In the paper, the authors discussed the numerical method solving generalized Sylvester equation least square problems with the nuclear norm and spectral norm:
,where X ∈ S is a closed convex set. They used inexact alternating direction method in combination with threshold algorithm, Moreau - Yosida regularization algorithm, spectrum projection algorithm,LSQR algorithm and SPG algorithm. Based on, we introduce a new variable and use the alternating direction method to simplify the algorithm. Each subproblem can be solved exactly. More important, each variable has its own explicit solution expression. We prove the convergence of the proposed algorithm. The numerical tests show that the improved algorithm can be improved greatly in both time and iteration.
. AN IMPROVED ALGORITHM FOR LEAST SQUARES PROBLEM OF GENERALIZED SYLVESTER EQUATION UNDER NUCLEAR NORM AND SPECTRAL NORM[J]. Mathematica Numerica Sinica, 2018, 40(4): 387-401.
Lei Y, Liao A P. A minimal residual algorithm for the inconsistent matrix equation AXB=C over symmetrices[J]. Applied Mathematricx and Computation, 2007, 188:499-513.
Michael NG K, Wang F, Yuan X M. Inexact alternating direction methods for image recovery[J]. SIAM Journal on Scientific Computing, 2011, 33:1643-1668.
Peng Y X, Hu X Y, Zhang L. An iteration method for the symmetric solutions and the optimal approximation solution of the matrix equation AXB=C[J]. Applied Mathematics and Computation, 2005, 160:763-777.
Yang J F. Yuan X M. Linearized augmented Lagrangian and alternating direction methods for nuclear norm minimization[J]. Mathematrics of Computation, 2012, 82:301-329.
Li J F, Hu X Y, Zhang L. Dykstra's algorithm for constrained least-squares doubly symmetric matrix problems[J]. Theoretical Computer Science, 2010, 411:2818-2826.
Han D R, Yuan X M. Local linear convergence of the alternating direction method of multipliers for quadratic programs[J]. SIAM Journal on Numerical Analysis, 2013, 51:3446-3457.
Li Q N. Alternating direction method for a class of constrained matrix approximation problems[J]. Pacific Journal of Optimization, 2012, 8:765-778.
Brigin E G, Martinezand J M, Raydan M. Inexact Spectral Projected Gradient methods on convex sets[J]. SIMA Journal on Numerical Analysis, 2003, 23:539-559.
Ding F, Chen T W. On iterative solutions of general coupled matrix equations[J]. SIAM Journal on Control and Optimization, 2006, 44:2269-2284.
Bouhamidi A, Jbilou K, Raydan M. Convex constrained optimization for large-scale generalized Sylvester equations[J]. Computational Optimization and Applications, 2011, 48:233-253.
Boyd S, Parikh N, Chu E, Peleato B, Eckstein J. Distributed Optimization and Statistical Learning via the Alternating Direction Method of Multipliers[J]. Foundations and Trends in Machine Learning[J]. 2011, 3:1-122.
He B S, Tao M and Yuan X M. Alternating direction method with Gaussian back substitution for separable convex programming[J]. SIAM Journal Optimization[J]. 2012, 22:313-340.