In this paper, the multiple constraint least squares solution of Sylvester equation AX+XB=C is discussed. The necessary and sufficient conditions for the existence of solutions to the considered problem are given. Noting that the alternating direction method of multipliers (ADMM) is an one-step iterative method, a multi-step alternating direction method of multipliers (MSADMM) to solve the considered problem is proposed and some convergence results of the proposed algorithm are proved. Problems which should be studied in near future are listed. Numerical comparisons between MSADMM and ADMM are included.
2010 Mathematics subject Classiflcations. 15A39, 65F30.