4OR

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

A reliable affine relaxation method for global optimization

Research paper

Abstract

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.

Keywords

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

Mathematics Subject Classification

90C26 65H20 65G30 65G40 49M20 

References

  1. Androulakis IP, Maranas CD, Floudas CA, Alpha BB (1995) A global optimization method for general constrained nonconvex problems. J Global Optim 7(4):337–363CrossRefGoogle Scholar
  2. Audet C, Hansen P, Jaumard B, Savard G (2000) A branch and cut algorithm for nonconvex quadratically constrained quadratic programming. Mathe Program Ser A 87(1):131–152Google Scholar
  3. Belotti P, Lee J, Liberti L, Margot F, Waechter A (2009) Branching and bounds tightening techniques for non-convex MINLP. Optim Methods Softw 24(4–5):597–634CrossRefGoogle Scholar
  4. Comba JLD, Stolfi J (1993) Affine arithmetic and its applications to computer graphics. In: Proceedings of SIBGRAPI’93 - VI Simpósio Brasileiro de Computação Gráfica e Processamento de Imagens, pp 9–18Google Scholar
  5. de Figueiredo L (1996) Surface intersection using affine arithmetic. In: Proceedings of graphics interface’96, pp 168–175Google Scholar
  6. de Figueiredo L, Stolfi J (2004) Affine arithmetic: concepts and applications. Numer Algorithms 37(1–4):147–158CrossRefGoogle Scholar
  7. Dolan ED, Moré JJ (2002) Benchmarking optimization software with performance profiles. Math Program Ser A 91(2):201–213CrossRefGoogle Scholar
  8. Du K, Kearfott RB (1994) The cluster problem in multivariate global optimization. J Global Optim 5(3):253265CrossRefGoogle Scholar
  9. Fitan E, Messine F, Nogarède B (2004) The electromagnetic actuator design problem: a general and rational approach. IEEE Trans Magn 40(3):1579–1590CrossRefGoogle Scholar
  10. Fontchastagner J, Messine F, Lefevre Y (2007) Design of electrical rotating machines by associating deterministic global optimization algorithm with combinatorial analytical and numerical models. IEEE Trans Magn 43(8):3411–3419CrossRefGoogle Scholar
  11. Granvilliers L, Benhamou F (2006) RealPaver: an interval solver using constraint satisfaction techniques. ACM Trans Mathe Softw 32:138156Google Scholar
  12. Hansen ER (1975) A generalized interval arithmetic. Lect Notes Comput Sci 29:7–18CrossRefGoogle Scholar
  13. Hansen ER, Walster WG (2004) Global optimization using interval analysis, 2nd edn. Marcel Dekker Inc., New YorkGoogle Scholar
  14. Horst R, Tuy H (1996) Global optimization: deterministic approaches, 3rd edn. Springer, BerlinCrossRefGoogle Scholar
  15. Jansson C (2003) Rigorous lower and upper bounds in linear programming. SIAM J Optim 14(3):914–935CrossRefGoogle Scholar
  16. IBEX (2014) a C++ numerical library based on interval arithmetic and constraint programming. http://www.ibex-lib.org
  17. Kearfott RB (1996) Rigorous global search: continuous problems. Kluwer Academis Publishers, DordrechtCrossRefGoogle Scholar
  18. Kearfott RB (2006) Discussion and empirical comparisons of linear relaxations and alternate techniques in validated deterministic global optimization. Optim Methods Softw 21(5):715–731CrossRefGoogle Scholar
  19. Kearfott RB (Jan 2009) GlobSol user guide. Optim Methods Softw 24(4–5):687–708Google Scholar
  20. Kearfott RB, Hongthong S (2005) Validated linear relaxations and preprocessing: some experiments. SIAM J Optim 16(2):418–433CrossRefGoogle Scholar
  21. Keil C (2006) Lurupa - Rigorous Error Bounds in Linear Programming. In: Buchberger B, Oishi S, Plum M, Rump SM (eds) Algebraic and Numerical Algorithms and Computer-assisted Proofs. Dagstuhl Seminar Proceedings 05391. Internationales Begegnungs- und Forschungszentrum für Informatik (IBFI), Schloss Dagstuhl, GermanyGoogle Scholar
  22. Lasserre J-B (2001) Global optimization with polynomials and the problem of moments. SIAM J Optim 11(3):796–817CrossRefGoogle Scholar
  23. Lebbah Y, Michel C, Rueher M (2005) Efficient pruning technique based on linear relaxations. In: Proceedings of global optimization and constraint satisfaction, vol 3478, pp 1–14Google Scholar
  24. Maranas CD, Floudas CA (1997) Global optimization in generalized geometric programming. Comput Chem Eng 21(4):351–369CrossRefGoogle Scholar
  25. Markot MC, Fernandez J, Casado LG, Csendes T (2006) New interval methods for constrained global optimization. Math Program 106(2):287–318CrossRefGoogle Scholar
  26. Messine F (2002) Extensions of affine arithmetic: application to unconstrained global optimization. J Univ Comput Sci 8(11):992–1015Google Scholar
  27. Messine F (2004) Deterministic global optimization using interval constraint propagation techniques. RAIRO Oper Res 38(4):277–293CrossRefGoogle Scholar
  28. Messine F (2005) A deterministic global optimization algorithm for design problems. In: Audet C, Hansen P, Savard G (eds) Essays and Surveys in Global Optimization. Springer, New York, pp 267–294CrossRefGoogle Scholar
  29. Messine F, Nogarède B, Lagouanelle J-L (1998) Optimal design of electromechanical actuators: a new method based on global optimization. IEEE Trans Magn 34(1):299–308CrossRefGoogle Scholar
  30. Messine F, Touhami A (2006) A general reliable quadratic form: an extension of affine arithmetic. Reliable Comput 12(3):171–192CrossRefGoogle Scholar
  31. Mitsos A, Chachuat B, Barton PI (2009) McCormick-based relaxations of algorithms. SIAM J Optim 20(2):573–601CrossRefGoogle Scholar
  32. Moore RE (1966) Interval analysis. Prentice-Hall Inc., Englewood CliffsGoogle Scholar
  33. Neumaier A, Shcherbina O (2004) Safe bounds in linear and mixed-integer linear programming. Math Program Ser A 99(2):283–296CrossRefGoogle Scholar
  34. Neumaier A, Shcherbina O, Huyer W, Vinko T (2005) A comparison of complete global optimization solvers. Math Program Ser B 103(2):335–356CrossRefGoogle Scholar
  35. Ninin J (2010) Optimisation Globale basé sur l’Analyse d’Intervalles: relaxation affine et limitation de la mémoire. PhD thesis, Institut National Polytechnique de ToulouseGoogle Scholar
  36. Perron S (2004) Applications jointes de l’optimisation combinatoire et globale. PhD thesis, École Polytechnique de MontréalGoogle Scholar
  37. Ratschek H, Rokne J (1988) New computer methods for global optimization. Ellis Horwood Ltd, ChichesterGoogle Scholar
  38. Schichl H, Markót MC, Neumaier A (2014) Exclusion regions for optimization problems. J Global Optim 59(2–3):569–595. doi:10.1007/s10898-013-0137-z
  39. Schichl H, Neumaier A (2005) Interval analysis on directed acyclic graphs for global optimization. J Global Optim 33(4):541–562CrossRefGoogle Scholar
  40. Shcherbina O, Neumaier A, Sam-Haroud D, Vu X-H, Nguyen T-V (2003) Set of test problems COCONUT, Benchmarking global optimization and constraint satisfaction codes. Springer.http://www.mat.univie.ac.at/~neum/glopt/coconut/Benchmark/Benchmark.html
  41. Sherali HD, Adams WP (1999) A reformulation-linearization technique for solving discrete and continuous nonconvex problems. Kluwer Academis Publishers, DordrechtCrossRefGoogle Scholar
  42. Stolfi J, de Figueiredo L (1997) Self-validated numerical methods and applications. Monograph for 21st Brazilian Mathematics Colloquium. IMPA/CNPq, Rio de Janeiro, BrazilGoogle Scholar
  43. Tawarmalani M, Sahinidis NV (2004) Global optimization of mixed-integer nonlinear programs: a theoretical and computational study. Math Program Ser A 99(3):563–591CrossRefGoogle Scholar
  44. Van Hentenryck P, Michel L, Deville Y (1997) Numerica: a modelling language for global optimization. MIT Press, MassachusettsGoogle Scholar
  45. Vu X-H, Sam-Haroud D, Faltings B (2009) Enhancing numerical constraint propagation using multiple inclusion representations. Ann Math Artif Intell 55(3–4):295–354CrossRefGoogle Scholar

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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