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Minimization of Isotonic Functions Composed of Fractions

Author

Listed:
  • J.-Y. Lin

    (National Chiayi University)

  • S. Schaible

    (Chung Yuan Christian University)

  • R.-L. Sheu

    (National Cheng Kung University)

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.

Suggested Citation

  • J.-Y. Lin & S. Schaible & R.-L. Sheu, 2010. "Minimization of Isotonic Functions Composed of Fractions," Journal of Optimization Theory and Applications, Springer, vol. 146(3), pages 581-601, September.
    • Handle: RePEc:spr:joptap:v:146:y:2010:i:3:d:10.1007_s10957-010-9684-3
    DOI: 10.1007/s10957-010-9684-3
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    References listed on IDEAS

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    1. H. P. Benson, 2002. "Global Optimization Algorithm for the Nonlinear Sum of Ratios Problem," Journal of Optimization Theory and Applications, Springer, vol. 112(1), pages 1-29, January.
    2. Y. Almogy & O. Levin, 1971. "A Class of Fractional Programming Problems," Operations Research, INFORMS, vol. 19(1), pages 57-67, February.
    3. Werner Dinkelbach, 1967. "On Nonlinear Fractional Programming," Management Science, INFORMS, vol. 13(7), pages 492-498, March.
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