Continuity of Solution Maps to Parametric Set Optimization Problems via Parametric Equilibrium Problems
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Continuity of Solution Maps to Parametric Set Optimization Problems via Parametric Equilibrium Problems Lam Quoc Anh1 · Nguyen Huu Danh2,3 · Tran Ngoc Tam4,5 Dedicated to Professor Hoang Tuy on the occasion of his 90th birthday Received: 7 July 2018 / Revised: 28 May 2019 / Accepted: 10 June 2019 / © Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Nature Singapore Pte Ltd. 2020
Abstract In this paper, we consider set optimization problems with respect to set less order relations. We introduce nonlinear scalarization functions for sets and study several properties of such functions. Using the concerning functions, we investigate relationships between set optimization problems and equilibrium problems. Sufficient conditions for the continuity of solution maps to such problems via equilibrium problems are established. Keywords Set optimization problem · Equilibrium problem · Nonlinear scalarization · Stability · Hausdorff continuity Mathematics Subject Classification (2010) 49K40 · 90C31 · 91B50 Tran Ngoc Tam
[email protected] Lam Quoc Anh [email protected] Nguyen Huu Danh [email protected] 1
Department of Mathematics, Teacher College, Cantho University, Cantho, Vietnam
2
Department of Mathematics, Taydo University, Cantho, Vietnam
3
Faculty of Mathematics and Computer Science, University of Science, Vietnam National University, Ho Chi Minh City, Vietnam
4
Division of Computational Mathematics and Engineering, Institute for Computational Science, Ton Duc Thang University, Ho Chi Minh City, Vietnam
5
Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam
L.Q. Anh et al.
1 Introduction It is well known that vector optimization problems with set-valued objective maps have received much attention of many authors due to their wide applications in various fields. For these problems, there are two types of criteria for solutions: the vectorial criterion and the set optimization criterion. The first criterion which is looking for efficient elements of the image set is the most commonly used in the literature (see, e.g., [11, 29, 30]). The second criterion was first introduced in [26]. This criterion is defined by an appropriate less order relation and based on comparisons among values in the whole image set. In [20], the author showed that the set criterion is more natural than the first criterion. There have been many studies of qualitative properties of solutions to set optimization problems via set less order relations: existence conditions for solutions (see, e.g., [1, 18, 27, 28]), stability and sensitivity analysis conditions (see, e.g., [16, 33, 34]), well-posedness for such problems (see, e.g., [15, 17, 35]). Recently, a lot of new order relations for sets together with their corresponding set optimization problems have been proposed. For a description of generalized models and applications of set optimization problems, we refer the interested readers to [5, 7, 14, 21–25] for more details. It is shown that among many tool
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