Variational Methods in Topological Vector Spaces

Abstract

In this chapter, we explain that the application of translation invariant functionals (or its modifications) is an important tool for deriving necessary optimality conditions in vector optimization, duality assertions, minimal point theorems and variational principles, necessary conditions for approximate solutions of vector optimization problems with respect to variable domination structure, existence results for solutions of vector variational inequalities and many other results in optimization theory and functional analysis. Furthermore, we derive certain relationships to the Extremal Principle by Kruger and Mordukhovich. For the proofs of these results, the separation properties of translation invariant functionals as well as the monotonicity, convexity and continuity properties are essential.