  Via a family of monotone scalar functions, a preorder on a set is extended to
its power set and then used to construct a hull operator and a corresponing
complete lattice of sets. A function mappping into the preordered set is
extended to a complete lattice-valued one, and concepts for exact and
approximate solutions for corresponding set optimization problems are
introduced and existence results are given. Well-posedness for complete
lattice-valued problems is introduced and characterized. The new approach is
compared to existing ones in vector and set optimization, and its relevance is
shown by means of many examples from vector optimization, statistics and
mathematical economics & finance.
