Journal of Global Optimization

, Volume 60, Issue 2, pp 145–164 | Cite as

Upper bounding in inner regions for global optimization under inequality constraints

  • Ignacio Araya
  • Gilles Trombettoni
  • Bertrand Neveu
  • Gilles Chabert
Article

Abstract

In deterministic continuous constrained global optimization, upper bounding the objective function generally resorts to local minimization at several nodes/iterations of the branch and bound. We propose in this paper an alternative approach when the constraints are inequalities and the feasible space has a non-null volume. First, we extract an inner region, i.e., an entirely feasible convex polyhedron or box in which all points satisfy the constraints. Second, we select a point inside the extracted inner region and update the upper bound with its cost. We describe in this paper two original inner region extraction algorithms implemented in our interval B&B called IbexOpt (AAAI, pp 99–104, 2011). They apply to nonconvex constraints involving mathematical operators like , \( +\; \bullet ,\; /,\; power,\; sqrt,\; exp,\; log,\; sin\). This upper bounding shows very good performance obtained on medium-sized systems proposed in the COCONUT suite.

Keywords

Global optimization Upper bounding Intervals  Branch and bound  Inner regions Interval Taylor 

References

  1. 1.
    Araya, I., Trombettoni, G., Neveu, B.: Exploiting monotonicity in interval constraint propagation. In: Proceedings of AAAI, pp. 9–14 (2010)Google Scholar
  2. 2.
    Araya, I., Trombettoni, G., Neveu, B.: A contractor based on convex interval Taylor. In: CPAIOR, pp. 1–16. LNCS 7298 (2012)Google Scholar
  3. 3.
    Belotti, P.: Couenne, a user’s manual (2013). http://www.coin-or.org/Couenne/
  4. 4.
    Benhamou, F., Goualard, F.: Universally quantified interval constraints. In: Proceedings of CP, Constraint Programming, LNCS 1894, pp. 67–82 (2004)Google Scholar
  5. 5.
    Benhamou, F., Goualard, F., Granvilliers, L., Puget, J.F.: Revising hull and box consistency. In: Proceedings of ICLP, pp. 230–244 (1999)Google Scholar
  6. 6.
    Bliek, C.: Computer methods for design automation. Ph.D. thesis, MIT (1992)Google Scholar
  7. 7.
    Chabert, G., Beldiceanu, N.: Sweeping with continuous domains. In: Proceedings of CP, LNCS 6308, pp. 137–151 (2010)Google Scholar
  8. 8.
    Chabert, G., Jaulin, L.: Contractor programming. Artif. Intell. 173, 1079–1100 (2009)CrossRefGoogle Scholar
  9. 9.
    Collavizza, H., Delobel, F., Rueher, M.: Extending consistent domains of numeric CSP. In: Proceedings of IJCAI, pp. 406–413 (1999)Google Scholar
  10. 10.
    Goldsztejn, A.: Définition et applications des extensions des fonctions réelles aux intervalles généralisés: nouvelle formulation de la théorie des intervalles modaux et nouveaux résultats. Ph.D. thesis, University of Nice Sophia Antipolis (2005)Google Scholar
  11. 11.
    Hansen, E.: Global Optimization Using Interval Analysis. Marcel Dekker, New York (1992)Google Scholar
  12. 12.
    Kearfott, R.B.: Rigorous Global Search: Continuous Problems. Kluwer Academic, Dordrecht (1996)CrossRefGoogle Scholar
  13. 13.
    Kreinovich, V., Lakeyev, A., Rohn, J., Kahl, P.: Computational Complexity and Feasibility of Data Processing and Interval Computations. Kluwer, Dordrecht (1997)Google Scholar
  14. 14.
    Lebbah, Y., Michel, C., Rueher, M., Daney, D., Merlet, J.: Efficient and safe global constraints for handling numerical constraint systems. SIAM J. Numer. Anal. 42(5), 2076–2097 (2005)CrossRefGoogle Scholar
  15. 15.
    Lin, Y., Stadtherr, M.: LP strategy for the interval-Newton method in deterministic global optimization. Ind. Eng. Chem. Res. 43, 3741–3749 (2004)CrossRefGoogle Scholar
  16. 16.
    McAllester, D., Van Hentenryck, P., Kapur, D.: Three cuts for accelerated interval propagation. tech. rep. AI Memo 1542, Massachusetts Institute of Technology (1995)Google Scholar
  17. 17.
    Messine, F., Laganouelle, J.L.: Enclosure methods for multivariate differentiable functions and application to global optimization. J. Univ. Comput. Sci. 4(6), 589–603 (1998)Google Scholar
  18. 18.
    Messine, F.: Méthodes d’optimisation globale basées sur l’analyse d’intervalle pour la résolution des problèmes avec contraintes. Ph.D. thesis, LIMA-IRIT-ENSEEIHT-INPT, Toulouse (1997)Google Scholar
  19. 19.
    Moore, R., Kearfott, R.B., Cloud, M.: Introduction to Interval Analysis. SIAM, Philadelphia (2009)Google Scholar
  20. 20.
    Moore, R.E.: Interval Analysis. Prentice-Hall, Englewood Cliffs (1966)Google Scholar
  21. 21.
    Neumaier, A.: Interval Methods for Systems of Equations. Cambridge University Press, Cambridge (1990)Google Scholar
  22. 22.
    Ninin, J., Messine, F., Hansen, P.: A reliable affine relaxation method for global optimization. tech. rep. RT-APO-10-05, IRIT (2010)Google Scholar
  23. 23.
    Oettli, W.: On the solution set of a linear system with inaccurate coefficients. SIAM J. Numer. Anal. 2(1), 115–118 (1965)Google Scholar
  24. 24.
    Rohn, J.: Inner solutions of linear interval systems. In: Proceedings of Interval Mathematics 1985, LNCS 212, pp. 157–158 (1986)Google Scholar
  25. 25.
    Shary, S.: Solving the linear interval tolerance problem. Math. Comput. Simul. 39, 53–85 (1995)CrossRefGoogle Scholar
  26. 26.
    Tawarmalani, M., Sahinidis, N.V.: A polyhedral branch-and-cut approach to global optimization. Math. Program. 103(2), 225–249 (2005)CrossRefGoogle Scholar
  27. 27.
    Trombettoni, G., Araya, I., Neveu, B., Chabert, G.: Inner regions and interval linearizations for global optimization. In: AAAI, pp. 99–104 (2011)Google Scholar
  28. 28.
    Trombettoni, G., Chabert, G.: Constructive interval disjunction. In: Proceedings of CP, LNCS 4741, pp. 635–650 (2007)Google Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • Ignacio Araya
    • 1
  • Gilles Trombettoni
    • 2
  • Bertrand Neveu
    • 3
  • Gilles Chabert
    • 4
  1. 1.UTFSMUniversitad Federico Santa MariaValparaisoChile
  2. 2.IRIT, LIRMMUniversité Montpellier 2MontpellierFrance
  3. 3.Imagine LIGM Université Paris–EstChamps-sur-MarneFrance
  4. 4.LINAEcole de Mines de NantesNantesFrance

Personalised recommendations