Second-order optimality conditions for set optimization using coradiant sets
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Second-order optimality conditions for set optimization using coradiant sets Bin Yao1,2 · Shengjie Li1 Received: 25 June 2019 / Accepted: 30 December 2019 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract The aim of this paper is to study second-order optimality conditions for a set-valued optimization problem with set criterion, where the order relation is induced by a set belong to a class of specific coradiant sets and is not necessarily a preorder. We introduce a notion of generalized second-order radial set, different from the classical second-order radial set, it is defined on a set, not on a point. Using the generalized second-order radial set, we introduce a new type of generalized second-order radial derivatives for set-valued maps and apply them to establish some necessary and sufficient conditions for the lower strict minimal solution of the set optimization problem. Keyword Second-order radial derivatives · Optimality conditions · Set optimization · Coradiant set
1 Introduction Set-valued optimization, which arise from the theory of vector optimization, have been investigated by many scholars due to its extensive applications in many fields, see [1–3]. There are two criteria to define the solution of a set-valued optimization problem: the vector criterion and the set criterion. The vector criterion, introduced in [4], consists of looking for efficient points of the union of the image sets of the feasible region under the set-valued objective map. The set criterion, also called set optimization, introduced by Kuroiwa [5], bases on some order relations among sets and consists of looking for minimal elements of the family of the image sets of the feasible region under the set-valued objective map. Therefore, the set criterion seems
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Shengjie Li [email protected] Bin Yao [email protected]
1
College of Mathematics and Statistics, Chongqing University, Chongqing 401331, China
2
College of Science, Shihezi University, Shihezi 832003, China
123
B. Yao, S. Li
to be more natural and interesting than the vector criterion, whenever one needs to consider preferences over sets. For further information about these two criteria, see [6–9]. It is well-known that optimality conditions is one of the most important topics in optimization problems. Since Aubin introduced the notion of the contingent derivatives of a set-valued map in [10], various notions of generalized derivatives, which are based on different kinds of tangency or linear approximations, have been introduced and applied to establish optimality conditions of set-valued optimization problems under the vector criterion. Among these generalized derivatives, radial derivatives, introduced by Taa [11], have received more attention than the others as the radial derivatives of a set carries global information about the set, see [12–18]. However, in order to establish the first-order or second-order optimality conditions of set optimization problems, we need to introduce new generalized derivatives of set-valued maps. In [19], Jahn us
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