Convergence analysis for variational inequalities and fixed point problems in reflexive Banach spaces

Abstract

In this paper, using a Bregman distance technique, we introduce a new single projection process for approximating a common element in the set of solutions of variational inequalities involving a pseudo-monotone operator and the set of common fixed points of a finite family of Bregman quasi-nonexpansive mappings in a real reflexive Banach space. The stepsize of our algorithm is determined by a self-adaptive method, and we prove a strong convergence result under certain mild conditions. We further give some applications of our result to a generalized Nash equilibrium problem and bandwidth allocation problems. We also provide some numerical experiments to illustrate the performance of our proposed algorithm using various convex functions and compare this algorithm with other algorithms in the literature.

An inertial‐like Tseng's extragradient method for solving pseudomonotone variational inequalities in reflexive Banach spaces

“…1. We point out that the sequentially weakly continuity condition often assumed by authors in solving pseudomonotone VIP is dispensed with in our proposed method (e.g., see [19,[22][23][24][25]). 2.…”

“…), method by Tan and Cho [54] (Tan & Cho Alg. ), method by Reich et al in [55], method by Jolaoso and Aphane [22], and method by Jolaoso et al [23]. All numerical computations were carried out using Matlab version R2022(b).…”

Inertial self-adaptive Bregman projection method for finite family of variational inequality problems in reflexive Banach spaces

“…2), then the CSVIP (1.2) redues to the VIP (1.1). The motivation for studying the CSVIP stems from its applications in studying several problems in applied sciences whose constraint can be modelled as CSVIP, for instance, generalized Nash equilibrium problem and utility-based bandwidth allocation problems (see, e.g., Jolaoso et al 2021).…”

“…Gibali (2018) also proposed a Bregman projection algorithm for solving the VIP (1.1) in finite dimensional space using the extragradient technique. Moreover, other methods involving the Bregman distance for solving the VIP (1.1) can be found in Jolaoso and Aphane (2020), Jolaoso et al (2021) and references therein. Motivated by the above results, we introduce a self adaptive algorithm for approximating a common solution of a finite family of VIPs.…”

Iterative Design for the Common Solution of Monotone Inclusions and Variational Inequalities

“…The theory of variational inequality draws much attention of mathematicians due to its wide application in several branches of pure and applied sciences [1]. Until now, it is still a hot topic (see [2][3][4][5][6] and the references therein).…”