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Analysis of a new algorithm for one-dimensional minimization

Analyse eines neuen Algorithmus zur Bestimmung von Minima von Funktionen einer Variablen

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Abstract

Davidon has recently introduced a new approach to optimization using the idea of nonlinear scaling. In this paper we study the algorithm that results when applying his ideas to the one-dimensional case. We show that the algorithm is locally convergent withQ-order equal 2 and compare it with the method of cubic interpolation.

Zusammenfassung

Kürzlich wurde von Davidon für Optimierungsprobleme ein neuer Weg vorgeschlagen, bei dem die Idee der nichtlinearen Skalierung verwendet wird. Der Algorithmus wird in der vorliegenden Arbeit analysiert für den eindimensionalen Fall. Es wird gezeigt, daß der Algorithmus lokal konvergiert mit quadratischerQ-Konvergenz und die Konvergenzeigenschaften werden mit denjenigen der Methode der kubischen Interpolation verglichen.

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References

  1. Davidon, W.: Optimization by non-linear scaling. Institutt for Teoretisk Fysikk, Trondheim NTH, Norway.

  2. Ortega, J., Rheinboldt, W.: Iterative solution of nonlinear equations in several variables. Ch. 9. Academic Press 1970.

  3. Gill, P., Murray, W.: Safeguarded steplength algorithms for optimization using descent methods. NPL Report NAC 37, August 1974.

  4. MACSYMA, Reference Manual. Version 9, July 1977. The Mathlab Group, Laboratory for Computer Science, MIT.

  5. Tamir, A.: Rates of convergence of a one-dimensional search based on interpolating polynomials, #143. The Center for Mathematical Studies in Economics and Management Science, Northwestern University, May 1975.

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Authors and Affiliations

  1. Computer Science Department Numerical Analysis, Stanford University, Serra House, Serra Street, 94305, Stanford, CA, USA

    Petter Bjørstad & Jorge Nocedal

Authors
  1. Petter Bjørstad
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  2. Jorge Nocedal
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Additional information

Supported by the Norwegian Research Council for Science and the Humanities and in part by the U.S. Army Research Grant No. DAHCO4-75-G-0185.

Supported by the Universidad Nacional Autonoma de Mexico and Banco de Mexico.

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Bjørstad, P., Nocedal, J. Analysis of a new algorithm for one-dimensional minimization. Computing 22, 93–100 (1979). https://doi.org/10.1007/BF02246561

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  • DOI: https://doi.org/10.1007/BF02246561

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