CONVERGENCE PROPERTIES OF DFP METHOD WITH MODIFIED QUASI-NEWTON EQUATIONS

1 IntroductionWe know that in order to obtain a superlinearly convergent method it is necessary to approximate the Newton step asymptotically (see ll). How can we do this without actually evaluatingthe Hessian matrix by ally approximate to the Hessian matrix at every iteration? The answerwas discovered by Davidonl2] and was subsequently developed and popularized by Fletcher andPowell3l. It consists of starting with any approximation to the Hessian matrir, and at eachiteration, updating th…

CONVERGENCE PROPERTIES OF DFP METHOD WITH MODIFIED QUASI-NEWTON EQUATIONS

Quasi-Newton (QN) equation plays a core role in contemporary nonlinear optimization. The traditional QN equation employs only the gradients, but ignores the function value information, which seems unreasonable. In this paper, we consider a class of DFP method with new QN equations which use both gradient and function value infor- mation and ask very little additional computation. We give the condition of convergence and superlinear convergence for these methods. We also prove that under some line search conditions the DFP method with new QN equations is convergeot and superlinearly con- vergent.