The Global Convergence of SelfScaling BFGS Algorithm with Nonmonotone Line Search for Unco.pdf

Acta Mathematica Sinica, English Series Jul., 2007, Vol. 23, No. 7, pp. 1233–1240 Published online: Sep. 11, 2006 DOI: 10.1007/s10114-005-0837-5 Http://www.ActaM The Global Convergence of Self-Scaling BFGS Algorithm with Nonmonotone Line Search for Unconstrained Nonconvex Optimization Problems Hong Xia YIN Chinese Academy of Sciences Research Center on Data Technology and Knowledge Economy, Department of Mathematics, Graduate University of the Chinese Academy of Sciences, Beijing 100049, P. R. China E-mail: hxyin@ Dong Lei DU Faculty of Administration, University of New Brunswick, P.O. Box 4400, Fredericton, NB E3B 5A3, New Brunswick, Canada E-mail: ddu@unb.ca Abstract The self-scaling quasi-Newton method solves an unconstrained optimization problem by scaling the Hessian approximation matrix before it is updated at each iteration to avoid the possible large eigenvalues in the Hessian approximation matrices of the objective function. It has been proved in the literature that this method has the global and superlinear convergence when the objective func- tion is convex (or even uniformly convex). We propose to solve unconstrained nonconvex optimization problems by a self-scaling BFGS algorithm with nonmonotone linear search. Nonmonotone line search has been recognized in numerical practices as a competitive approach for solving large-scale nonlinear problems. We consider two different nonmonotone line search forms and study the global conver- gence of these nonmonotone self-scale BFGS algorithms. We prove that, under some weaker condition than t