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Robust Fractional Programming

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  • Bram L. Gorissen

    (Tilburg University)

Abstract

We extend robust optimization (RO) to fractional programming, where both the objective and the constraints contain uncertain parameters. Earlier work did not consider uncertainty in both the objective and the constraints, or did not use RO. Our contribution is threefold. First, we provide conditions to guarantee that either a globally optimal solution, or a sequence converging to the globally optimal solution, can be found by solving one or more convex optimization problems. Second, we identify two cases for which an exact solution can be obtained by solving a single optimization problem: (1) when uncertainty in the numerator is independent from the uncertainty in the denominator, and (2) when the denominator does not contain an optimization variable. Third, we show that the general problem can be solved with an (iterative) root finding method. The results are demonstrated on a return on investment maximization problem, data envelopment analysis, and mean-variance optimization. We find that the robust optimal solution is only slightly more robust than the nominal solution. As a side-result, we use RO to show that two existing methods for solving fractional programs are dual to each other.

Suggested Citation

  • Bram L. Gorissen, 2015. "Robust Fractional Programming," Journal of Optimization Theory and Applications, Springer, vol. 166(2), pages 508-528, August.
    • Handle: RePEc:spr:joptap:v:166:y:2015:i:2:d:10.1007_s10957-014-0633-4
    DOI: 10.1007/s10957-014-0633-4
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    1. H. J. Chen & S. Schaible & R. L. Sheu, 2009. "Generic Algorithm for Generalized Fractional Programming," Journal of Optimization Theory and Applications, Springer, vol. 141(1), pages 93-105, April.
    2. Charnes, A. & Cooper, W. W. & Rhodes, E., 1978. "Measuring the efficiency of decision making units," European Journal of Operational Research, Elsevier, vol. 2(6), pages 429-444, November.
    3. J. Y. Lin & R. L. Sheu, 2005. "Modified Dinkelbach-Type Algorithm for Generalized Fractional Programs with Infinitely Many Ratios," Journal of Optimization Theory and Applications, Springer, vol. 126(2), pages 323-343, August.
    4. Jeyakumar, V. & Li, G.Y. & Srisatkunarajah, S., 2013. "Strong duality for robust minimax fractional programming problems," European Journal of Operational Research, Elsevier, vol. 228(2), pages 331-336.
    5. Siegfried Schaible, 1976. "Fractional Programming. II, On Dinkelbach's Algorithm," Management Science, INFORMS, vol. 22(8), pages 868-873, April.
    6. Emrouznejad, Ali & Parker, Barnett R. & Tavares, Gabriel, 2008. "Evaluation of research in efficiency and productivity: A survey and analysis of the first 30 years of scholarly literature in DEA," Socio-Economic Planning Sciences, Elsevier, vol. 42(3), pages 151-157, September.
    7. Dimitris Bertsimas & Melvyn Sim, 2004. "The Price of Robustness," Operations Research, INFORMS, vol. 52(1), pages 35-53, February.
    8. Schaible, Siegfried & Ibaraki, Toshidide, 1983. "Fractional programming," European Journal of Operational Research, Elsevier, vol. 12(4), pages 325-338, April.
    9. Aharon Ben-Tal & Dick den Hertog & Anja De Waegenaere & Bertrand Melenberg & Gijs Rennen, 2013. "Robust Solutions of Optimization Problems Affected by Uncertain Probabilities," Management Science, INFORMS, vol. 59(2), pages 341-357, April.
    10. Sadjadi, S.J. & Omrani, H., 2008. "Data envelopment analysis with uncertain data: An application for Iranian electricity distribution companies," Energy Policy, Elsevier, vol. 36(11), pages 4247-4254, November.
    11. Bram L. Gorissen & Hans Blanc & Dick den Hertog & Aharon Ben-Tal, 2014. "Technical Note---Deriving Robust and Globalized Robust Solutions of Uncertain Linear Programs with General Convex Uncertainty Sets," Operations Research, INFORMS, vol. 62(3), pages 672-679, June.
    12. Yanikoglu, I. & den Hertog, D. & Kleijnen, Jack P.C., 2013. "Adjustable Robust Parameter Design with Unknown Distributions," Other publications TiSEM 47fec228-1ffe-4803-8e97-5, Tilburg University, School of Economics and Management.
    13. SHOKOUHI, Amir H. & HATAMI-MARBINI, Adel & TAVANA, Madjid & SAATI, Saber, 2010. "A robust optimization approach for imprecise data envelopment analysis," LIDAM Reprints CORE 2215, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    14. Werner Dinkelbach, 1967. "On Nonlinear Fractional Programming," Management Science, INFORMS, vol. 13(7), pages 492-498, March.
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    Cited by:

    1. van Eekelen, Wouter, 2023. "Distributionally robust views on queues and related stochastic models," Other publications TiSEM 9b99fc05-9d68-48eb-ae8c-9, Tilburg University, School of Economics and Management.
    2. Mostafa Mardani Najafabadi & Niloofar Ashktorab, 2023. "Mathematical programming approaches for modeling a sustainable cropping pattern under uncertainty: a case study in Southern Iran," Environment, Development and Sustainability: A Multidisciplinary Approach to the Theory and Practice of Sustainable Development, Springer, vol. 25(9), pages 9731-9755, September.
    3. Jiawei Chen & Suliman Al-Homidan & Qamrul Hasan Ansari & Jun Li & Yibing Lv, 2021. "Robust Necessary Optimality Conditions for Nondifferentiable Complex Fractional Programming with Uncertain Data," Journal of Optimization Theory and Applications, Springer, vol. 189(1), pages 221-243, April.
    4. Yichun Chi & Zuo Quan Xu & Sheng Chao Zhuang, 2022. "Distributionally Robust Goal-Reaching Optimization in the Presence of Background Risk," North American Actuarial Journal, Taylor & Francis Journals, vol. 26(3), pages 351-382, August.

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