Problems of sciences concerned with minimizing an objective function that depends on real
values without restrictions on there is called unconstrained optimization problems. Quasi-
Newton methods are one of the most common approaches to solve unconstrained
optimization problems. Mainly, the quasi-Newton equation is the focus of quasi-Newton
methods. In this paper, we extended the quasi-Newton equation introduced by Razieh et
al.[1] and some new quasi-Newton methods are presented. The convergence behaviors of …

In this paper, we propose a derivative-free quasi-Newton condition, which results in a new
class of quasi-Newton updating formulas for unconstrained optimization. Each updating
formula in this class is a rank-two updating formula and preserves the positive definiteness
of the second derivative matrix of the quadratic model. Its first two terms are the same as the
first two terms of the BFGS updating formula. We establish global convergence of quasi-
Newton methods based upon the updating formulas in this class, and superlinear …