A New Generalized Quasi-Newton Equation And Its Updates

The quasi-Newton equation plays a central role in the quasi-Newton methods for solving systems of nonlinear equations and/or unconstrained optimization problems. In stead of the equation, which was shown in certain sense to be of the first order, a second order quasi-Newton equation was suggested (Pan [1], 1984). In this paper, a generalization of the latter is made and discussed in the first part.In the second part, a DFP-like updating formula derived from it makes this generalized quasi-Newton equation and its relevant updates attractive since the new update is not only of some properties—the property of hereditary positive definiteness of {B_k}, the locally linear convergence and superlinear convergence of {x_k}—similar to those of such conventional ones as DFP and BFGS, but also of merit in enlarging the domain of convergence by properly choosing the value of a parameter(θ∈R) contained in it.To prove these, the algorithm(DFP-like) is implemented and tested on some problems from [17](Moré, etc., 1981). The computational results firstly verify the theoretical properties of DFP-like update. Moreover, when comparing to other updates such as the BFGS, DFP, 2-order BFGS, we find the DFP-like algorithm with θ = 0.85 outperforms the DFP not only in its stability but in the convergent rate as well.