Dykstra’s Algorithm for the Optimal Approximate Symmetric Positive Semidefinite Solution of a Class of Matrix Equations

Chunmei Li,Zhuling Jiang,Xuefeng Duan

doi:10.4236/alamt.2016.61001

Abstract

Dykstra’s alternating projection algorithm was proposed to treat the problem of finding the projection of a given point onto the intersection of some closed convex sets. In this paper, we first apply Dykstra’s alternating projection algorithm to compute the optimal approximate symmetric positive semidefinite solution of the matrix equations AXB = E, CXD = F. If we choose the initial iterative matrix X0 = 0, the least Frobenius norm symmetric positive semidefinite solution of these matrix equations is obtained. A numerical example shows that the new algorithm is feasible and effective.

Highlights

Throughout this paper, we use Rm×n and SPn×n to stand for the set of m × n real matrices and n × n symmetric positive semidefinite matrices, respectively
We propose a new algorithm to compute the optimal approximate symmetric positive semidefinite solution of Equation (1.3)
C1,C2,Cn, Dykstra [18] proposed Dykstra’s alternating projection algorithm which can be stated as follows

Summary

Introduction

Throughout this paper, we use Rm×n and SPn×n to stand for the set of m × n real matrices and n × n symmetric positive semidefinite matrices, respectively. (2016) Dykstra’s Algorithm for the Optimal Approximate Symmetric Positive Semidefinite Solution of a Class of Matrix Equations. If X = 0, the solution Xof Problem I is just the least Frobenius norm symmetric positive semidefinite solution of the matrix equations (1.3). Dykstra’s alternating projection algorithm was proposed by Dykstra [18] to treat the problem of finding the projection of a given point onto the intersection of some closed convex sets. We propose a new algorithm to compute the optimal approximate symmetric positive semidefinite solution of Equation (1.3). We state Problem I as the minimization of a convex quadratic function over the intersection of three closed convex sets in the vector space Rn×n From this point of view, Problem I can be solved by the Dykstra’s alternating projection algorithm. We use a numerical example to show that the new algorithm is feasible and effective

Dykstra’s Algorithm for Solving Problem I

Numerical Experiments

Conclusion