约束向量优化问题若干对偶的一致性及其应用研究

In this project, the unified property of some dualities for constrained vector optimization problems with applications to the penalty function method will be investigated. Considering the highly unified property of the image space analysis approach following classical Lagrange ideas to deal with related dual problems, we introduce a class of regular weak separation functions and establish a unified duality scheme in the image space of the constrained vector optimization problem by using the separation of sets. On the one hand, by means of appropriate assumptions, we make a detailed discussion on the relationships among the generalized Lagrange multiplier and saddle point, the lower semicontinuity of the canonical perturbation function at the origin, the regular weak separation of corresponding sets in the image space and the zero duality gap property in a unified frame. On the other hand, for some special classes of regular weak separation functions, including the separable functions with respect to the object function and constrained conditions, the augmented Lagrange functions and the nonlinear Lagrange functions, we further verify the validity of related hypothesis and give a unified interpretation of corresponding duality schemes. Simultaneously, we will apply the obtained results to the penalty function method for constrained vector optimization problems, and establish some geometric interpretations and convergence analysis based on the separation of sets.

本课题主要研究约束向量优化问题若干对偶的一致性及其在罚函数法中的应用。鉴于像空间分析方法在沿用Lagrange思想处理相关对偶问题的高度一致性，拟在约束向量优化问题的像空间中，从集合分离的角度引入正则弱分离函数类并建立统一的对偶模型。一方面，借助适当的假设条件，在统一框架下详细讨论广义Lagrange乘子和鞍点、正规扰动函数在零点处的下半连续性以及像空间中相应集合的正则弱分离性与零对偶间隙性质之间的联系。另一方面，针对特殊的正则弱分离函数类，包括目标函数与约束条件可分离的函数类、增广Lagrange函数类和非线性Lagrange函数类，进一步验证相关假设条件的合理性并给出相应对偶模型的一致性解释。同时，将相关研究成果应用到约束向量优化问题的罚函数法中，并从集合分离的角度，给出相应的几何解释以及收敛性分析。

本课题主要研究了约束向量优化问题若干对偶的一致性及其在罚函数法中的应用。鉴于像空间分析方法在沿用Lagrange思想处理相关对偶问题的高度一致性，我们在约束向量优化问题的像空间中，从集合分离的角度引入了向量值形式的正则弱分离函数类并建立了统一的对偶模型。一方面，借助适当的假设条件，在统一框架下详细讨论广义Lagrange乘子和鞍点以及像空间中相应集合的正则弱分离性与零对偶间隙性质之间的联系。另一方面，针对特殊的正则弱分离函数类，包括目标函数与约束条件可分离的函数类、增广Lagrange函数类和非线性Lagrange函数类，进一步验证了相关假设条件的合理性并给出了相应对偶模型的一致性解释。同时，将相关研究成果应用到了约束向量优化问题的罚函数法中，在局部像正则性条件下建立了精确罚函数的存在性。