Study On The Iteration Methods Of The Matrix Equations Problems In Mixed Constraints

The linear and nonlinear matrix equation problems are the important research prob-lems in numerical algebra field.Both of them have many important practical applications in modern finance theory,systems engineering,optimization method,statistical analy-sis,stability theory,time series analysis,control theory and information theory.In this paper,the following kinds of problems are considered.Problem ??Given matrices A ? Rm×n,B ? Rn×p,C ?Rm×p,L?Rn×n,U? Rn×n and a real number ? ? 0,find a matrix X such that min 1/2||AXB-C||2 s.t.XT =X,L?X?U,Amin(X)???0,where ?min(X)denotes the minimum eigenvalue of the matrix X.Problem ??Given matrices A ? Rm×n,B?Rn×p,C?Rm×p,E? Rq×n,F ? Rn×t,D?Rq×t,find a matrix X such that min f(X)= 1/2||AXB-C||2 s.t.EXF?D,X?SRn×n.Problem ??Given matrices A,B,C ? Rm×n L1,U1 ? Rn×n,L2,U2? Rm×m and real numbers ?1,?2 ? 0,find the matrices X,Y such that min 1/2||AX+YB-C||2 s.t.XT =X,L1?X?U1,?min(X)??1?0,YT=Y,L2?Y?U2,?min(Y)??2?0.Problem ??Given matrices Q?SR+m×m,A ? Rmxmand the positive integer n,find a matrix X ? SR+ m×m such that X+A*X-nA= Q.In Chapter 2,the properties of the matrix function f(X)= ||AXB-C||2,and the existence and uniqueness of the solution of the Problem ? are discussed.The basic steps of using alternating projection algorithm to solve Problem ? and the numerical examples to illustrate the effectiveness of the method are given.In Chapter 3,based on the ideal of augmented Lagrange multiplier algorithm,the matrix form augmented Lagrange multiplier algorithms to solve Problem ? and Problem? are given,and the convergence results of the algorithm are proved.The numerical examples to illustrate the effectiveness of the method are given.In Chapter 4,based on the ideal of alternating direction multiplier algorithm,the matrix form alternating direction multiplier algorithm to solve Problem ? is given,and the convergence results of the algorithm are proved.The numerical examples to illustrate the effectiveness of the method are given.In Chapter 5,approximation alternating gradient iterative algorithm to solve Prob-lem ? is given,and the convergence results of the algorithm are proved.The numerical examples to illustrate the effectiveness of the method are given.In Chapter 6,alternating direction multiplier algorithm to solve Problem ? is giv-en,and the convergence results of the algorithm are proved firstly.Then,based on the general inner product theory,the direct method to solve subproblem in iteration is giv-en.Finally,the numerical examples to illustrate the effectiveness of the method are also given.In Chapter 7,Newton's iterative method to solve Problem ? is considered.When the symmetric positive semidefinite matrix Q satisfy matrix inequality 0<?=(n+1)||Q-1||n||A||2/1-n||Q-1||n+1||A||2<1-(n||Q-1||2?2)1/(n+2)/||Q-1||,the results that the matrix sequences {Xk}k=0 ?,X0 = Q generated by the iterative method are contained in the fixed closed ball B(X0,?)which include the only solution of the matrix equation,and that the matrix sequences {Xk}k=0 ? generated by the iterative method converges to the only solution of the matrix equation in the fixed closed ball B(X0,?)are proved.In addition,the error estimate of the approximate solution in the fixed closed ball B(X0,?)is presented.