Research On Algorithms For Low-rank And Sparse Matrix Optimization Problems

In this paper,we study the low-rank and sparse matrix optimization problems.In the first part,we consider the composite norms least squares problem with the l₁norm and the nuclear norm.We use a dual inexact symmetric Gauss-Seidel alternating direction method of multipliers(dsGS-ADMM)to solve the problem,and analyze the global convergence under certain assumptions.In the second part,we consider low-rank and sparse matrix optimization problems related to rank constraints and l₀norm.We use a two-stage algorithm to solve this type of problem:in the first stage,a good initial point is generated by solving the approxima-tion convex problem;in the second stage,we use the SCAD function to approximate the l₀norm and rank-constrained transformations to construct a DC programming.Then we use the sequential convex algorithm(DCA)to solve the constructed DC program-ming problem.For the convex subproblem of the second-stage,we use a dual inexact symmetric Gauss-Seidel alternating direction method of multipliers(dsGS-ADMM)to solve.Numerical experiments are performed on the two models to illustrate the stability and efficiency of the optimization algorithm in this paper.