求解低秩密度矩阵约束最小二乘问题的优函数罚方法

本文应用优函数罚方法求解具有低秩密度矩阵约束的最小二乘问题. 首先用凸差方法处理非凸的低秩约束，并结合罚方法和优函数方法将原问题转化为一系列具有密度矩阵约束的凸优化问题，然后给出求解该优化问题的优函数罚方法，并对该方法进行收敛性分析. 之后，运用半光滑牛顿增广拉格朗日算法求解优函数罚方法的子问题. 最后，合成数据集和真实数据集上的数值结果表明了优函数罚方法有效地求解了具有低秩密度矩阵约束的最小二乘问题.

The majorized penalty algorithm for the least squares problem with the low rank density matrix constraint

In this paper, a majorized penalty algorithm was proposed to solve the least squares problem with the low rank density matrix constraint. We first used the difference of convex function to deal with the low rank constraint and then proposed the majorization approach to the penalized problem by solving a sequence of convex optimization without the rank constraint. Then the algorithm framework and convergence of the majorized penalty algorithm was given and analyzed. The subproblem was solved by a recently developed semismooth newton-based augmented lagrangian method. The experimental results demonstrated the efficiency of our approach on the least squares problem with the low rank density matrix constraint.