The Second-order Subdifferential Characterization of the Strong Efficient Solution of Set-valued Optimization Problems
1. College of Mathematics and Statistics, Chongqing Jiaotong University, Chongqing 400074, China
2. Key Laboratory of Complex Systems Optimization and Intelligent Control of Chongqing Municipal Education Commission, Chongqing Jiaotong University, Chongqing 400074, China
| Abstract: | This paper discusses the optimality conditions for the strong efficient solutions of set-valued optimization problems. Firstly, by using the second-order weak subdifferential of set-valued mappings and the relationship between the strong efficient solutions of set-valued optimization problems and the optimal solutions of composite optimization problems, one establishes the optimality necessary and sufficient conditions of the strong efficient solutions for unconstrained set-valued optimization problems. Meanwhile, by using the second-order weak subdifferential of set- valued mappings, several optimality necessary and sufficient conditions of unconstrained composite optimization problems are obtained under the assumption of generalized cone sub-convexlikeness. Secondly, one introduces a class of saddle points for constrained set-valued optimization problems. Then, several sufficient optimality conditions for strong efficient solutions of constrained set-valued optimization problems are established under the assumptions of generalized cone sub-convexlikeness and saddle points for constrained set-valued optimization problems. Additionally, by using the second-order weak subdifferential of set-valued mappings, the sufficient optimality conditions for strong efficient solutions of constrained set-valued optimization problems are established under the assumption of generalized cone sub-convexlikeness. Finally, two optimality sufficient conditions for the optimal solutions of the constrained composite optimization problem are obtained under the assumption of generalized cone sub-convexlikeness. |
| Keywords: | Set-value Optimization Problems; Strong Efficient Solutions; Second-order Subdifferentials; Optimality Conditions |
| DOI: | 10.57237/j.wjms.2026.01.003 |
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Received: 21 January 2026
Accepted: 10 February 2026
Published Online: 26 February 2026
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