Two Step Algorithm for Solving Regularized Generalized Mixed Variational Inequality Problem

Abstract

Abstract. In this paper, we consider a new class of regularized (nonconvex) generalized mixed variational inequality problems in real Hilbert space. We give the concepts of partially relaxed strongly mixed monotone and partially relaxed strongly θ-pseudomonotone mappings, which are extension of the concepts given by Xia and Ding [19], Noor [13] and Kazmi et al. [9]. Further we use the auxiliary principle technique to suggest a two-step iterative algorithm for solving regularized (nonconvex) generalized mixed variational inequality problem. We prove that the convergence of the iterative algorithm requires only the continuity, partially relaxed strongly mixed monotonicity and partially relaxed strongly θ-pseudomonotonicity. The theorems presented in this paper represent improvement and generalization of the previously known results for solving equilibrium problems and variational inequality problems involving the nonconvex (convex) sets, see for example Noor [13], Pang et al. [14], and Xia and Ding [19].

Existence Results for Vector Mixed Quasi-Complementarity Problems

“…Every monotone with respect to is pseudomonotone with respect to but converse does not hold in general. Definition 3 is vector version of -pseudomonotonicity studied by Kazmi et al in [23,24].…”

System of implicit nonconvex variationl inequality problems A projection method approach

“…In recent years, Bounkhel et al [6], Moudafi [20], Wen [28], Kazmi et al [17], Noor [[23, 24, 22] and the relevant references cited therein], Alimohammady et al [1], Balooee et al [4] suggested and analyzed iterative algorithms for solving some classes (systems) of nonconvex variational inequality problems in the setting of uniformly prox-regular sets.…”