On the Applications of Nonsmooth Vector Optimization Problems to Solve Generalized Vector Variational Inequalities Using Convexificators

Abstract

In this paper, we employ the characterization for an approximate convex function in terms of its convexificator to establish the relationships between the solutions of Stampacchia type vector variational inequality problems in terms of convexificator and quasi efficient solution of a nonsmooth vector optimization problems involving locally Lipschitz functions. We identify the vector critical points, the weak quasi efficient points and the solutions of the weak vector variational inequality problem under generalized approximate convexity assumptions. The results of the paper extend, unify and sharpen corresponding results in the literature. In particular, this work extends and generalizes earlier works by Giannessi [11], Upadhyay et al. [31], Osuna-Gomez et al. [30], to a wider class of functions, namely the nonsmooth approximate convex functions and its generalizations. Moreover, this work sharpens earlier work by Daniilidis and Georgiev [5] and Mishra and Upadhyay [23], to a more general class of subdifferentials known as convexificators.