On the Local and Superlinear Convergence of a Parameterized DFP Method.

A new parameterized DFP method is proposed in (Zhang and Pan [35]) via parameterizing the traditional DFP updating formula by a parameter θkin eachk-th iteration. Preliminary but favorable computational experiments are reported and indicate that the parameterized DFP method has a better numerical performance than the traditional DFP method which corresponds to θk ≡ 1 in each iteration. To lay a solid ground for the parameterized DFP method, in this article, we provide the rigorous theoretical analysis for the parameterized DFP method by showing that the positive definiteness of the updating matrices {Bk} is retained, the local linear and superlinear convergence of the generated sequence {xk} are achievable if θkis chosen in the intervals [0, 2], (0, 1] and, respectively. We also discuss some practical strategies in selecting the parameter θk, which are helpful in stabilizing the traditional DFP method. [ABSTRACT FROM PUBLISHER]