A Survey of Set Optimization Problems with Set Solutions

This paper presents a state-of-the-art survey on set-valued optimization problems whose solutions are defined by set criteria. It provides a general framework that allows to give an overview about set-valued optimization problems according to decision concepts based on certain set relations. The first part of this paper (Sects. 1 and 2) motivates and describes the set-valued optimization problem (in short, SVOP). The present survey deals with general problems of set-valued optimization and recall its main properties in order to establish the differences between vector set-valued optimization problems (VOP) and set optimization problems (SOP). In this context, in the second part (Sects. 3–5) we focus on those results existing in the literature related with optimality conditions by using a set approach. We list and quote references devoted to (SOP) from the beginning up to now. In Sect. 5, a particular attention is paid to applications of the set relations considered in other fields as fixed point theory. The last section provides some conclusions and suggestions for further study.