基于二阶变分分析的非凸优化问题的扰动分析

Perturbation analysis is an important part in optimization theory, which has close relations with bilevel programs and numerical algorithm analysis. The literature on perturbation of twice continuously differentiable nonlinear optimization problems, under ordinary constraint qualifications, is enormous, but the perturbation for many non-convex optimization problems which do not satisfy such constraint qualifications or whose problem functions are not twice continuously differentiable remians to be investigated. This project will develop the second-order variational analysis; Based on the second-order variational analysis, this project will study the perturbation theory for several important non-convex optimization problems. The contents in this project include: 1) development of the second-order variational analysis, consisting of calculating second-order subdifferentials for several important nonsmooth extended real-valued functions and second-order tangent sets of several complementarity sets, etc; 2)analyzing upper Lipschitz continuity properties of KKT solutions for non-convex cone constrained optimization and outer semi-continuity properties of stationary points for MPECs; 3)second-order optimality conditions and perturbation analysis for the optimization problems whose functions have semi-smooth derivatives; 4)optimality theory for conic MPEC problems including second-order necessary and second-order sufficient optimality conditions; 5)the perturbation analysis for conic MPEC problems. The study on these topics will make important contributions to the research on the perturbation theory for non-convex optimization.

扰动分析是最优化理论的重要组成部分，它与双层规划和数值算法分析都有密切关系。已有大量文献研究了在通常约束规范下，二次连续可微的非线性优化问题的扰动性分析，但很多重要的不满足这些约束规范以及问题函数并非二次连续可微的非凸约束优化问题的扰动性分析还有待于深入研究。本项目将发展二阶变分分析，并基于二阶变分分析研究若干重要的非凸优化问题的扰动性理论。研究的内容有：1)发展二阶变分分析，包括计算若干重要非光滑增广实值函数的二阶次微分与若干互补集合的二阶切集，等等； 2)分析非凸锥约束优化的KKT解的上Lipschitz连续性和MPEC稳定点的外半连续性；3)问题函数具有半光滑导数的优化问题的二阶最优性条件与这类问题的扰动分析；4)包括二阶必要性与二阶充分性最优条件在内的均衡锥约束优化问题的最优性理论；5)均衡锥约束优化的扰动性分析。这些专题的研究将对非凸最优化的扰动分析的理论研究做出重要贡献。

本项目研究内容包括包括：锥约束优化的Karush-Kuhn-Tucker解的稳定性，分布鲁棒优化若干问题的稳定性，结构凸优化半邻近方法的研究,等等。取得的成果具体描述为.1. 证明了非凸半定规划KKT系统孤立平稳性的刻画定理，即KKT系统的孤立平稳性等价于严格Robinson约束规范与二阶充分性条件;.2. 同样证明了非凸二阶锥优化KKT系统的孤立平稳性的刻画定理，即KKT系统的孤立平稳性等价于严格Robinson约束规范与二阶充分性条件;.3. 解决了一类(二次光滑简约的)锥优化问题稳健孤立平稳性的刻画，即KKT系统的稳健孤立平稳性等价于严格Robinson约束规范与二阶充分性条件..4. 证明了结构凸优化问题的semi-proximal乘子交替方法的线性收敛率。.5. 提出求解复合凸优化问题的半光滑Newton方法,收敛性分析用到关于邻近映射的一阶变分分析,用于求解重要的实际问题,取得非常好的数值结果..6. 对一般的DC优化问题，给出基于次梯度的序列凸优化方法，证明了方法的收敛性质并讨论了应用。.7. 用稳定性理论,在分布鲁棒优化稳定性分析方面做出几项有理论价值的工作,包括分布机会约束鲁棒优化问题的收敛性分析和随机拟变分不等式的定量稳定性分析, 等等..8. 对矩约束的分布鲁棒优化问题，以概率测度为参数，给出最优解映射的定量的稳定性分析。