The Global Convergence of Selfscale BFGS Algorithm with Nonmonotone Line Search for Uncons.pdfVIP

The Global Convergence of Self-scale BFGS Algorithm with Nonmonotone Line Search for Unconstrained Nonconvex Optimization Problems Hongxia Yin∗ and Donglei Du † (January 2005) Abstract The self-scaling quasi-Newton method solves an unconstrained optimiza- tion problem by scaling the Hessian approximation matrix before it is updated at each iteration to avoid the possible large eigenvalues in the Hessian approximation matri- ces of the objective function. It has been proved in the literature that this method has the global and superlinear convergence when the objective function is convex (or even uniformly convex). We propose to solve unconstrained nonconvex optimization problems by a self-scaling BFGS algorithm with nonmonotone linear search. Non- monotone line search has been recognized in numerical practices as a competitive approach for solving large-scale nonlinear problems. We consider two different non- monotone line search forms and study the global convergence of these nonmonotone self-scale BFGS algorithms. We prove that, under some weaker condition than that in the literature, both forms of the self-scaling BFGS algorithm are globally convergent for unconstrained nonconvex optimization problems. Key Words. Nonmonotone line search, self-scaling BFGS method, global conver- gence. AMS(MOS) subject classifications. 90C26, 90C30, 65H10. Abbreviated title. Nonmonotone Self-scaling BFGS Method. ∗Department of Mathematics, Graduate School of Chinese Academy of Sciences, Beijing 100039, P. R. China. E-mail: hxyin@gscas.ac.cn. The author’s work is supported by NSFCand †Faculty of Administration, University of New Brunswick, P.O. Box 4400, Fredericton, NB E3B 5A3, New Brunswick, Canada, E-ma