Iterative methods for solving consistent or inconsistent matrix inequality AXB⩾CAXB⩾C with linear constraints

A matrix iteration algorithm was proposed by Peng (2012) for solving unconstrained matrix inequality AXB⩾CAXB⩾C, and it can be amplified to solve the matrix inequality with various linear constraints. However, a great deal of computation for this algorithm is required because a least squares subproblem with or without constraints should be solved accurately at each iteration. In this paper, we present a modified iteration algorithm, which is a generalization of Peng’s algorithm. In this iteration process, the matrix-form LSQR method is implemented to determine an approximate solution of each least squares subproblem with less computational effort. The convergence of this algorithm is analyzed and several numerical examples are presented to illustrate the efficiency of our algorithm.