Minty Variational Inequalities, Increase-Along-Rays Property and Optimization¹

Abstract

Let E be a linear space, let K $\subseteq$ E and f:K→ℝ . We formulate in terms of the lower Dini directional derivative problem GMVI (f ^′,K), which can be considered as a generalization of MVI (f ^′,K), the Minty variational inequality of differential type. We investigate, in the case of K star-shaped (SS), the existence of a solution x ^* of GMVI (f ^′K) and the property of f to increase-along-rays starting at x ^*, f∈IAR (K,x ^*). We prove that the GMVI (f ^′,K) with radially l.s.c. function f has a solution x ^*∈ ker K if and only if f∈IAR (K,x ^*). Further, we prove that the solution set of the GMVI (f ^′,K) is a convex and radially closed subset of ker K. We show also that, if the GMVI (f ^′,K) has a solution x ^*∈K, then x ^* is a global minimizer of the problem min f(x), x∈K. Moreover, we observe that the set of the global minimizers of the related optimization problem, its kernel, and the solution set of the variational inequality can be different. Finally, we prove that, in the case of a quasiconvex function f, these sets coincide.