Several Types Of Linear Constrained Matrix Inequality And Its Least Squares Problem

Linear constrainted matrix inequality and the corresponding least squares problem is one of the important research topics in the field of numerical algebra.They have important ap-plications in image reconstruction,the inverse problem of radiation therapy and the matrix optimization problems.In the paper,we studied systematically several kinds of linear constrained matrix inequal-ity and the least squares problems.Described as follows:Problem Ⅰ A ∈ Rm×n,B ∈ Rn×q,C ∈Rm×q are given matrices,the solution X∈S satisfies:AXB ≥ C or minf(X)=‖(C-AXB)+‖Problem Ⅱ A ∈ Rm×n,B ∈ n×pp,C ∈Rm×q,D ∈Rq×p,E ∈m×o are given matrices,the solutions(X,Y)∈ Rn×n × Rq×q satisfies:AXB + CYD>E or min f(X,Y)= ‖(E-AXB-CYD)+‖Problem III A ∈Rm×n,B ∈Rp×n,C ∈ Rm×m,D ∈ Rp×p are given matrices,the solution X ∈Rn×n satisfies:(AXAT,BXBT)≥(C,D)where‖·‖ is Frobenius norm and S denotes a matrix set with linear constraints in Rn×n.In the paper,we study systematically several kinds of linear constrained matrix inequality and the least squares problems in problem Ⅰ-Ⅲ and raise the characterization of the solutions by using the projection theorem and the polar decomposition in Hilbert space.On the basis of the existing algorithm,we propose a modified matrix iteration algorithm for solving problem I-III,and further the convergence analysis of this algorithm is given.Finally,we present several examples to show the correctness of the theoretical results and the feasibility of the iteration method.