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
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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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