Annals of Operations Research

, Volume 156, Issue 1, pp 25–44 | Cite as

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

  • Mirjam Dür
  • Charoenchai Khompatraporn
  • Zelda B. Zabinsky
Article

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.

Keywords

Fractional programming Stochastic algorithms Global optimization Improving hit-and-run Lipschitz properties Multistart Pure adaptive search 

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Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  • Mirjam Dür
    • 1
  • Charoenchai Khompatraporn
    • 2
  • Zelda B. Zabinsky
    • 3
  1. 1.Department of MathematicsDarmstadt University of TechnologyDarmstadtGermany
  2. 2.Department of Production EngineeringKing Mongkut’s University of Technology ThonburiBangkokThailand
  3. 3.Industrial EngineeringUniversity of WashingtonSeattleUSA

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