Points of efficiency in vector optimization with increasing-along-rays property and minty variational inequalities

Abstract

Minty variational inequalities are studied as a tool for vector optimization. Instead of focusing on vector inequalities, we propose an approach through scalarization which allows to construct a proper variational inequality type problem to study any concept of efficiency in vector optimization. This general scheme gives an easy and consistent extension of scalar results, providing also a notion of increasing along rays vector function. This class of generalized convex functions seems to be intimately related to the existence of solutions to a Minty variational inequality in the scalar case, we now extend this fact to vector case. Finally, to prove a reversal of the main theorem, generalized quasiconvexity is considered and the notion of*-quasiconvexity plays a crucial role to extend scalar evidences. This class of functions, indeed, guarantees a Minty-type variational inequality is a necessary and sufficient optimality condition for several kind of efficient solution. © 2006 Springer-Verlag Berlin Heidelberg.