Abstract
In this paper we propose a new line search algorithm that ensures global convergence of the Polak-Ribière 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-Ribière 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 retain the same good features of the Polak-Ribière method, while avoiding pathological situations.
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This research was supported by Agenzia Spaziale Italiana, Rome, Italy.
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Grippo, L., Lucidi, S. A globally convergent version of the Polak-Ribière conjugate gradient method. Mathematical Programming 78, 375–391 (1997). https://doi.org/10.1007/BF02614362
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DOI: https://doi.org/10.1007/BF02614362