Superlinear convergence of nonlinear conjugate gradient method and scaled memoryless BFGS method based on assumptions about the initial point

摘要

The Perry nonlinear conjugate gradient method and scaled memoryless BFGS method are two quasi-Newton methods for unconstrained minimization. All convergence theory in the literature assume existence of a minimizer and bounds on the objective function in a neighbourhood of the minimizer. These conditions cannot be checked in practice. The motivation of this work is to derive a convergence theory where all assumptions can be verified, and the existence of a minimizer and its superlinear rate of convergence are consequences of the theory. Only the basic versions of these methods without line search are considered. The theory is simple in the sense that it contains as few constants as possible.