Stability of efficient solutions to set optimization problems
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Stability of efficient solutions to set optimization problems L. Q. Anh1 · T. Q. Duy2,3
· D. V. Hien4,5,6
Received: 23 December 2019 / Accepted: 13 July 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract This article deals with considering stability properties of Pareto minimal solutions to set optimization problems with the set less order relation in real topological Hausdorff vector spaces. We focus on studying the Painlevé–Kuratowski convergence of Pareto minimal elements in the image space. Employing convexity properties, we study the external stability of Pareto minimal solutions via weak ones. Then, we use converse properties to investigate external stability conditions to such problems where Pareto minimal solution sets and weak/ideal ones are distinct. For the internal stability, we propose a concept of compact convergence in the sense of Painlevé–Kuratowski and use it together with a domination property to analyze stability conditions for the reference problems. Keywords Set optimization problem · Pareto minimal solution · Internal and external stability · Compact convergence · Domination property Mathematics Subject Classification 49K40 · 65K10 · 90C29 · 90C30
1 Introduction In the last decades, set-valued optimization problems, where the objective functions are set-valued maps with ordered values, have received an increasing attention due to their
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T. Q. Duy [email protected] L. Q. Anh [email protected] D. V. Hien [email protected]
1
Department of Mathematics, Teacher College, Cantho University, Cantho, Vietnam
2
Division of Computational Mathematics and Engineering, Institute for Computational Science, Ton Duc Thang University, Ho Chi Minh City, Vietnam
3
Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam
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Faculty of Mathematics and Computer Science, University of Science, Ho Chi Minh City, Vietnam
5
Vietnam National University, Ho Chi Minh City, Vietnam
6
Department of Mathematics, Ho Chi Minh City University of Food Industry, Ho Chi Minh City, Vietnam
123
Journal of Global Optimization
extensive applications in many fields such as economics, optimal control, differential inclusions, bilevel optimization, robust optimization, interval optimization, game theory, see [5,19,22,26]. Because of generalizing the notions of solution already well considered within the framework of vector optimization, two well-known solution criteria, namely, vector criterion and set criterion of solution to such problems have been investigated. The first one consists of finding efficient solutions of the image set, and hence one usually called them by vector optimization problems with set-valued maps; see, for instance, [9,10,32,33,35]. In 1998, Kuroiwa [25] introduced the second criterion of solution to set-valued optimization problems, say set optimization criterion, which was based on a comparison among the values of the objective set-valued maps. In other words, the second one is equivalent to considering relationships
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