Quasiconvexity of set-valued maps assures well-posedness of robust vector optimization

Robust optimization is a fast growing methodology to study optimization problems with uncertain data. An uncertain vector optimization problem can be studied through its robust or optimistic counterpart, as in Ben-Tal and Nemirovski (Math Oper Res 23:769–805, 1998) and Beck and Ben-Tal (Oper Res Lett 37: 1–6, 2009). In this paper we formulate the counterparts as set optimization problems. This setting appears to be more natural, especially when the uncertain problem is a non-linear vector optimization problem. Under this setting we study the well-posedness of both the robust and the optimistic counterparts, using the embedding technique for set optimization developed in Kuroiwa and Nuriya (Proceedings of the fourth international conference on nonlinear and convex analysis, pp 297–304, 2006). To prove our main results we also need to study the notion of quasiconvexity for set-valued maps, that is the property of convexity of level set. We provide a general scheme to define the notion of level set and we study the relations among different subsequent definitions of quasi-convexity. We prove some existing notions arise as a special case in the proposed scheme.