In this paper we propose a new line search algorithm that ensures global convergence of the Polak-Ribi~re conjugate gradient method for the unconstrained minimization of nonconvex differentiable functions. In particular, we show that with this line search every limit point produced by the Polak-Ribi~re iteration is a stationary point of the objective function. Moreover, we define adaptive rules for the choice of the parameters in a way that the first stationary point along a search direction can be eventually accepted when the algorithm is converging to a minimum point with positive definite Hessian matrix. Under strong convexity assumptions, the known global convergence results can be reobtained as a special case. From a computational point of view, we may expect that an algorithm incorporating the step-size acceptance rules proposed here will retaLa the same good features of the Polak-Ribi~re method, while avoiding pathological situations.
A Globally Convergent Version of the Polak-Ribiere Conjugate Gradient Method / Grippo, Luigi; Lucidi, Stefano. - In: MATHEMATICAL PROGRAMMING. - ISSN 0025-5610. - STAMPA. - 78:(1997), pp. 375-391. [10.1007/BF02614362]
A Globally Convergent Version of the Polak-Ribiere Conjugate Gradient Method
GRIPPO, Luigi;LUCIDI, Stefano
1997
Abstract
In this paper we propose a new line search algorithm that ensures global convergence of the Polak-Ribi~re conjugate gradient method for the unconstrained minimization of nonconvex differentiable functions. In particular, we show that with this line search every limit point produced by the Polak-Ribi~re iteration is a stationary point of the objective function. Moreover, we define adaptive rules for the choice of the parameters in a way that the first stationary point along a search direction can be eventually accepted when the algorithm is converging to a minimum point with positive definite Hessian matrix. Under strong convexity assumptions, the known global convergence results can be reobtained as a special case. From a computational point of view, we may expect that an algorithm incorporating the step-size acceptance rules proposed here will retaLa the same good features of the Polak-Ribi~re method, while avoiding pathological situations.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.