An iterative method for solving minimization, variational inequality and fixed point problems in reflexive Banach spaces

Abstract

<jats:title>Abstract</jats:title> <jats:p>In this paper, we propose an iterative algorithm for approximating a common fixed point of an infinite family of quasi-Bregman nonexpansive mappings which is also a solution to finite systems of convex minimization problems and variational inequality problems in real reflexive Banach spaces. We obtain a strong convergence result and give applications of our result to finding zeroes of an infinite family of Bregman inverse strongly monotone operators and a finite system of equilibrium problems in real reflexive Banach spaces. Our result extends many recent corresponding results in literature.</jats:p>

On Mixed Equilibrium Problems in Hadamard Spaces

“…In this section, we establish the existence of solution for MEP (4). We also establish the unique existence of the resolvent operator associated with the bifunction F and the convex functional Ψ.…”

“…In this case, x * is called a minimizer of Ψ and argmin y∈C Ψ(y) denotes the set of minimizers of Ψ. MPs are very useful in optimization theory and convex and nonlinear analysis. One of the most popular and effective methods for solving MPs is the proximal point algorithm (PPA) which was introduced in Hilbert space by Martinet [1] in 1970 and was further extensively studied in the same space by Rockafellar [2] in 1976. e PPA and its generalizations have also been studied extensively for solving MP (1) and related optimization problems in Banach spaces and Hadamard manifolds (see [3][4][5][6][7] and the references therein), as well as in Hadamard and p-uniformly convex metric spaces (see [8][9][10][11][12][13] and the references therein).…”

A viscosity-proximal gradient method with inertial extrapolation for solving certain minimization problems in Hilbert space

“…Consequently, the sequence {x n } generated by (1.5) may fail to converge strongly (see Section 4 in [49]). The gradient projection algorithm (1.5) has been studied extensively by many authors, see for instance [8,9,19,20,21,44,49] and reference therein. In 2000, Moudafi [29] introduced the viscosity approximation method for approximating fixed points of nonexpansive mappings.…”

Algorithm for the generalized &#934;-strongly monotone mappings and application to the generalized convex optimization problem

“…The problem of finding zero points for maximal monotone operators plays an important role in optimizations because it can be reduced to a convex minimization problem and a variational inequality problem. The approximation of solutions of these problems has also been studied by numerous authors (see for examples, [1,22,23,30,31,41].…”