New Trends in General Variational Inequalities

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, the auxiliary principle and the 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.

Novel forward–backward algorithms for optimization and applications to compressive sensing and image inpainting

“…Remark 3.2 For variational inequality problem, this kind of method is firstly appeared in Noor [18,19,22] Lemma 3.3 ([14]) Linesearch (3.1) stops after finitely many steps.…”

“…It is well known that (1.1) is equivalent to the problem of finding the zero of subdifferentials of f + g at x. This problem is called the variational inclusion problem, see [19]. We denote by argmin(f + g) the solution set of (1.1).…”

Strongly log-biconvex Functions and Applications

“…Clearly every strongly convex function is a convex function, but the converse is not true. For the applications of strongly convex functions in variational inequalities, differential equations and equilibrium problems, see [6,7,8,11,12,13,14,15,16,19,20,22,24] and the references therein.…”

“…The log-parallelogram law (5.2) characterizes the inner product spaces involving exponentially biconvex function. Also, see [6,20] for the derivation and other properties of the inner product spaces.…”

“…It is an interesting problem to study the existences of a solution of the log-bivariational inequalities and develop some numerical methods. For the applications, formulations, numerical methods and other aspects of variational inequalities, see [7,8,15,16,17,19,20,24] and the references therein…”