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Minimization of Isotonic Functions Composed of Fractions

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Abstract

In this paper, we introduce a class of minimization problems whose objective function is the composite of an isotonic function and finitely many ratios. Examples of an isotonic function include the max-operator, summation, and many others, so it implies a much wider class than the classical fractional programming containing the minimax fractional program as well as the sum-of-ratios problem. Our intention is to develop a generic “Dinkelbach-like” algorithm suitable for all fractional programs of this type. Such an attempt has never been successful before, including an early effort for the sum-of-ratios problem. The difficulty is now overcome by extending the cutting plane method of Barros and Frenk (in J. Optim. Theory Appl. 87:103–120, 1995). Based on different isotonic operators, various cuts can be created respectively to either render a Dinkelbach-like approach for the sum-of-ratios problem or recover the classical Dinkelbach-type algorithm for the min-max fractional programming.

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Authors and Affiliations

  1. Department of Applied Mathematics, National Chiayi University, Chiayi, Taiwan

    J.-Y. Lin

  2. Department of Applied Mathematics, Chung Yuan Christian University, Chung-Li, Taiwan

    S. Schaible

  3. Department of Mathematics, National Cheng Kung University, Tainan, Taiwan

    R.-L. Sheu

Authors
  1. J.-Y. Lin
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  2. S. Schaible
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  3. R.-L. Sheu
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Correspondence to S. Schaible.

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Lin, JY., Schaible, S. & Sheu, RL. Minimization of Isotonic Functions Composed of Fractions. J Optim Theory Appl 146, 581–601 (2010). https://doi.org/10.1007/s10957-010-9684-3

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  • DOI: https://doi.org/10.1007/s10957-010-9684-3

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