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Solving fractional problems with dynamic multistart improving hit-and-run

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  • Mirjam Dür
  • Charoenchai Khompatraporn
  • Zelda Zabinsky

Abstract

Fractional programming has numerous applications in economy and engineering. While some fractional problems are easy in the sense that they are equivalent to an ordinary linear program, other problems like maximizing a sum or product of several ratios are known to be hard, as these functions are highly nonconvex and multimodal. In contrast to the standard Branch-and-Bound type algorithms proposed for specific types of fractional problems, we treat general fractional problems with stochastic algorithms developed for multimodal global optimization. Specifically, we propose Improving Hit-and-Run with restarts, based on a theoretical analysis of Multistart Pure Adaptive Search (cf. the dissertation of Khompatraporn ( 2004 )) which prescribes a way to utilize problem specific information to sample until a certain level α of confidence is achieved. For this purpose, we analyze the Lipschitz properties of fractional functions, and then utilize a unified method to solve general fractional problems. The paper ends with a report on numerical experiments. Copyright Springer Science+Business Media, LLC 2007

Suggested Citation

  • Mirjam Dür & Charoenchai Khompatraporn & Zelda Zabinsky, 2007. "Solving fractional problems with dynamic multistart improving hit-and-run," Annals of Operations Research, Springer, vol. 156(1), pages 25-44, December.
    • Handle: RePEc:spr:annopr:v:156:y:2007:i:1:p:25-44:10.1007/s10479-007-0232-y
    DOI: 10.1007/s10479-007-0232-y
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    References listed on IDEAS

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    1. Chadha, S. S., 2002. "Fractional programming with absolute-value functions," European Journal of Operational Research, Elsevier, vol. 141(1), pages 233-238, August.
    2. 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.
    3. Robert L. Smith, 1984. "Efficient Monte Carlo Procedures for Generating Points Uniformly Distributed over Bounded Regions," Operations Research, INFORMS, vol. 32(6), pages 1296-1308, December.
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