The inf-translation for solving set minimization problems

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 of 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.