Local and superlinear convergence of quasi-Newton methods based on modified secant conditions

For solving unconstrained minimization problems, quasi-Newton methods are popular iterative methods. The secant condition which employs only the gradient information is imposed on these methods. Several researchers paid attention to other secant conditions to get a better approximation of the Hessian matrix of the objective function. Recently, Zhang et al. [New quasi-Newton equation and related methods for unconstrained optimization, J. Optim. Theory Appl. 102 (1999) 147–167] and Zhang and Xu [Properties and numerical performance of quasi-Newton methods with modified quasi-Newton equations, J. Comput. Appl. Math. 137 (2001) 269–278] proposed the modified secant condition which uses both gradient and function value information in order to get a higher order accuracy in approximating the second curvature of the objective function. They showed the local and q-superlinear convergence property of the BFGS-like and DFP-like updates based on their proposed secant condition. In this paper, we incorporate one parameter into this secant condition to smoothly switch the standard secant condition and the secant condition of Zhang et al. We consider a modified Broyden family which includes the BFGS-like and the DFP-like updates proposed by Zhang et al. We prove the local and q-superlinear convergence of our method.