Minimizers of Translation Invariant Functions

Abstract

When applying a translation invariant function for the separation of sets or the scalarization of problems, then it usually has to be minimized over a given set. In the subsequent chapters, we will characterize solutions to vector optimization problems by minimizers of translation invariant functions on a feasible set. Good-deal bounds of cash streams turn out to be infima of translation invariant functions. In this chapter, we study minimization problems having a translation invariant objective function. Such problems are connected with a generalization of an optimization problem introduced by Pascoletti and Serafini. The existence of optimal solutions and properties of the solution set are investigated. Relationships to problems with an altered feasible set are proved. The considered problems depend on certain parameters when they are applied to scalarization. We examine the related parameter control. Assumptions are given under which the considered problems are equivalent to the minimization of special Minkowski functionals and norms.