Set Relations and Approximate Solutions in Set Optimization

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.