Global Convergence of a Modified Two-Parameter Scaled BFGS Method with Yuan-Wei-Lu Line Search for Unconstrained Optimization

Abstract

The BFGS method is one of the most efficient quasi-Newton methods for solving small- and medium-size unconstrained optimization problems. For the sake of exploring its more interesting properties, a modified two-parameter scaled BFGS method is stated in this paper. The intention of the modified scaled BFGS method is to improve the eigenvalues structure of the BFGS update. In this method, the first two terms and the last term of the standard BFGS update formula are scaled with two different positive parameters, and the new value of yk is given. Meanwhile, Yuan-Wei-Lu line search is also proposed. Under the mentioned line search, the modified two-parameter scaled BFGS method is globally convergent for nonconvex functions. The extensive numerical experiments show that this form of the scaled BFGS method outperforms the standard BFGS method or some similar scaled methods.