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Mathematical Programming Computation

, Volume 8, Issue 1, pp 83–111 | Cite as

Minimizing the sum of many rational functions

  • Florian BugarinEmail author
  • Didier Henrion
  • Jean Bernard Lasserre
Full Length Paper

Abstract

We consider the problem of globally minimizing the sum of many rational functions over a given compact semialgebraic set. The number of terms can be large (10 to 100), the degree of each term should be small (up to 10), and the number of variables can be relatively large (10 to 100) provided some kind of sparsity is present. We describe a formulation of the rational optimization problem as a generalized moment problem and its hierarchy of convex semidefinite relaxations. Under some conditions we prove that the sequence of optimal values converges to the globally optimal value. We show how public-domain software can be used to model and solve such problems. Finally, we also compare with the epigraph approach and the BARON software.

Keywords

Rational optimization Global optimization Semidefinite relaxations Sparsity 

Mathematics Subject Classification

46N10 65K05 90C22 90C26 

Notes

Acknowledgments

We are grateful to Michel Devy, Jean-José Orteu, Tomáš Pajdla, Thierry Sentenac and Rekha Thomas for insightful discussions on applications of real algebraic geometry and SDP in computer vision, and to Josh Taylor for his feedback on the example of Sect. 4.2.

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Copyright information

© Springer-Verlag Berlin Heidelberg and The Mathematical Programming Society 2015

Authors and Affiliations

  • Florian Bugarin
    • 1
    Email author
  • Didier Henrion
    • 2
    • 3
    • 4
  • Jean Bernard Lasserre
    • 2
    • 3
    • 5
  1. 1.Université de Toulouse; UPS, ICA (Institut Clément Ader)ToulouseFrance
  2. 2.CNRS, LAASToulouseFrance
  3. 3.Université de Toulouse, LAASToulouseFrance
  4. 4.Faculty of Electrical EngineeringCzech Technical University in PraguePragueCzech Republic
  5. 5.Institut de Mathématiques de ToulouseUniversité de ToulouseToulouseFrance

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