Vectorial penalisation in vector optimisation in real linear-topological spaces

Abstract

The aim of this paper is to present a vectorial penalisation approach for vector optimisation problems in which the vector-valued objective function acts between real linear-topological spaces X and Y, where the image space Y is partially ordered by a pointed convex cone. In essence, the approach replaces the original constrained vector optimisation problem (with not necessarily convex feasible set) by two unconstrained vector optimisation problems, where in one of the two problems a penalisation term (function) with respect to the original feasible set is added to the vector objective function. To derive our main results, we use a generalised convexity (quasiconvexity) notion for vector functions in the sense of Jahn. Our results extend/generalise known results in the context of vectorial penalisation in multiobjective/vector optimisation. We put a special emphasis on the construction of appropriate penalisation functions for several popular classes of (vector) optimisation problems (e.g., semidefinite/copositive programming, second-order cone programming, optimisation in function spaces).