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Modification of the Wolfe Line Search Rules to Satisfy the Descent Condition in the Polak-Ribière-Polyak Conjugate Gradient Method

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Abstract

This paper proposes a line search technique to satisfy a relaxed form of the strong Wolfe conditions in order to guarantee the descent condition at each iteration of the Polak-Ribière-Polyak conjugate gradient algorithm. It is proved that this line search algorithm preserves the usual convergence properties of any descent algorithm. In particular, it is shown that the Zoutendijk condition holds under mild assumptions. It is also proved that the resulting conjugate gradient algorithm is convergent under a strong convexity assumption. For the nonconvex case, a globally convergent modification is proposed. Numerical tests are presented.

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Authors and Affiliations

  1. Maître de Conférences, Université de Limoges, Limoges, France

    P. Armand

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  1. P. Armand
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Communicated by G. Di Pillo

This paper is based on an earlier work presented at the International Symposium on Mathematical Programming in Lausanne in 1997. The author thanks J. C. Gilbert for his advice and M. Albaali for some recent discussions which motivated him to write this paper. Special thanks to G. Liu, J. Nocedal, and R. Waltz for the availability of the software CG+ and to one of the referees who indicated to him the paper of Grippo and Lucidi (Ref. 1).

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Armand, P. Modification of the Wolfe Line Search Rules to Satisfy the Descent Condition in the Polak-Ribière-Polyak Conjugate Gradient Method. J Optim Theory Appl 132, 287–305 (2007). https://doi.org/10.1007/s10957-006-9123-7

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