The inf-translation for solving set minimization problems

TL;DR

This paper introduces the inf-translation method for set minimization problems, providing a unified framework for optimality conditions and scalarization in lattice-valued optimization, with applications to convex problems and vector calculus of variations.

Contribution

It develops a novel inf-translation approach for set minimization, enabling reduction to single points and facilitating solution procedures within an abstract lattice framework.

Findings

Optimality conditions based on inf-translation are established.

Scalarization results for convex lattice-valued functions are provided.

Application to vector calculus of variations demonstrates the method's versatility.

Abstract

Set- and vector-valued optimization problems can be re-formulated as complete lattice-valued problems. This has several advantages, one of which is the existence of a clear-cut solution concept which includes the attainment as the infimum (not present in traditional vector optimization theory) and minimality as two potentially different features. The task is to find a set which is large enough to generate the infimum but at the same time small enough to include only minimizers. In this paper, optimality conditions for such sets based on the inf-translation are given within an abstract framework. The inf-translation reduces the solution set to a single point which in turn admits the application of more standard procedures. For functions with values in complete lattices of sets, scalarization results are provided where the focus is on convex problems. Vector optimization problems, in particular a vectorial calculus of variations problem, are discussed as examples.