参数约束优化问题的若干对偶以及灵敏性研究

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, multiobjective optimization theory. Also,it can be widely applied to economic systems, communication systems and defense systems, which provide a reference for decision analysis.

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。

本课题主要研究了参数约束优化问题的若干对偶性、最优性以及灵敏性。借助共轭函数的上图性质引入了一些更弱的正则性条件，并用其等价刻画了DC无限规划问题、复合优化问题以及带有复合函数的DC规划问题的强对偶、全对偶、稳定强对偶和稳定全对偶等对偶性理论，同时借助标量化方法研究了多目标分式优化问题的序列最优性条件；分析了经典的次微分计算法则，提出了几类新的鲁棒型次微分约束品性，并用其等价刻画了不确定性凸优化问题与不确定分式优化问题的鲁棒最优解、鲁棒最优解集以及鲁棒对偶性，同时借助标量化技巧研究了不确定多目标分式优化问题的鲁棒有效解以及鲁棒有效解集；在不借助任何标量化技巧的前提下，研究了广义向量变分不等式的间隙函数与正则间隙函数，并首次在不借助投影算子方法的前提下，刻画了广义向量变分不等式间隙函数的误差界，阐明了所用方法在计算上更具有可行性；借助相依导数和Hadamard方向导数研究了一类集值映射的可微性性质，随后借助集值映射的间隙函数，研究了弱向量变分不等式问题的灵敏性性质。本课题的研究不仅具有重要的理论意义，而且在工程设计、图像恢复、数据拟合、最优控制等领域有着广泛的应用前景。