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Journal of Optimization Theory and Applications

, Volume 126, Issue 2, pp 323–343 | Cite as

Modified Dinkelbach-Type Algorithm for Generalized Fractional Programs with Infinitely Many Ratios

  • J. Y. Lin
  • R. L. Sheu
Article

Abstract

In this paper, we extend the Dinkelbach-type algorithm of Crouzeix, Ferland, and Schaible to solve minmax fractional programs with infinitely many ratios. Parallel to the case with finitely many ratios, the task is to solve a sequence of continuous minmax problems,
$$P(\alpha_{k})=\min_{x\in X}\left(\max_{t\in T}\left[f_{t}(x)-\alpha_{k}g_{t}(x) \right]\right)$$
, until {α k } converges to the root of P(α)=0. The solution of P k ) is used to generate αk+1. However, calculating the exact optimal solution of P k ) requires an extraordinary amount of work. To improve, we apply an entropic regularization method which allows us to solve each problem P k ) incompletely, generating an approximate sequence \(\{\tilde{\alpha}_{k}\}\), while retaining the linear convergence rate under mild assumptions. We present also numerical test results on the algorithm which indicate that the new algorithm is robust and promising.

Keywords

Generalized fractional programming minmax problems entropic regularization 

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References

  1. 1.
    Crouzeix, J. P., Ferland, J. A., Schaible, S. 1983Duality in Generalized Linear Fractional ProgrammingMathematical Programming27342354Google Scholar
  2. 2.
    Crouzeix, J. P., Ferland, J. A., Schaible, S. 1985An Algorithm for Generalized Fractional ProgramsJournal of Optimization Theory and Applications473549CrossRefGoogle Scholar
  3. 3.
    Crouzeix, J. P., Ferland, J. A. 1991Algorithms for Generalized Fractional ProgrammingMathematical Programming52191207CrossRefGoogle Scholar
  4. 4.
    Schaible, S., Fractional Programming, Handbook of Global Optimization, Edited by R. Horst and P. M. Pardalos, Kluwer Academic Publishers, Dordrecht, Netherlands, pp. 495–608, 1995.Google Scholar
  5. 5.
    Stancu-Minasian, I. M. 1997Fractional Programming: Theory, Methods, and ApplicationsKluwer Academic PublishersDordrecht NetherlandsGoogle Scholar
  6. 6.
    Gugat, M. 1996A Fast Algorithm for a Class of Generalized Fractional ProgramsManagement Science4214931499Google Scholar
  7. 7.
    Barros, A. I., Frenk, J. B. G. 1995Generalized Fractional Programming and Cuttting-Plane AlgorithmsJournal of Optimization Theory and Applications87103120Google Scholar
  8. 8.
    Barros, A. I., Frenk, J. B. G., Schaible, S., Zhang, S. 1996A New Algorithm for Generalized Fractional ProgramsMathematical Programming72147175CrossRefGoogle Scholar
  9. 9.
    Gwinner, J., Jeyakumar, V. 1993A Solvability Theorem and Minimax Fractional ProgrammingZeitschrift für Operations Research37112Google Scholar
  10. 10.
    Shi, J. 2001A Combined Algorithm for Fractional ProgrammingAnnals of Operations Research103135147CrossRefGoogle Scholar
  11. 11.
    Abbe, L. 2001A Logarithmic Barrier Approach and Its Regularization Applied to Convex Semi-Infinite Programming ProblemsUniversität TrierTrier, GermanyPhD DissertationGoogle Scholar
  12. 12.
    Fang, S. C., Wu, S. Y. 1996Solving Min-Max Problems and Linear Semi-Infinite ProgramsComputers and Mathematics with Applications328793CrossRefGoogle Scholar
  13. 13.
    Li, X. S., Fang, S. C. 1997On the Entropic Regularization Method for Solving Min-Max Problems with ApplicationsZeischrift für Operations Research46119130Google Scholar
  14. 14.
    Lin, J. Y., Sheu, R. L. 2004Solving Continuous Min-Max Problems Using Iteratively Entropic Regularization MethodJournal of Optimization Theory and Applications121597612CrossRefGoogle Scholar
  15. 15.
    Auslender, A. 1999Penalty and Barrier Methods: A Unified FrameworkSIAM Journal on Optimization10211230CrossRefGoogle Scholar
  16. 16.
    Ben-Tal, A., and Teboulle, M., A Smoothing Technique for Nondifferentiable Optimization Problems, Lecture Notes in Mathematics, Springer-Verlag, Berlin, Germany, Vol. 1405, pp. 1–11, 1989.Google Scholar
  17. 17.
    Chang P. L., A Minimax Approach to Nonlinear Programming, Doctoral Dissertation, Department of Mathematics, University of Washington, 1980.Google Scholar
  18. 18.
    Crouzeix, J. P., Ferland, J. A., Schaible, S. 1986A Note on an Algorithm for Generalized Fractional ProgramsJournal of Optimization Theory and Applications50183187CrossRefGoogle Scholar
  19. 19.
    Dieudonné, J. 1969Foundations of Modern Analysis 10-1Academic PressNew York, NYGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2005

Authors and Affiliations

  • J. Y. Lin
    • 1
  • R. L. Sheu
    • 1
  1. 1.Department of MathematicsNational Cheng-Kung UniversityTainanTaiwan

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