Existence results and optimization over the set of efficient solutions in vector-valued approximation theory

Abstract

Continuity of the objective functions and compactness of their domain are classical assumptions widely used to obtain existence results for solutions of optimization problems. Due to the lack of compactness in general spaces, under some moderate assumptions concerning the objective function and the feasible set, we derive existence results for vector-valued optimization problems and corresponding results for associated scalarized problems in this paper. Furthermore, we apply our results to special vector-valued approximation problems, especially to multi-objective location problems where the whole set of efficient solutions can be generated by a geometric primal-dual algorithm. Moreover, by using the nonlinear scalarizing functional introduced by Gerstewitz, we perform an optimization according to the preferences of a decision maker on the generated set of efficient solutions from which we derive a single solution of this set that corresponds to the preferences of the decision maker.