稀疏矩阵锥约束优化问题的最优化理论与数值算法

Sparse optimization problems with matrix conic constraints have become a hot topic in numerical optimization. Reflecting structural features objectively in practical applications such as computational vision, statistical analysis, machine learning, data mining, image processing, this model has attracted much attention in the fields of mathematical programming and engineering. Therefore, it is of great importance to the study of theory and algorithms for sparse optimization problems with matrix conic constraints. Based on the varational analysis and perturbation analysis, this project is devote to the descriptions of tangent cone, normal cone and second-order tangent set for the epigraph of matrix spectral functions as well as the first-order and second-order optimality conditions of sparse optimization problems with matrix conic constraints. With the help of the recent achievements in stability analysis, we also study the necessary and sufficient condition for the existence of stable locally optimal solution as well as the Aubin property of the first-order optimal system. Under the guidance of the iteration-complexity theory, we design a two-phase algorithm for solving sparse optimization problems with matrix conic constraints, and establish the corresponding convergence results as well as the iteration-complexity. The aim of this project is to establish the optimality conditions and design numerical algorithms for sparse optimization problems with matrix conic constraints. We hope the results obtained will make a contribution to the theory and algorithms of matrix optimization.

带矩阵锥约束的稀疏优化问题已经成为数值最优化领域一个研究热点，由于该模型客观地刻画了计算视觉、统计分析、机器学习、数据挖掘、图像处理等实际应用的结构性特征，引起了优化界和工程界的广泛关注，因此深入研究其理论与算法有着十分重要的意义。本项目理论上以变分分析和最优化问题的扰动分析为基础，研究内容包括给出矩阵谱函数上图的切锥、法锥、二阶切集的描述，构建稀疏矩阵锥约束优化问题的一阶和二阶最优性条件。借助稳定性分析的最新研究成果，建立问题局部最优解处稳定点存在的充要条件，同时分析一阶最优系统的Aubin性质；在算法设计上，以迭代复杂度理论为指导，设计求解稀疏矩阵锥约束优化问题的两阶段算法，并对算法的收敛性和迭代复杂度进行分析。本项目旨在建立稀疏矩阵锥约束优化问题的最优化理论并设计实现数值算法，期望取得的结果对矩阵优化的理论与算法研究做出贡献。

本项目考虑带矩阵锥约束的稀疏优化问题，旨在研究该问题的最优性条件，稳定性分析和两阶段算法。由于新情况和新想法的不断产生，我们对研究计划进行了一定的调整。首先，本项目研究了一类非线性对称矩阵最大特征值函数复合优化问题，给出了问题的一阶和二阶最优性条件的刻画及对应的UV分解算法；其次，研究了一类结构化稀疏半正定矩阵二次规划问题的逆问题的非凸交替方向方法的收敛性和复杂度分析，再次，研究了一类结构化的非凸非光滑约束优化问题并设计了光滑化增广拉格朗日算法求解。以此同时，本项目刻画了圆锥集合的切锥、法锥和二阶切集的刻画，为进一步研究问题的稳定性奠定了基础，同时引出了对几类非对称锥理论问题的研究。针对信号重建问题，设计了基于光滑化l1范数的共轭梯度方法。