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New quasi-Newton equation and related methods for unconstrained optimization. (English) Zbl 0991.90135
Summary: In unconstrained optimization, the usual quasi-Newton equation is $$B_{k+1} s_k= y_k$$, where $$y_k$$ is the difference of the gradients at the last two iterates. In this paper, we propose a new quasi-Newton equation, $$B_{k+1} s_k+\widetilde y_k$$, in which $$\widetilde y_k$$ is based on both the function values and gradients at the last two iterates. The new equation is superior to the old equation in the sense that $$\widetilde y_k$$ better approximates $$\nabla^2 f(x_{k+1})s_k$$ than $$y_k$$. Modified quasi-Newton methods based on the new quasi-Newton equation are locally and superlinearly convergent. Extensive numerical experiments have been conducted which show that the new quasi-Newton methods are encouraging.

##### MSC:
 90C53 Methods of quasi-Newton type 65K05 Numerical mathematical programming methods
##### Keywords:
quasi-Newton methods
CUTEr
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##### References:
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