矩阵不等式约束下矩阵方程最小二乘问题的增广Lagrangian方法  

Augmented Lagrangian Method for Matrix Equation Least-squares Problem Under a Matrix Inequality Constraint

作　　者：李姣芬[1] 宋丹丹[1] 周学林[2] 邢雨蒙 

机构地区：[1]桂林电子科技大学数学与计算科学学院,广西高校数据分析与计算重点实验室,广西桂林541004 [2]桂林电子科技大学教务处,广西桂林541004

出　　处：《数学物理学报（A辑）》2017年第3期562-576,共15页Acta Mathematica Scientia

基　　金：国家自然科学基金(11301107,11261014,11561015);广西自然科学基金(2016GXNSFAA380074,2016GXNSFFA380009)~~

摘　　要：称X∈R^(m×n)为实(R,S)对称矩阵,若满足X=RXS,其中R∈R^(m×m)和S∈R^(n×n)为非平凡实对合矩阵,即R=R^(-1)≠±I_m,S=S^(-1)≠±I_n.该文将优化理论中求凸集上光滑函数最小值的增广Lagrangian方法应用于求解矩阵不等式约束下实(R,S)对称矩阵最小二乘问题,即给定正整数m,n,p,t,q和矩阵A_i∈R^(m×m),B_i∈R^(n×n)(i=1,2,…,q),C∈R^(m×m),E∈R^(p×m),F∈R^(n×t)和D∈R^(p×t),求实(R,S)对称矩阵X∈R^(m×m)且在满足相容矩阵不等式EXF≥D约束下极小化‖∑_(i=1)~qA_iXB_i-C‖,其中EXF≥D表示矩阵EXF-D非负,‖·‖为Frobenius范数.该文给出求解问题的矩阵形式增广Lagrangian方法的迭代格式,并用数值算例验证该方法是可行且高效的.We say that a matrix X∈Rm×n is real （R, S） symmetric matrix if X = RXS, where R ∈Rm×m and S ∈Rn×n are real nontrivial involutions; thus R = R-1 ≠±Im, S = S-1 ≠ ±In. In this paper we apply the augmented Lagrangian method, for minimizing general smooth functions on convex sets in optimization theory, to solve the （R, S） symmetric matrix least squares problem under a linear inequality constraint. That is, given positive integers m,n,p,t,q, matrices Ai ∈ Rm×m, Bi ∈Rn×n （i = 1,2,-..,q）, C ∈m×n E ∈Rp×m F ∈Rn×t q and D ∈Rp×t find a （R, S） symmetric matrix X ∈Rm×n that minimize ｜｜q∑i=1 AiXBi - C｜｜ under matrix inequality constraint EXF 〉 D, where EXF ≥ D means that matrix EXF - D nonnegative. We present matrix-form iterative format basing on the augmented Lagrangian method to solve the proposed problem and give some numerical examples to show that the iterative method is feasible and effective.

关 键 词：矩阵不等式 最小二乘问题 实(R  S)对称矩阵 增广Lagrangian方法. 

分 类 号：O241.2[理学—计算数学]