Research On Numerical Algorithms For Solving Constrained Matrix Optimization Problems

Constraint matrix optimization is one of the important topics in the field of numerical algebra and nonlinear optimization.It has a wide range of applications in the fields of image processing,pattern recognition,machine learning and complex systems.In this paper,the theoretical and numerical algorithms for the following constrained matrix optimization problems are systematically studied.In Chapter 2,the matrix optimization problem in the unsupervised feature selection(?)1/2‖A-AXY‖F2,s.t.X≥0,Y≥0,XTX=Ip is studied,where X∈R+m×p,Y∈R+p×m represent feature weight matrix and coefficient matrix respectively.Firstly,the problem is reconstructed to a non-negative constrained matrix optimization problem using relaxation technique,and then the problem is solved with the aid of Lagrange multiplier to design the modified multiplicative update algorithm,and the convergence analysis is given.Finally,numerical experiments show that the method is feasible and effective.In Chapter 3,the matrix optimization problem in self-representation feature selection(?)‖X-XW‖2,1+λΩ(W),s.t.W≥0 is studied,where X∈Rn×m is a high-dimension matrix and W ∈ Rm×m is a feature weight matrix.First,an optimization model of self-representation feature selection is proposed,thus the problem is transformed into unconstrained matrix optimization problem,and the new problem can be solved by the multiplicative update algorithm.Finally,the numerical experiment is carried out on the data set.Compared with the traditional algorithm,the numerical effect is better.In Chapter 4,the complex system matrix optimization problem(?)‖X-SΓ‖F2,ΩΓ={Γ∈RK×T|(?)Γk,t=1,Γk,t≥0}is studied,where X∈Rn×T is a data matrix,S={S1,S2,…,SK} stands for K different discrete states and Γ∈ RK×T stands for the probability matrix.Based on the Karush-Kuhn-Tucker condition of the problem,the proximal alternative nonnegative least square method is designed to solve the problem.Then the optimality condition of the unconstrained problem and the quadratic programming algorithm are used to solve the two sub-problems respectively.Finally,the convergence of the algorithm is analyzed.The numerical experiments show that the algorithm is effective.