Minty variational inequalities, increase-along-rays property and optimization

Abstract

Let E be a linear space, K E and f : K ? R. We put in terms of the lower Dini directional derivative a problem, referred to as GMV I(f ,K), which can be considered as a generalization of the Minty variational inequality of differential type (for short, MV I(f ,K)). We investigate, in the case of K star-shaped (for short, st-sh), the existence of a solution x of GMV I(f ,K) and the property of f to increase-along-rays starting at x (for short, f IAR(K, x )). We prove that GMV I(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 GMV I(f ,K) is a convex and radially closed subset of kerK. We show also that, if GMV I(f ,K) has a solution x K, then x is a global minimizer of the problem f(x) ? min, 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 case of a quasi-convex function f, these sets coincide. Key words: Minty variational inequality, Generalized variational inequality, Existence of solutions, Increase along rays, Quasi-convex functions.