Journal of Global Optimization

, Volume 58, Issue 4, pp 653–672 | Cite as

First order rejection tests for multiple-objective optimization

  • Alexandre Goldsztejn
  • Ferenc Domes
  • Brice Chevalier


Three rejection tests for multi-objective optimization problems based on first order optimality conditions are proposed. These tests can certify that a box does not contain any local minimizer, and thus it can be excluded from the search process. They generalize previously proposed rejection tests in several regards: Their scope include inequality and equality constrained smooth or nonsmooth multiple objective problems. Reported experiments show that they allow quite efficiently removing the cluster effect in mono-objective and multi-objective problems, which is one of the key issues in continuous global deterministic optimization.


Multi-objective deterministic global optimization First order optimality conditions Interval analysis Branch and bound algorithm Cluster effect 



This work was partially funded by the Région Pays de la Loire of France, and the National Institute of Informatics of Japan. The authors are much grateful to Christophe Jermann for his valuable help in experimenting and analyzing the cluster effect of the presented academic examples.


  1. 1.
    Alefeld, G., Herzberger, J.: Introduction to interval computations. Comput. Sci. Appl. Math. (1974)Google Scholar
  2. 2.
    Barichard, V., Deleau, H., Hao, J.K., Saubion, F.: A hybrid evolutionary algorithm for CSP. In: Artificial Evolution, volume 2936 of LNCS, pp. 79–90. Springer, Berlin (2004)Google Scholar
  3. 3.
    Barichard, V., Hao, J.K.: A population and interval constraint propagation algorithm. In: Evolutionary Multi-Criterion Optimization, volume 2632 of LNCS, pp. 72–72. Springer, Berlin (2003)Google Scholar
  4. 4.
    Clarke, F.H.: Optimization and Nonsmooth Analysis. Society for Industrial and Applied Mathematics, Philadephia (1990)CrossRefGoogle Scholar
  5. 5.
    Domes, F.: Gloptlab: a configurable framework for the rigorous global solution of quadratic constraint satisfaction problems. Optim. Methods Softw. 24(4–5), 727–747 (2009)CrossRefGoogle Scholar
  6. 6.
    Du, K., Kearfott, R.B.: The cluster problem in multivariate global optimization. J. Glob. Optim. 5, 253–265 (1994)CrossRefGoogle Scholar
  7. 7.
    Duran, M.A., Grossman, I.E.: An outer-approximation algorithm for a class of mixed-integer nonlinear programs. Math. Program. 36, 307–339 (1986)CrossRefGoogle Scholar
  8. 8.
    Garloff, J.: Interval gaussian elimination with pivot tightening. SIAM J. Matrix Anal. Appl. 30(4), 1761–1772 (2009)CrossRefGoogle Scholar
  9. 9.
    Goldberg, D.: What every computer scientist should know about floating-point arithmetic. Comput. Surv. 23(1), 5–48 (1991)CrossRefGoogle Scholar
  10. 10.
    Goualard, F.: GAOL 3.1.1: Not Just Another Interval Arithmetic Library, 4th edn. Laboratoire d’Informatique de Nantes-Atlantique (2006)Google Scholar
  11. 11.
    Granvilliers, L., Goldsztejn, A.: A branch-and-bound algorithm for unconstrained global optimization. In: Proceedings of the 14th GAMM-IMACS International Symposium on Scientific Computing, Computer Arithmetic and Validated Numerics (SCAN) (2010)Google Scholar
  12. 12.
    Hansen, E.: Global Optimization Using Interval Analysis, 2nd edn. Marcel Dekker, NY (1992)Google Scholar
  13. 13.
    Hansen, E.R., Walster, G.W.: Bounds for Lagrange multipliers and optimal points. Comput. Math. Appl. 25(1011), 59–69 (1993)CrossRefGoogle Scholar
  14. 14.
    Ichida, K., Fujii, Y.: Multicriterion optimization using interval analysis. Computing 44, 47–57 (1990)CrossRefGoogle Scholar
  15. 15.
    Jahn, J.: Multiobjective search algorithm with subdivision technique. Comput. Optim. Appl. 35(2), 161–175 (2006)CrossRefGoogle Scholar
  16. 16.
    Jaulin, L., Kieffer, M., Didrit, O., Walter, E.: Applied Interval Analysis with Examples in Parameter and State Estimation, Robust Control and Robotics. Springer, Berlin (2001)Google Scholar
  17. 17.
    Kearfott, R.B.: Interval computations: introduction, uses, and resources. Euromath Bull. 2(1), 95–112 (1996)Google Scholar
  18. 18.
    Kearfott, R.B.: An interval branch and bound algorithm for bound constrained optimization problems. J. Glob. Optim. 2, 259–280 (1992)CrossRefGoogle Scholar
  19. 19.
    Kearfott, R.B.: Interval computations, rigour and non-rigour in deterministic continuous global optimization. Optim. Methods Softw. 26(2), 259–279 (2011)CrossRefGoogle Scholar
  20. 20.
    Knueppel, O.: PROFIL/BIAS—a fast interval library. Computing 53(3–4), 277–287 (1994)CrossRefGoogle Scholar
  21. 21.
    Kubica, BJ., Niewiadomska-Szynkiewicz, E.: An improved interval global optimization method and its application to price management problem. In: Applied Parallel Computing. State of the Art in Scientific Computing, volume 4699 of LNCS, pp. 1055–1064. Springer, Berlin (2007)Google Scholar
  22. 22.
    Kubica, B.J., Wozniak, A.: Interval methods for computing the pareto-front of a multicriterial problem. In: Parallel Processing and Applied Mathematics, volume 4967 of LNCS, pp. 1382–1391. Springer, Berlin (2008)Google Scholar
  23. 23.
    Kubica, B.J., Wozniak, A.: Using the second-order information in pareto-set Computations of a multi-criteria problem. In: Applied Parallel and Scientific Computing, volume 7134 of LNCS, pp. 137–147. Springer, Berlin (2012)Google Scholar
  24. 24.
    Lin, Y., Stadtherr, M.A.: Encyclopedia of Optimization. In: Floudas, C.A., Pardalos, P.M. (eds.) LP Strategy for Interval-Newton Method in Deterministic Global Optimization, pp. 1937–1943. Springer, USA (2009)Google Scholar
  25. 25.
    Makino, K., Berz, M.: Automatic differentiation of algorithms. In: Corliss, G., Faure, C., Griewank, A., Hascoët, L., Naumann, U. (eds.) New Applications of Taylor Model Methods, Automatic Differentiation of Algorithms. Springer, New York (2002)Google Scholar
  26. 26.
    Miettinen, K.M.: Nonlinear Multiobjective Optimization. Kluwer, Dordrecht (1998)CrossRefGoogle Scholar
  27. 27.
    Moore, R.: Interval Analysis. Prentice-Hall, Englewood Cliffs, NJ (1966)Google Scholar
  28. 28.
    Neumaier, A.: Interval Methods for Systems of Equations. Cambridge Univ Press, Cambridge (1990)Google Scholar
  29. 29.
    Nocedal, J., Wright, S.J.: Numerical Optimization. Springer, Berlin (2006)Google Scholar
  30. 30.
    Rohn, J.: Forty necessary and sufficient conditions for regularity of interval matrices: a survey. Electron. J. Linear Algebra 18, 500–512 (2009)Google Scholar
  31. 31.
    Ruetsch, G.R.: An interval algorithm for multi-objective optimization. Struct. Multidiscip. Optim. 30, 27–37 (2005)CrossRefGoogle Scholar
  32. 32.
    Ruetsch, G.R.: Using interval techniques of direct comparison and differential formulation to solve a multi-objective optimization problem (patent US-7742902), (2010)Google Scholar
  33. 33.
    Rump, S.M.: Developments in reliable computing. In: Csendes, T. (ed.) INTLAB—Interval Laboratory. Kluwer, Dordrecht (1999)Google Scholar
  34. 34.
    Schichl, H., Neumaier, A.: Exclusion regions for systems of equations. SIAM J. Numer. Anal. 42(1), 383–408 (2004)CrossRefGoogle Scholar
  35. 35.
    Schichl, H., Neumaier, A.: Interval analysis on directed acyclic graphs for global optimization. J. Glob. Optim. 33(4), 541–562 (2005)CrossRefGoogle Scholar
  36. 36.
    Schichl, H., Neumaier, A.: Transposition theorems and qualification-free optimality conditions. SIAM J. Optim. 17(4), 1035–1055 (2006)CrossRefGoogle Scholar
  37. 37.
    Shcherbina, O., Neumaier, A., Sam-Haroud D., Vu, X.-H., Nguyen, T.-V.: Benchmarking global optimization and constraint satisfaction codes. In: Bliek, C., Jermann, C., Neumaier, A. (eds.) Global Optimization and Constraint Satisfaction, volume 2861 of Lecture Notes in Computer Science, pp. 211–222. Springer, Berlin (2003)Google Scholar
  38. 38.
    Soares, G.L., Parreiras, R.O., Jaulin, L., Vasconcelos, J.A., Maia, C.A.: Interval robust multi-objective algorithm. Nonlinear Anal. Theor Methods Appl. 71(12), 1818–1825 (2009)CrossRefGoogle Scholar
  39. 39.
    Tóth, B.G., Hernndez, J.F.: Interval Methods for Single and Bi-objective Optimization Problems Applied to Competitive Facility Location Models. LAP Lambert Academic Publishing (2010)Google Scholar
  40. 40.
    Vu, X.-H., Schichl, H., Sam-Haroud, D.: Interval propagation and search on directed acyclic graphs for numerical constraint solving. J. Glob. Optim. 45(4), 499–531 (2009)CrossRefGoogle Scholar
  41. 41.
    Wolfram Research Inc., Mathematica. Champaign, Illinois, Version 7.0 (2008)Google Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Alexandre Goldsztejn
    • 1
  • Ferenc Domes
    • 2
  • Brice Chevalier
    • 3
  1. 1.CNRS, LINA (UMR 6241)NantesFrance
  2. 2.LINA (UMR 6241)Université de NantesNantesFrance
  3. 3.Université de NantesNantesFrance

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