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.


90C53 Methods of quasi-Newton type
65K05 Numerical mathematical programming methods


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