一类线性约束矩阵不等式及其最小二乘问题

A CLASS OF LINEAR CONSTRAINED MATRIX INEQUALITY AND ITS LEAST SQUARES PROBLEM

本文主要考虑一类线性矩阵不等式及其最小二乘问题,它等价于相应的矩阵不等式最小非负偏差问题.之前相关文献提出了求解该类最小非负偏差问题的迭代方法,但该方法在每步迭代过程中需要精确求解一个约束最小二乘子问题,因此对规模较大的问题,整个迭代过程需要耗费巨大的计算量.为了提高计算效率,本文在现有算法的基础上,提出了一类修正迭代方法.该方法在每步迭代过程中利用有限步的矩阵型LSQR方法求解一个低维矩阵Krylov子空间上的约束最小二乘子问题,降低了整个迭代所需的计算量.进一步运用投影定理以及相关的矩阵分析方法证明了该修正算法的收敛性,最后通过数值例子验证了本文的理论结果以及算法的有效性.

A class of linear constrained matrix inequality and its least squares problem, which is equivalent to the corresponding matrix inequality smallest nonnegative deviation problem, is considered in this paper. Some related literatures have proposed an iteration method to solve the smallest nonnegative deviation problem, however, a great deal of computation for this algorithm is required for large scale problems because a constrained least squares subproblem should be solved exactly at each iteration. Based on the existing algorithm,a modified iteration method is proposed to improve the computational efficiency. In this iteration process, the whole required computation has been reduced by implementing the matrix form LSQR method to solve a constrained least squares subproblem over a Krylov subspace of low dimension in finite steps. Furthermore, the convergence of the modified algorithm is analyzed by using the projection theorem and related matrix analysis methods. Finally, several numerical experiments are presented to verify the theoretical results and the effectiveness of the iteration method.