Abstract:

We say that a matrix X∈R^m×n is real (R, S) symmetric matrix if X=RXS, where R∈R^m×m and S∈R^n×n are real nontrivial involutions; thus R=R^-1≠±I_m, S=S^-1≠±I_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 A_i∈R^m×m, B_i∈R^n×n (i=1,2,…,q), C∈R^m×n, E∈R^p×m, F∈R^n×t and D∈R^p×t, find a (R, S) symmetric matrix X∈R^m×n that minimize||A_iXB_i-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.

Key words: Matrix inequality, Least-squares problem, (R,S) Symmetric matrix, Augmented Lagrangian method