A GLOBALLY CONVERGENT VERSION OF THE POLAK-RIBIERE CONJUGATE-GRADIENTMETHOD

In this paper we propose a new line search algorithm that ensures glob al convergence of the Polak-Ribiere conjugate gradient method for the unconstrained minimization of nonconvex differentiable functions. In p articular, we show that with this line search every limit point produc ed by the Polak-Ribiere iteration is a stationary point of the objecti ve function. Moreover, we define adaptive rules for the choice of the parameters in a way that the first stationary point along a search dir ection can be eventually accepted when the algorithm is converging to a minimum point with positive definite Hessian matrix. Under strong co nvexity assumptions, the known global convergence results can be reobt ained as a special case. From a computational point of view, we may ex pect that an algorithm incorporating the step-size acceptance rules pr oposed here will retain the same good features of the Polak-Ribiere me thod, while avoiding pathological situations. (C) 1997 The Mathematica l Programming Society, Inc. Published by Elsevier Science B.V.