Multiobjective Optimization

Abstract

This chapter focuses on scalarization methods in multiobjective optimization and on properties of Geoffrion’s properly efficient point set. Geoffrion’s properly efficient point set is described by efficient and weakly efficient point sets related to cones, especially to different types of polyhedral cones. Moreover, existence results and the density in the Pareto optimal point set are proved under rather mild assumptions. The properly efficient point set is characterized by minimizers of strictly monotone functionals and by minimizers of translation invariant functions. Conditions for the coincidence of Geoffrion’s proper efficiency with Nehse–Iwanow’s proper efficiency are given. Statements which can be deduced from the previous chapters and from the investigation of Geoffrion’s proper efficiency are applied to scalarization procedures in multiobjective optimization. Beside relationships between the solution sets of the scalarizing problems and optimal point sets of the multiobjective optimization problem, the results for the scalar problems include statements about the existence and uniqueness of their solutions as well as about the parameter control. The weighted Chebyshev norm minimization and its extension by Choo and Atkins, Wierzbicki’s reference point projection, the $\varepsilon$-constraint method, and the Hurwicz Rule for decision making under uncertainty belong to the examined scalarizations. The results are relevant to many other scalarization methods since a general framework for the systematic investigation of such methods is presented. This framework is based on translation invariant functions.