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IMA Journal of Mathematical Control and Information 1998 15(2):185-214; doi:10.1093/imamci/15.2.185
© 1998 by Institute of Mathematics and its Applications
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A generalized golden-section algorithm for line search

LUC PRONZATO, HENRY P. WYNN and ANATOLY A. ZHIGLJAVSKY

Laboratoire 13S, bat. 4, 250 rue A. Einstein, CNRS-URA 1376, Sophia Antipolis, 06560 Valbonne, France E-mail: pronzato{at}unice.fr
Department of Statistics, University of Warwick Coventry CV4 7AL, UK
Department of Mathematics, St. Petersburg University Bibliotechnaya sq. 2, 198904, Russia

This paper aims at promoting a generalization of the golden-section line-search algorithm, with better performance for functions locally symmetric around their optimum. Any line-search algorithm can be represented as a nonconvergent dynamic system when a suitable renormalization of the uncertainty interval is performed at each iteration. This allows a detailed study of the finite-sample and asymtotic behaviour of the algorithm through a Markov-chain representation. We show that an expansion of the initial uncertainty interval improves this behavior. Some asymptotic characteristics based on the evolution of the length of the uncertainty interval are shown to be related to classical ergodic characteristics of dynamic systems, such as Lyapunov exponent and Kolmogorov entropy. However, other ergodic characteristics, related to the Renyi entropy, are suggested as being more suitable in this context.


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