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Convergence of a Dinkelbach-type algorithm in generalized fractional programming

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Abstract

The convergence of a Dinkelbach-type algorithm in generalized fractional programming is obtained by considering the sensitivity of a parametrized problem. We show that the rate of convergence is at least equal to (1+√5)/2 when regularity conditions hold in a neighbourhood of the optimal solution. We give also a necessary and sufficient condition for the convergence to be quadratic (which will be verified in particular in the linear case) and an idea of its implementation in the convex case.

Zusammenfassung

Die Konvergenz eines Verfahrens i. S. von Dinkelbach zur Lösung verallgemeinerter Quotientenprogramme wird durch Untersuchung der Sensitivität eines parametrisierten Problems abgeleitet. Es wird gezeigt, daß die Konvergenzrate durch (1+√5)/2 nach unten beschränkt ist, falls gewisse Regularitätsbedingungen in einer Umgebung der Optimallösung erfüllt sind. Ferner wird eine notwendige und hinreichende Bedingung zur quadratischen Konvergenz hergeleitet. Es wird gezeigt, wie diese im Falle konvexer Probleme implementiert werden kann.

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Author notes

  1. J. Borde

    Present address: Mathematical Studies Section, European Space Agency, ESTEC, Noordwijk, The Netherlands

Authors and Affiliations

  1. Département de Mathématiques Appliquées, Université de Clermont II Complexe Scientifique des Cézeaux, BP 45, F-63170, Aubière, France

    J. Borde & J. P. Crouzeix

Authors
  1. J. Borde
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  2. J. P. Crouzeix
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Borde, J., Crouzeix, J.P. Convergence of a Dinkelbach-type algorithm in generalized fractional programming. Zeitschrift für Operations Research 31, A31–A54 (1987). https://doi.org/10.1007/BF01258607

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  • DOI: https://doi.org/10.1007/BF01258607

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