Research On Indefinite Region Algorithm Based On Differential Equation Model

Trust region algorithm has drawn the comprehensive attention of the optimize researchers owing to its posedness and global convergence properties.As a sort of siginificant numerical methods for solving unconstrained optimization questions,the trust region methods have been one of the discussion issues on the way of non-linear programming problems.Further,the solution of trust region subproblem is the crucial part of effective implemention of trust region methods,and it is the way to solve trust region subproblem that affects stability and convergence of the implemention of the algorithm.The trust region algorithm can be transformed by mading a new model.For example,the quadratic model,conic model,novel conic model and tensor model.Amongst the models,the quadratic model is of the significance owing to its brief form and convenience when calculate.It is the differential equation model that has been widely used which belongs to quadratic models.At present,most of the algorithms are based on the definite trust region subproblems,and the study of indefinite trust region subproblems is less.Basing on the trust region subproblem with differential equation models,we analyze the algorithm to solve the trust region subproblem in which the Hessian martrix is indefinite,and develop a new method which can extend present methods,which consummate the method to solve trust region subproblems.Modifying the indefinite matrix takes great part in solving the indefinite trust region subproblems,which are significant on the process of converting the indefinite trust region subproblems into definite subproblems.This paper outlines the Bunch-Parlett decomposition and modified-Cholesky decomposition method,which are efficient to modify indefinite matrix into definite matrix.Basing on Adams in fourth-order method,Heun in third-order method,the piecewise cubic Hermite interpolation method and the explicit Euler method,which is used widely in address definite matrix trust region subproblems,this paper puts forward four kinds of amendatory differential equation model of trust region subproblem algorithms.Numerical results show that the four algorithms are better than some methods which is used to solve the indefinite trust region subproblems.In addition,this paper also demonstrates the fitness of the modified Heun in third-order algorithm.