Alternating Direction Methods For Nuclear Norm Minimization

There are a lot of problems from practical applications,such as restoring all the correct information from incomplete or polluted information,whose solution is generally sparse or low-rank. Such problems can be transformed to nuclear norm minimization problems with certain constraints.This dissertation is devoted to the study of two kinds of nuclear norm minimization prob-lem,one is with affine constraint and positive semidefinite constraint,another is under the affine constraint in S".Firstly,we use alternating direction method to solve these two kinds of optimiza-tion problems.In the subproblems which are obtained by applying alternating direction methods to nuclear norm minimization,we give the analytical solution of every subproblem through prox-imal mapping of nuclear norm on S+n and Sn respectively and projection onto a closed convex set.Then,we give the convergence theory of alternating direction method.Finally,we demonstrate some numerical results of matrix complete problem and general nuclear norm minimization problem.