Convergence results for a common solution of a finite family of variational inequality problems for monotone mappings with Bregman distance function

In this paper, we introduce an iterative process which converges strongly to a common solution of a finite family of variational inequality problems for monotone mappings with Bregman distance function. Our convergence theorem is applied to the convex minimization problem. Our theorems extend and unify most of the results that have been proved for the class of monotone mappings.MSC:47H05, 47J05, 47J25.

[1]  Dan Butnariu,et al.  Bregman distances, totally convex functions, and a method for solving operator equations in Banach spaces , 2006 .

[2]  Wataru Takahashi,et al.  APPROXIMATION OF SOLUTIONS OF VARIATIONAL INEQUALITIES FOR MONOTONE MAPPINGS , 2004 .

[3]  Meijuan Shang,et al.  Strong Convergence Theorems for a Finite Family of Nonexpansive Mappings , 2007 .

[4]  Habtu Zegeye,et al.  A hybrid scheme for finite families of equilibrium, variational inequality and fixed point problems , 2011 .

[5]  S. Reich,et al.  Two strong convergence theorems for Bregman strongly nonexpansive operators in reflexive Banach spaces , 2010 .

[6]  Shoham Sabach,et al.  Existence and Approximation of Fixed Points of Bregman Firmly Nonexpansive Mappings in Reflexive Banach Spaces , 2011, Fixed-Point Algorithms for Inverse Problems in Science and Engineering.

[7]  R. Phelps Convex Functions, Monotone Operators and Differentiability , 1989 .

[8]  Shoham Sabach,et al.  A projection method for solving nonlinear problems in reflexive Banach spaces , 2011 .

[9]  Habtu Zegeye,et al.  A hybrid approximation method for equilibrium, variational inequality and fixed point problems , 2010 .

[10]  P. Maingé Strong Convergence of Projected Subgradient Methods for Nonsmooth and Nonstrictly Convex Minimization , 2008 .

[11]  Hong-Ku N Xu Another control condition in an iterative method for nonexpansive mappings , 2002, Bulletin of the Australian Mathematical Society.

[12]  Habtu Zegeye,et al.  Convergence theorems for equilibrium problem, variational inequality problem and countably infinite relatively quasi-nonexpansive mappings , 2010, Appl. Math. Comput..

[14]  Heinz H. Bauschke,et al.  Bregman Monotone Optimization Algorithms , 2003, SIAM J. Control. Optim..

[15]  Anna Nagurney,et al.  Variational Inequalities , 2009, Encyclopedia of Optimization.

[16]  Y. Alber Metric and Generalized Projection Operators in Banach Spaces: Properties and Applications , 1993, funct-an/9311001.

[17]  Wataru Takahashi,et al.  Weak convergence of a projection algorithm for variational inequalities in a Banach space , 2008 .

[18]  Y. Censor,et al.  An iterative row-action method for interval convex programming , 1981 .

[19]  Shoham Sabach,et al.  Bregman strongly nonexpansive operators in reflexive Banach spaces , 2013 .

[20]  Habtu Zegeye,et al.  An Iteration to a Common Point of Solution of Variational Inequality and Fixed Point-Problems in Banach Spaces , 2012, J. Appl. Math..

[21]  Wataru Takahashi,et al.  Strong convergence studied by a hybrid type method for monotone operators in a Banach space , 2008 .

[22]  Habtu Zegeye,et al.  Approximating common solution of variational inequality problems for two monotone mappings in Banach spaces , 2011, Optim. Lett..

[23]  Wataru Takahashi,et al.  Strong Convergence of a Proximal-Type Algorithm in a Banach Space , 2002, SIAM J. Optim..

[24]  J. Frédéric Bonnans,et al.  Perturbation Analysis of Optimization Problems , 2000, Springer Series in Operations Research.

[25]  C. Zălinescu,et al.  On Uniform Convexity, Total Convexity and Convergence of the Proximal Point and Outer Bregman Projection Algorithms in Banach Spaces , 2003 .

[26]  S. Reich,et al.  Two Strong Convergence Theorems for a Proximal Method in Reflexive Banach Spaces , 2010 .

[27]  A. Iusem,et al.  Totally Convex Functions for Fixed Points Computation and Infinite Dimensional Optimization , 2000 .

[28]  A. Kartsatos Theory and applications of nonlinear operators of accretive and monotone type , 1996 .

[29]  N. Shahzad,et al.  Strong convergence theorems for a common point of solution of variational inequality, solutions of equilibrium and fixed point problems , 2012 .

[30]  Heinz H. Bauschke,et al.  Fixed-Point Algorithms for Inverse Problems in Science and Engineering , 2011, Springer Optimization and Its Applications.

[31]  J. Hiriart-Urruty,et al.  Convex analysis and minimization algorithms , 1993 .

[32]  L. Bregman The relaxation method of finding the common point of convex sets and its application to the solution of problems in convex programming , 1967 .

[33]  Habtu Zegeye,et al.  Strong convergence theorems for monotone mappings and relatively weak nonexpansive mappings , 2009 .

[34]  Heinz H. Bauschke,et al.  ESSENTIAL SMOOTHNESS, ESSENTIAL STRICT CONVEXITY, AND LEGENDRE FUNCTIONS IN BANACH SPACES , 2001 .