New quasi-Newton equation and related methods for unconstrained optimization

In unconstrained optimization, the usual quasi-Newton equation is B-k + S-1(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 + S-1(k) = (y) over tilde(k), in which (y) over tilde(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 (y) over tilde(k) better approximates del(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.