矩阵秩优化问题的一种分离算法

Rank optimization of matrix via splitting technique

具有线性约束的最小矩阵秩优化问题在控制、信号处理、系统识别等领域都有着广泛的应用。在矩阵优化问题中,矩阵的秩能够反应数据的稀疏性,但由于矩阵秩函数的非凸性,矩阵秩优化问题一般解决起来比较困难。目前,矩阵核范数的应用对于解决矩阵秩优化问题提供了有效的工具。具有线性约束的最小核范数问题为最小秩问题最紧的凸松弛问题,对于最小核范数问题,如今已存在大量的算法,而可以解决最小化2个下半连续凸函数之和这一类优化问题的Douglas-Rachford分离技巧也同样可以用于此类问题的研究,运用此类技巧得到的算法具有良好的稳健性、有效性和收敛性。

The linearly constrained matrix rank minimization problem is widely applicable in many fields such as control,signal processing and system identification.For matrix-valued data,the rank of a matrix is a good notion of sparsity.As it is a non-convex function,matrix rank is difficult to minimize in general.Recently,the nuclear norm was advocated to be used to solve the rank optimization.The tightest convex relaxation of the minimization rank problem is the linearly constrained nuclear norm minimization.At present there are many algorithms to solve it.The Douglas-Rachford splitting technique,which can solve the minimization of the sum of two lower semicontinuous convex functions,can also use in nuclear norm minimization.The algorithm based on this splitting technique is robust,effectively and convergent.