Dirichlet problem fixed point heat flow Hadamard space nonlinear Markov operator nonlinear semigroup proximal point algorithm resolvent

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proximal Bundle algorithms for Nonlinearly Constrained Convex Minimax Fractional Programs

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 2018年第1期179卷 212-239页

作者： Addoune, Smail Boufi, Karima Roubi, Ahmed Univ Bordj Bou Arreridj Fac MI Dept Math Bordj Bou Arreridj Algeria Univ Hassan 1 Fac Sci & Tech Lab MISI Settat Morocco

A generalized fractional programming problem is defined as the problem of minimizing a nonlinear function, defined as the maximum of several ratios of functions on a feasible domain. In this paper, we propose new methods based on the method of centers, on the proximal point algorithm and on the idea of bundle methods, for solving such problems. First, we introduce proximal point algorithms, in which, at each iteration, an approximate prox-regularized parametric subproblem is solved inexactly to obtain an approximate solution to the original problem. Based on this approach and on the idea of bundle methods, we propose implementable proximal bundle algorithms, in which the objective function of the last mentioned prox-regularized parametric subproblem is replaced by an easier one, typically a piecewise linear function. The methods deal with nondifferentiable nonlinearly constrained convex minimax fractional problems. We prove the convergence, give the rate of convergence of the proposed procedures and present numerical tests to illustrate their behavior.

关键词： Generalized fractional programs Method of centers proximal point algorithm Bundle methods Quadratic programming

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