Special issue on `Set-valued optimization': Editors' preface

Abstract

This special issue on Set-Valued Optimization treats set-valued analysis and its application to optimization. On the one hand optimization problems are investigated with constraints and/or an objective function described by setvalued maps, and on the other hand investigations in set-valued analysis are applied to standard optimization problems. In the last decade there has been an increasing interest in set-valued optimization. General optimization problems with set-valued constraints or a set-valued objective function are closely related to problems in stochastic programming, fuzzy programming and optimal control. If the values of a given function vary in a speci®ed region, this fact could be described using a membership function in the theory of fuzzy sets or using information on the distribution of the function values. In this general setting probability distributions or membership functions are not needed because only sets are considered. Optimal control problems with di erential inclusions belong to this class of set-valued optimization problems as well. Set-valued optimization seems to have the potential to become a bridge between di erent areas in optimization. And it is a substantial extension of standard optimization theory. Set-valued analysis is the most important tool for such an advancement in continuous optimization. And conversely, the development of set-valued analysis receives important impulses from optimization. This issue makes evident the interplay between set-valued analysis and optimization. The articles in this special issue throw light on various facets of set-valued analysis in optimization. These contributions re ect scienti®c activities in Europe, China, Australia and North America. The papers may be divided into two groups: