, Volume 13, Issue 3, pp 247–277 | Cite as

A reliable affine relaxation method for global optimization

  • Jordan Ninin
  • Frédéric Messine
  • Pierre Hansen
Research paper


An automatic method for constructing linear relaxations of constrained global optimization problems is proposed. Such a construction is based on affine and interval arithmetics and uses operator overloading. These linear programs have exactly the same numbers of variables and inequality constraints as the given problems. Each equality constraint is replaced by two inequalities. This new procedure for computing reliable bounds and certificates of infeasibility is inserted into a classical branch and bound algorithm based on interval analysis. Extensive computation experiments were made on 74 problems from the COCONUT database with up to 24 variables or 17 constraints; 61 of these were solved, and 30 of them for the first time, with a guaranteed upper bound on the relative error equal to \(10^{-8}\). Moreover, this sample comprises 39 examples to which the GlobSol algorithm was recently applied finding reliable solutions in 32 cases. The proposed method allows solving 31 of these, and 5 more with a CPU-time not exceeding 2 min.


Affine arithmetic Interval analysis Linear relaxation Branch and bound algorithm Global optimization 

Mathematics Subject Classification

90C26 65H20 65G30 65G40 49M20 


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Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Jordan Ninin
    • 1
  • Frédéric Messine
    • 2
  • Pierre Hansen
    • 3
    • 4
  1. 1.LabSTICC, IHSEV TeamENSTA-BretagneBrestFrance
  2. 2.ENSEEIHT-IRITToulouseFrance
  3. 3.Studies and Research in Decision AnalysisHEC MontréalMontrealCanada
  4. 4.LIXÉcole PolytechniquePalaiseauFrance

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