The Approximate Solution And Optimization Calculation Of Cubic Regularisation Model

In this paper we discuss the approximate solving method for unconstrained optimization problems,on the basis of approximate solving method for the quadratic model, we discuss a kind of trust region problems for a cubic model. Trust region algorithm is one of the methods for solving unconstrained optimization. By using Taylor expansion to approximate the objective function, the problem of solving objective function is transformed into a problem of solving the trust region subproblem,in the process of solving the trust region subproblem,Shultz put forward dogleg path algorithm,to obtain the approximate solution of approximate function,such as Powell’s single dogleg algorithm,Dennis and Mei’s double dogleg algorithm.We use a cubic model to approximate the objective function, by using the dogleg path algorithm, we solve the trust region sub-problem approximately to obtain the minimum value of the objective function. The main idea of this dogleg path algorithm is:(1) When Hessian matrix or its approximate matrix is positive definite,we use single dogleg path algorithm or double dogleg path algorithm to solve the trust region sub-problem; (2) When Hessian matrix or its approximate matrix is non-positive definite,through the matrix decomposition of Hessian matrix or its approximate matrix,it produce an iteration direction, generating iteration point by using indefinite dogleg path algorithm.The adaptive cubic model improves the approximation accuracy of objective function, combine the dogleg path algorithm,it open up a new prospect of solving the cubic model approximately.This algorithm is easy to implement, and under the condition of conventional assumptions, we prove its good convergence properties, a lot of numerical experiments are conducted to compare this method with some existing methods, it is proved the effectiveness and reasonability of the algorithm in this paper.