Interval variational inequalities and their relationship with interval optimization problems

This article is devoted to studying Stampacchia and Minty variational inequalities for interval-valued functions (IVFs). In the sequel, it is observed that conventional Stampacchia and Minty variational inequalities are special cases of the proposed inequalities. The relation between solution sets of these two variational inequalities is analyzed. Existence and uniqueness results are provided for the solutions of the proposed variational inequalities. Moreover, a necessary optimality condition is given for a constrained interval optimization problem (IOP) using the generalized Hukuhara differentiability. It is observed that the first-order characterization of convex IVFs given in the literature is not true and a new first-order necessary condition is given for convex IVFs. By using this new first-order necessary condition for convex IVFs, as an application of the proposed study, a necessary and sufficient optimality condition for a constrained IOP is provided in terms of Stampacchia IVI. Further, a sufficient optimality condition for a constrained IOP is provided in terms of Minty IVI.