Research On Algorithms For Low Rank Matrix Optimization Problem

In this paper,we propose two efficient algorithms for solving the low rank matrix optimization problem.The convergence analysis of the proposed methods is given as well.Numerical results show the efficiency of the proposed algorithms.In the first chapter,we introduce the relative models of low rank matrix optimization problem,the current research situation,and the main work of this paper.In chapter 2,some preliminary knowledge used in this paper is introduced.In chapter 3,we propose an efficient penalty decomposition?PD?method to solve the nuclear norm minimization problem with7)₁norm fidelity term.This al-gorithm transforms a hard constrained optimization problem into two easy penalty subproblems.Then the block coordinate descent method?BCD?is used to solve each subproblem alternatively and the optimal solution of the original problem can be obtained.In chapter 4,we propose an alternating direction method of multiplier-s?ADMM?to solve the nuclear norm square root model.By introducing an aux-iliary variable,the algorithm takes the original problem into a variable separable problem,then solves the subproblems alternatively with respect to different vari-ables and both subproblems have explicit solutions.Numerical results illustrate that the nuclear norm square root model can effectively solve the high dimensional matrix completion problem with unknown noise level.In chapter 5,we give a summary of this paper and put forward some problems which need further research in the future.