The Second-order Subdifferential Characterization of the Strong Efficient Solution of Set-valued Optimization Problems

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. 主要讨论集值优化问题强有效解的最优性条件。首先，借助于集值映射的二阶弱次微分，利用集合优化问题强有效解与复合优化问题最优解的关系，在广义锥次似凸假设条件下建立了无约束集值优化问题强有效解的最优性必要条件和充分条件。同时，利用集值映射的二阶弱次微分，在广义锥次似凸假设条件下也获得了无约束复合优化问题的几个最优性必要条件和充分条件。其次，引入了约束集值优化问题的一类鞍点，并在广义锥次似凸和约束集值优化问题鞍点的假设条件下，建立了约束集值优化问题强有效解的最优性充分条件。另外，借助于集值映射的二阶弱次微分，在广义锥次似凸假设条件下，建立了约束集值优化问题强有效解的最优性充分条件。最后，在广义锥次似凸假设条件下，获得了约束复合优化问题最优解的2个最优性充分条件。