## Combining quasi-Newton and Cauchy directions.(English)Zbl 1042.90041

Authors’ abstract: We propose a line search algorithm for unconstrained optimization in which the search direction is a combination of the quasiNewton and the Cauchy directions. We prove that our algorithm is globally convergent for general objective functions even when inexact line searches are used. We also prove that the new method with the update from the restricted Broyden family is superlinearly convergent when the objective function is uniformly convex. We test the algorithm on the standard test problems which come from the Moré-Garbow-Hillström collection. Numerical results comparing the new algorithm with usual line search and dogleg quasi-Newton methods show that the new approach is encouraging.

### MSC:

 90C30 Nonlinear programming 90C53 Methods of quasi-Newton type 65K10 Numerical optimization and variational techniques