Directional variational principles and applications to the existence study in optimization

Abstract

The purpose of this paper is to present a refinement of earlier directional variational principles and solution existence in optimization. By using the so-called directional minimal time function, we first provide a directional invariant point theorem. As direct consequences, we obtain several directional variational principles (with quite different formulations). Then, applying these results, we establish sufficient conditions for the existence of solutions for two general models of directional variational relation and inclusion problems. We also include corresponding consequences for particular cases.