Alternating Direction Method For Matrix Nuclear Norm Minimization

In this thesis, we propose two alternating direction methods for solving the matrixnuclear norm minimization problem. Global convergence is analyzed. Numerical exper-iments are also reported, which illustrate that the proposed methods are efcient andpromising.In the frst chapter, we introduce some preliminaries of the matrix nuclear normproblem including problems’ formulations, developments and some existing methods. Webriefy review the basic knowledge of optimization and list some important notations,symbols and defnitions which used in the thesis.In chapter2, we propose an alternating direction method for linear-constrained ma-trix nuclear norm minimization. At each iteration, the method involves a singular valuedecomposition and a linear system. In order to broaden the utility of the method, wesolve the resulting linear system inexactly by the Barzilai-Borwein method. Moreover, theproposed method is extended to solve the linear inequality constrained problem and theregularized least squares problem. The numerical experiments which show the efciencyare also reported, which illustrate that the proposed method is competitive with FPCA.In chapter3, we improve the method which given in the previous chapter by usingconjugate gradient method to solve the subproblem of linear systems. Due to the lowerstorage and simple of conjugate gradient algorithm, the proposed method improve theperformance. Moreover, numerical comparisons illustrate that the proposed method isvery efcient.In chapter4, we give a summary of this thesis and list some further research topics.