参数约束优化问题的若干对偶以及灵敏性研究

项目名称： 参数约束优化问题的若干对偶以及灵敏性研究

项目编号： No.11301570

项目类型： 青年科学基金项目

立项/批准年度： 2014

项目学科： 数理科学和化学

项目作者： 孙祥凯

作者单位： 重庆工商大学

项目金额： 23万元

中文摘要： 本课题主要研究参数约束优化问题的若干对偶理论以及二阶灵敏性。借助共扼函数上图性质引入一些更弱的上图类正则性条件并用其刻画约束优化问题的强对偶、稳定强对偶、全对偶、稳定全对偶以及逆对偶；研究不确定DC优化问题以及不确定多目标优化问题的鲁棒对偶，特别是研究如何通过选择合适的不确定集将其不确定对偶问题的最优对应表示成计算上易处理的优化问题，并将其应用到数据分类、最佳逼近以及自动控制等领域，为解决这些实际问题提供技术保障和理论依据；研究参数向量变分不等式问题和参数向量平衡问题的间隙函数以及解集映射的二阶相依导数和二阶上图导数的具体二阶计算公式。本课题的研究不仅涉及到凸分析、非光滑分析、集值分析、多目标优化理论等多个学科的集成和综合应用，而且能够广泛应用到经济系统、通讯系统和国防系统中，为决策分析提供参考。

中文关键词： 参数约束优化；对偶性；最优性；灵敏性；

英文摘要： In this project, some duality and second order sensitivity for parametric constrained optimization problems are investigated. Firstly, by using the properties of the epigraph of the conjugated functions, we introduce some weaker constraint qualifications which completely characterize the strong duality, the stable strong duality, the stable total duality and the converse duality of constrained optimization problems. Secondly, we investigate robust duality results for DC and multiobjective optimization problems with data uncertainty. And we also provide some simple cases of the uncertain set such that the optimistic counterpart of the uncertain dual is computationally tractable. Moreover, we apply the robust duality results to many areas,such as data classification, best approximation problem and automatic control,providing technical support and theoretical basis to solve these practical problems. Finally, we investigate some explicit expressions of second order contingent derivatives and second order contingent epiderivatives for gap functions and solution mappings in parametric vector variational inequalities and parametric vector equilibrium problems. This research is an integration and comprehensive applications of multiple disciplines such as convex analysis, nonsmooth analysis, set value analysis, multiobje

英文关键词： Paramatric constrained optimization；Duality；Optimality；Sensitivity；