矫正低秩相关系数矩阵的松弛序列凸近似方法  

A relaxed SCA approach for calibrating low rank correlation matrix problem

作　　者：王博[1] 张立卫[1] 初丽[1,2] 

机构地区：[1]大连理工大学数学科学学院,大连116024 [2]大连理工大学城市学院,大连116600

出　　处：《中国科学：数学》2015年第4期411-426,共16页Scientia Sinica：Mathematica

基　　金：国家自然科学基金(批准号:11071029和91130007)资助项目

摘　　要：本文主要讨论带有秩约束以及简单上下界约束的相关系数矩阵矫正问题的求解方法.该问题可以写成一个含有DC(两个凸函数之差)约束的优化问题,于是考虑利用求解DC优化问题的序列凸近似(SCA)方法求解.然而对本文讨论的问题,经典的序列凸近似方法收敛所需的约束规范不成立,于是,本文提出一种松弛的序列凸近似方法.本文证明当松弛参数趋于零时,松弛的DC问题的稳定点趋于原问题的稳定点.另一方面,可以利用序列凸近似方法求解松弛的DC问题.可以证明,序列凸近似方法生成的一系列凸子问题的解的聚点就是该松弛DC问题的稳定点.数值实验验证了该方法的有效性.In this paper, we consider a class of rank constrained correlation matrix calibrating problems with simple upper and lower bounds. This problem can be reformulated into a DC（difference of convex） constrained problem, thus sequential convex approximation（SCA） type approaches can be considered. However, classical SCA approach cannot be directly applied since the constraint qualification needed in convergence theorem does not hold for this problem. This motivates us to develop a relaxed SCA approach. We prove that for relaxed DC problems, the sets of their stationary points converge to the set of stationary points of the original problem as the relaxation parameter approaching to zero. For each relaxed DC problem, we apply the SCA approach to generatea sequence of convex subproblems. We show that all cluster points of optimal solutions of these subproblems are stationary points of the relaxed DC problem. Numerical results verify the efficiency of our approach.

关 键 词：DC优化 相关系数矩阵 低秩问题 

分 类 号：O151.21[理学—数学]