Some global convergence properties of a conic-variable metric algorithm for minimization with inexact line searches

Abstract

Among the most interesting methods for finding the minimizer of a function f{x) of several variables when gradients are available, are conic-variable metric methods [1,4,11,12]. Much of the published theory only studies the local convergence properties of this class, save that a global convergence theorem depending on exact line searches is given in [5].In practice, it is best to be quite tolerant in the termination criterion of line searches. Therefore, this paper studies the global convergence of this method with inexact line searches. It is shown that, if / f(x) is uniformly convex, then convergence to the minimizer is obtained, and the rate of convergence is superlinear. Moreover, we prove the convergence of a more general variable metric method with two parameters proposed by Spedicato [13]