Numerical Methods For Several Matrix Optimization Problems

Matrix optimization is one of the most important topics in the field of numerical algebra and numerical optimization.It is widely used in the signal processing,image processing,machine learning,data analysis,financial engineering,quantum computing,systems and control theory,high-dimensional statistics and other scientific and engineering fields.This dissertation systematically studies the following numerical methods of matrix optimization problems.In order to improve the precision of clustering,the Q-weighted norm is applied to the nonnegative matrix factorization problem(?)Based on the additive representation method of Q-weighted norm,this problem is transformed into a class of trace function minimization problem.The bivariate nonlinear conjugate gradient method is constructed to solve this problem,the convergence analysis of the algorithm is given,and numerical examples are used to verify the feasibility and effectiveness of the new algorithm,especially the example in the clustering analysis proves that the new algorithm is more accurate than the EM-WNMF algorithm and the ANLS-WNMF algorithm.Consider the nonnegative matrix tri-factorisation problem(?)Firstly,by using the property of weighted norm,this problem is transformed into a matrix optimization problem,and then the necessary condition for the existence of a solution is given.We construct the proximal alternating nonnegative least squares method to solve the problem and give the convergence theorem.In order to improve the convergence speed,the enhanced line search technology is applied to the proximal alternating nonnegative least squares method for acceleration.Numerical experiments show that the proximal alternating nonnegative least squares method and its acceleration method converge faster and have higher clustering accuracy than the traditional WNMTF algorithm.Consider the nonnegative matrix multi-factorization problem(?)Firstly,the problem is transformed into the trace function minimization problem.Based on the KKT condition,the multiplicative update method is designed to solve this problem.The convergence of the new method is given by introducing an auxiliary functions.The numerical results show that this algorithm is feasible,especially,the convergence rate is faster than the alternating nonnegative least squares method.A class of matrix convex feasible problem in the quantum computation is studied.Namely,find an mn?mn positive semidefinite definite matrixX=(X _ij)_i,j=1,2,,n,whereX_ij?C^m?mis the block matrices,satisfiesⁿ_i,j?₌₁(A _l)_ijX_ij=B _l,l=1,2,,k,tr(X _ij)=d_ij,i,j=1,2,,n,where(?)Consider the optimal matrix approximation problem(?)where(?).This problem is converted into the problem of finding the projection of a given point onto the three closed convex sets.The projection formulas of a points into the three closed convex sets are given,and then the Dykstra's alternating projection method is designed to solve it.Numerical experiments show that this method is feasible and effective.