Journal of Optimization Theory and Applications https://doi.org/10.1007/s10957-020-01657-2 Convergence of Solutions to Set Optimization Problems with the Set Less Order Relation Lam Quoc Anh 1 · Tran Quoc Duy 2,3 · Dinh Vinh Hien 4,5 · Daishi Kuroiwa 6 · Narin Petrot 7 Received: 26 March 2019 / Accepted: 12 March 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020 Abstract This article investigates stability conditions for set optimization problems with the set less order relation in the senses of Panilevé–Kuratowski and Hausdorff convergence. Properties of various kinds of convergences for elements in the image space are dis- cussed. Taking such properties into account, formulations of internal and external stability of the solutions are studied in the image space in terms of the convergence of a solution sets sequence of perturbed set optimization problems to a solution set of the given problem. Keywords Set optimization · Set less order relation · Internal stability · External stability Mathematics Subject Classification 49M37 · 90C30 · 65K05 · 47J20 1 Introduction Set-valued optimization is an interesting and important branch of applied mathematics, that is designed to solve optimization problems, in which either the objective mapping or the constraint mappings are set-valued mappings acting between abstract spaces. Concerning set-valued optimization, there are two main approaches, depending on what the notion of minimality is considered. The classical one is vector approach, where the definition of a minimizer considers only an efficient point of the union of all images of the set-valued objective mapping and identifies the image set containing this minimal element as the “best” available [1,2]. However, a serious disadvantage of this approach is that, in general, only one element does not necessarily imply that the Communicated by Anil Aswani. B Tran Quoc Duy tranquocduy@tdtu.edu.vn Extended author information available on the last page of the article 123