New Trends in General Variational Inequalities

TL;DR

This paper introduces new numerical techniques and theoretical insights for solving general variational inequalities, improving convergence and efficiency, and extending to related equilibrium and complementarity problems.

Contribution

It presents novel iterative methods, introduces new classes of convex functions, and simplifies convergence proofs for solving a broad class of variational inequalities.

Findings

New iterative methods with proven convergence.

Introduction of strongly exponentially general convex functions.

Numerical results demonstrating improved efficiency.

Abstract

It is well known that general variational inequalities provide us with a unified, natural, novel and simple framework to study a wide class of unrelated problems, which arise in pure and applied sciences. In this paper, we present a number of new and known numerical techniques for solving general variational inequalities and equilibrium problems using various techniques including projection, Wiener-Hopf equations, dynamical systems, auxiliary principle and penalty function. General variational-like inequalities are introduced and investigated. Properties of higher order strongly general convex functions have been discussed. The auxiliary principle technique is used to suggest and analyze some iterative methods for solving higher order general variational inequalities. Some new classes of strongly exponentially general convex functions are introduced and discussed. Our proofs of convergence are very simple as compared with other methods. Our results present a significant improvement of previously known methods for solving variational inequalities and related optimization problems. Since the general variational inequalities include (quasi) variational inequalities and (quasi) implicit complementarity problems as special cases, these results continue to hold for these problems. Some numerical results are included to illustrate the efficiency of the proposed methods. Several open problems have been suggested for further research in these areas.