Research On The Matrix Valued Function Extremum Problem

Matrix optimization can be divided into two classes.One is the problem that the objective function value is a matrix,that is,the Matrix valued function extremum problem(referred to as MVFE),there are important applications in various fields such as the operator equation,the constraint matrix equation,the matrix inequality,the fields of least squares problem,etc.The other is the optimization problem with matrix as the variable,such as semi-definite programming,lowest rank correlation coefficient matrix problem,eigenvalue optimization problem,second-order cone constrained optimization problem,etc.The semi-definite planning problem is its typical representative.The research on these issues has been fruitful in recent years.This paper mainly studies the matrix valued function extremum problem.Since each element of the matrix is a function in the matrix value function,the matrix valued function extremum problem is same as a multi-objective optimization problem.The introduction to the Chapter 1 of this paper gives a brief introduction to the development of multiple problems such as multi-objective optimization,matrix theory,and matrix optimization.The Chapter 2 gives the definition of matrix valued function,and discusses the analytical properties of matrix valued functions related to this paper.The Chapter 3 gives several order relations,as well as the definition of effective matrix,weak effective matrix,weighted effective matrix and norm optimal matrix under order relationship.The Chapter 4firstly describes some problems about the generalized inverse matrix of matrices,and then gives the definition of the semi-negative(weak)efficient solutions,weighted effective solutions,?degree efficient solutions and norm optimal solutions under various specific order relations that the matrix valued functions extremal problems,and study some properties related to these solutions.In the last section of Chapter 4,an example of the application of the matrix valued function extremum problem is given,namely its application in the least squares related problem,and some conclusions about the matrix equationB~T(A-BX)0=are given.