Optim Lett DOI 10.1007/s11590-016-1079-4 Convergence of relaxed minimizers in set optimization Michel H. Geoffroy 1 · Yvesner Marcelin 1 · Diana Nedelcheva 2 Received: 21 January 2016 / Accepted: 1 September 2016 © Springer-Verlag Berlin Heidelberg 2016 Abstract In this paper we investigate, in a unified way, the stability of several relaxed minimizers of set optimization problems. To this end, we introduce a topology on vector ordered spaces from which we derive a concept of convergence that allows us to study both the upper and the lower stability of the sets of relaxed minimizers we consider. Keywords Variational convergence · Ŵ-convergence · Relative interior · Pseudo relative interior · Quasi relative interior. 1 Introduction Throughout, X is a general Banach space while Y is a Banach space ordered by a nonempty, closed, convex and pointed cone C ⊂ Y through a binary relation ≤ defined by y 1 ≤ y 2 if and only if y 2 − y 1 ∈ C . We recall that C is pointed whenever C ∩ (−C ) ={0 Y }. Let F be a set-valued mapping acting from X to the subsets of Y , indicated by F : X ⇒ Y the domain of which, denoted by dom F, is nonempty. We recall that B Michel H. Geoffroy michel.geoffroy@univ-ag.fr Yvesner Marcelin yvesner.marcelin@univ-ag.fr Diana Nedelcheva diana.nedelcheva@tu-varna.bg 1 LAMIA, Department of Mathematics, Université des Antilles, Guadeloupe, Pointe-à-Pitre, France 2 Department of Mathematics, Technical University, Varna, Bulgaria 123