  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.
