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Nonlinear biobjective optimization: improvements to interval branch & bound algorithms

  • Ignacio ArayaEmail author
  • Jose Campusano
  • Damir Aliquintui
Article
  • 33 Downloads

Abstract

Interval based solvers are commonly used for solving single-objective nonlinear optimization problems. Their reliability and increasing performance make them useful when proofs of infeasibility and/or certification of solutions are a must. On the other hand, there exist only a few approaches dealing with nonlinear optimization problems, when they consider multiple objectives. In this paper, we propose a new interval branch & bound algorithm for solving nonlinear constrained biobjective optimization problems. Although the general strategy is based on other works, we propose some improvements related to the termination criteria, node selection, upperbounding and discarding boxes using the non-dominated set. Most of these techniques use and/or adapt components of IbexOpt, a state-of-the-art interval-based single-objective optimization algorithm. The code of our plugin can be found in our git repository (https://github.com/INFPUCV/ibex-lib/tree/master/plugins/optim-mop).

Keywords

Interval methods Branch & bound Multiobjective optimization 

Notes

Acknowledgements

This work is supported by the Fondecyt Project 1160224.

References

  1. 1.
    Marler, R.T., Arora, J.S.: Survey of multi-objective optimization methods for engineering. Struct. Multidiscipl. Optim. 26(6), 369–395 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Deb, K.: Multi-objective optimization. In: Search Methodologies. Springer, pp. 403–449 (2014)Google Scholar
  3. 3.
    Przybylski, A., Gandibleux, X.: Multi-objective branch and bound. Eur. J. Oper. Res. 260(3), 856–872 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Redondo, J.L., Fernández, J., Ortigosa, P.M.: FEMOEA: a fast and efficient multi-objective evolutionary algorithm. Math. Methods Oper. Res. 85(1), 113–135 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Coello, C.A.C., Lamont, G.B., Van Veldhuizen, D.A., et al.: Evolutionary Algorithms for Solving Multi-objective Problems, vol. 5. Springer, Berlin (2007)zbMATHGoogle Scholar
  6. 6.
    Ruetsch, G.: An interval algorithm for multi-objective optimization. Struct. Multidiscip. Optim. 30(1), 27–37 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Fernández, J., Tóth, B.: Obtaining an outer approximation of the efficient set of nonlinear biobjective problems. J. Glob. Optim. 38(2), 315–331 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Fernández, J., Tóth, B.: Obtaining the efficient set of nonlinear biobjective optimization problems via interval branch-and-bound methods. Comput. Optim. Appl. 42(3), 393–419 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Kubica, B. J., Woźniak, A.: Tuning the interval algorithm for seeking pareto sets of multi-criteria problems. In: International Workshop on Applied Parallel Computing. Springer, pp. 504–517 (2012)Google Scholar
  10. 10.
    Martin, B., Goldsztejn, A., Granvilliers, L., Jermann, C.: On continuation methods for non-linear bi-objective optimization: towards a certified interval-based approach. J. Glob. Optim. 64(1), 3–16 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Martin, B., Goldsztejn, A., Granvilliers, L., Jermann, C.: Constraint propagation using dominance in interval branch & bound for nonlinear biobjective optimization. Eur. J. Oper. Res. 260(3), 934–948 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Niebling, J., Eichfelder, G.: A branch-and-bound based algorithm for nonconvex multiobjective optimization. Preprint-Series of the Institute for Mathematics (2018)Google Scholar
  13. 13.
    Goldsztejn, A., Domes, F., Chevalier, B.: First order rejection tests for multiple-objective optimization. J. Glob. Optim. 58(4), 653–672 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Araya, I., Trombettoni, G., Neveu, B., Chabert, G.: Upper bounding in inner regions for global optimization under inequality constraints. J. Glob. Optim. 60(2), 145–164 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Trombettoni, G., Araya, I., Neveu, B., Chabert, G.: Inner regions and interval linearizations for global optimization. In: AAAI Conference on Artificial Intelligence, pp. 99–104 (2011)Google Scholar
  16. 16.
    Araya, I., Neveu, B.: lsmear: a variable selection strategy for interval branch and bound solvers. J. Glob. Optim. 71(3), 483–500 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Benhamou, F., Goualard, F., Granvilliers, L., Puget, J.-F.: Revising hull and box consistency. In: International Conference on Logic Programming, Citeseer (1999)Google Scholar
  18. 18.
    Neveu, B., Trombettoni, G., et al.: Adaptive constructive interval disjunction. In: International Conference on Tools with Artificial Intelligence (ICTAI), pp. 900–906 (2013)Google Scholar
  19. 19.
    Ninin, J., Messine, F., Hansen, P.: A reliable affine relaxation method for global optimization. 4OR 13(3), 247–277 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Araya, I., Trombettoni, G., Neveu, B.: A contractor based on convex interval taylor. In: Integration of AI and OR Techniques in Contraint Programming for Combinatorial Optimzation Problems. Springer, pp. 1–16 (2012)Google Scholar
  21. 21.
    Martin, B.: Rigorous algorithms for nonlinear biobjective optimization. Ph.D. dissertation, Université de Nantes (2014)Google Scholar
  22. 22.
    Tóth, B., Fernández, J.: Interval Methods for Single and Bi-objective Optimization Problems-Applied to Competitive Facility Location Problems. Lambert Academic Publishing, Saarbrücken (2010)Google Scholar
  23. 23.
    Chabert, G., Jaulin, L.: Contractor programming. Artif. Intell. 173, 1079–1100 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Trombettoni, G., Chabert, G.: Constructive interval disjunction. In: Principles and Practice of Constraint Programming (CP 2007). Springer, pp. 635–650 (2007)Google Scholar
  25. 25.
    Csendes, T., Ratz, D.: Subdivision direction selection in interval methods for global optimization. SIAM J. Numer. Anal. 34(3), 922–938 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Moore, R.E.: Interval Analysis. Prentice-Hall, Englewood Cliffs, NJ (1966)Google Scholar
  27. 27.
    Zitzler, E., Thiele, L.: Multiobjective optimization using evolutionary algorithms: a comparative case study. In: International Conference on Parallel Problem Solving from Nature. Springer, pp. 292–301 (1998)Google Scholar
  28. 28.
    Neveu, B., Trombettoni, G., Araya, I.: Node selection strategies in interval branch and bound algorithms. J. Glob. Optim. 64(2), 289–304 (2016)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Pontificia Universidad Católica de ValparaísoValparaísoChile

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