Vector Optimization in General Spaces

Abstract

In this chapter, vector optimization problems in linear spaces are studied. Here the notion of vector optimization is used in a very general way. A vector optimization problem with a variable domination structure is equivalent to the problem of finding nondominated solutions in a set related to some arbitrary binary relation. Each decision problem can be shown to be of this type. We introduce the basic concept for variable domination structures and connect it to decision problems. Optimal decisions are investigated as optima w.r.t. (with regard to) the preference relation, and the results can straightforwardly be transferred to optima w.r.t. arbitrary binary relations. It will be proved that—even for variable domination structures—efficient points can be helpful in finding the best solutions. Later on, we investigate the case where the domination structure can be described by a single set, namely the domination set of the vector optimization problem. We define efficiency w.r.t. reference sets which are not necessarily domination sets and give a thorough motivation for this approach. Properties of such efficient point sets are investigated. We study efficient and weakly efficient solutions of vector optimization problems, including surrogates for the weakly efficient point set and problems with uncertainties or perturbations. Scalarization results are proved, with an emphasis on translation invariant functions and implications for norms. Beside this, some basic properties of efficient and weakly efficient solutions are examined, especially their existence. The results imply statements for properly efficient solutions and related subsets of the efficient point set.