摘要：
称X∈Rm×n为实(R,S)对称矩阵,若满足X=RXS,其中R∈Rm×m和S∈Rn×n为非平凡实对合矩阵,即R=R-1≠±Im,S=S-1≠±In.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Ai∈Rm×m,Bi∈Rn×n(i=1,2,…,q),C∈Rm×n,E∈Rp×m,F∈Rn×t和D∈Rp×t,求实(R,S)对称矩阵X∈Rm×n且在满足相容矩阵不等式EXF≥D约束下极小化‖q∑i=1AiXBi-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 ≠ ±n.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 ∈ Rm×n,E ∈ Rp×m,F ∈ Rn×t and D ∈ Rp×t.find a (R,S) symmetric matrix X ∈ Rm×n that minimize ‖q∑i=1AiXBi-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.
