Spectral-scaling quasi-Newton methods with updates from the one parameter of the Broyden family

In this paper, based on the spectral-scaling secant condition [W.Y. Cheng, D.H. Li, Spectral-scaling BFGS method, Journal of Optimization Theory and Applications, 146 (2010) 305–319], we propose spectral-scaling one parameter Broyden family methods which allow for negative values of the parameter. We show that the proposed methods possess some good properties such as quadratic termination property and single-step convergence rate not inferior to that of the steepest descent method when minimizing an n-dimensional quadratic function. Under appropriate conditions, we establish the global convergence of the proposed methods for uniformly convex functions. Numerical results from problems in the CUTE test set show that the proposed methods are promising.