Generalized Vector Variational-Like Inequalities

In this paper, we consider different types of generalized vector variational-like inequalities and study the relationships between their solutions. We study the general forms of Stampacchia and Minty type vector variational inequalities for bifunctions and establish the existence of their solutions in the setting of topological vector spaces. We extend these vector variational inequalities for the Clarke’s subdifferential of non-differentiable locally Lipschitz functions and prove the existence of their solutions. As applications, we establish some existence results for a solution of the vector optimization problem by using Stampacchia and Minty type vector variational inequalities.