结构低秩矩阵优化问题在统计、控制与系统辨识、信号与图像处理、机器学习、以及组合优化等诸多领域中有着广泛应用．本课题拟从结构低秩矩阵优化问题的等价MPEC入手，通过松弛其广义互补约束诱导的精确罚来开展该类非凸非光滑优化问题的多阶段凸松弛法研究，包括(1)研究将MPEC的广义互补约束罚到目标函数上所诱导的罚问题在全局意义下的精确性；(2)将MPEC的精确罚问题松弛为系列易处理的非光滑凸矩阵优化问题，设计多阶段凸松弛方法；(3)研究每阶段凸松弛问题的(近似)最优解与真实解(或原问题最优解)的误差和近似秩界，分析多阶段凸松弛方法的收敛速率；(4)设计求解凸松弛问题的有效收敛算法，研究其收敛速率；(5)编写多阶段凸松弛算法的程序代码，进行数值试验．该研究成果将为结构低秩矩阵优化问题和可分离非光滑凸矩阵优化问题提供有效的计算工具，丰富低秩矩阵优化和非光滑凸矩阵优化的理论，具有重要理论意义和应用价值．

Structured low-rank matrix optimization problems have a wide range of applications in a diverse set of fields such as statistics, control and system identification, signal and image processing, machine learning, combinatorial optimization, and so on. In this project, we conduct the research for the multi-stage convex relaxation approach to this class of nonconvex and nonsmooth optimization problems, starting with their equivalent MPECs and making suitable relaxations for the exact penalty problems yielded by the generalized complementarity constraint of the MPECs. It includes the following several aspects: (1) to study the exactness of the penalty problems yielded by penalizing the generalized complementarity constraint of MPECs to the objective, in the sense that the global optimal solution sets of the penalty problems coincide with those of the MPECs; (2) to design the multi-stage convex relaxation approach by relaxing the exact penalty problems of the MPECs into a sequence of tractable nonsmooth convex matrix optimization problems; (3) to study the error and approximate rank bounds for estimating the distance from the (approximate) optimal solution of convex relaxation problem in each stage to the true solution (or a global optimal solution of the original problem), and analyze the convergence rate of the multi-stage convex relaxation approach; (4) to design effective convergent algorithms for solving convex relaxation problems, and study the convergence rate of algorithms; (5) to write the codes for the multi-stage convex relaxation algorithms, and test the performance of new algorithms via numerical experiments. The research results will provide effective computational tools for structured low-rank matrix optimization problems and separable nonsmooth convex matrix optimization problems, and enrich the theory of low-rank matrix optimization and nonsmooth convex matrix optimization, having an important theoretical and practical significance.