集优化问题的最优性条件

Optimality Conditions for Set Optimization Problems

摘要: 集值优化问题是现代优化理论研究的重要分支，在工程设计、金融决策等领域有着广泛的应用。针对单值模型难以刻画复杂问题的局限性，本文以集优化方法为核心，在实分离拓扑线性空间框架下，基于集序关系${\ll }_{intC}^p$ 定义了集优化问题的解。通过引入连续仿射线性映射，构造拉格朗日集值映射与对偶集值映射，建立了对偶优化问题，明确了可行对与解的概念。本文借助锥的拓扑性质与序关系的传递性，推导出弱对偶定理及其推论，揭示了原问题极小解与对偶问题极大解的内在关联。本文结果完善了集序关系框架下的对偶理论，为带集值约束的优化问题求解提供了理论依据。

Abstract: Set-valued optimization is an important branch of modern optimization theory, with broad applications in fields such as engineering design and financial decision-making. In view of the limitations of single-valued models in describing complex problems, this paper adopts the set optimization approach as its core methodology. Within the framework of real separated topological linear spaces, and based on the set order relation ${\ll }_{intC}^p$ , we define solutions for the set optimization problem (SOP). By introducing continuous affine linear mappings, we construct the Lagrangian set-valued map and the dual set-valued map, establish the corresponding dual optimization problem (DSOP), and clarify the concepts of feasible pairs and solutions. Leveraging the topological properties of cones and the transitivity of the set order relation, we derive the weak duality theorem and its corollary, revealing the intrinsic relationship between minimal solutions of the primal problem and maximal solutions of the dual problem. The results obtained in this paper refine the duality theory within the framework of set order relations and provide a theoretical foundation for solving optimization problems with set-valued constraints.