A scalarization scheme for binary relations with applications to set-valued and robust optimization

Abstract

In this paper, a method for scalarizing optimization problems whose final space is endowed with a binary relation is stated without assuming any additional hypothesis on the data of the problem. By this approach, nondominated and minimal solutions are characterized in terms of solutions of scalar optimization problems whose objective functions are the post-composition of the original objective with scalar functions satisfying suitable properties. The obtained results generalize some recent ones stated in quasi ordered sets and real topological linear spaces. Besides, they are applied both to characterize by scalarization approximate solutions of set optimization problems with set ordering and to generalize some recent conditions on robust solutions of optimization problems. For this aim, a new robustness concept in optimization under uncertainty is introduced which is interesting in itself.