Nonlinear biobjective optimization: improving the upper envelope using feasible line segments


In this work, we propose a segment-based representation for the upper bound of the non-dominated set in interval branch & bound solvers for biobjective non linear optimization. We ensure that every point over the upper line segments is dominated by at least one point in the feasible objective region. Segments are generated by linear envelopes of the image of feasible line segments. Finally, we show that the segment-based representation together with methods for generating upper line segments allows us to converge more quickly to the desired precision of the whole strategy. The code of our solver can be found in our git repository (

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6


  1. 1.

    Due to the infinite precision of real numbers, computer programs use IEEE-754 floating point numbers [19] for representing reals. Floating point numbers are composed by two integers: a mantissa m and an exponent e which together represent the real number \(m\cdot 2^e\).

  2. 2.

    For the CTP instances, we consider a function \(\phi =1+\sum _{i=2}^n x_i\), otherwise we obtain only one \(\epsilon \)-efficient solution: \(x=(0,0,\ldots )\), \(y=(0,1)\). For the CF3 instances, we consider a size \(n=5\) instead of \(n=2\), because with \(n=2\), we obtain the undetermined term 2/0 in the first objective.


  1. 1.

    Deb, K.: Multi-objective evolutionary algorithms. In: Kacprzyk J., Pedrycz W. (eds.) Springer Handbook of Computational Intelligence. Springer Handbooks. Springer, Berlin, Heidelberg (2015)

  2. 2.

    Miettinen, K.: Nonlinear Multiobjective Optimization, vol. 12. Springer, Berlin (2012)

    Google Scholar 

  3. 3.

    Collette, Y., Siarry, P.: Multiobjective Optimization: Principles and Case Studies. Springer, Heidelberg (2013)

    Google Scholar 

  4. 4.

    Evtushenko, Y.G., Posypkin, M.A.: Nonuniform covering method as applied to multicriteria optimization problems with guaranteed accuracy. Comput. Math. Math. Phys. 53(2), 144–157 (2013)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Žilinskas, A., Žilinskas, J.: Adaptation of a one-step worst-case optimal univariate algorithm of bi-objective Lipschitz optimization to multidimensional problems. Commun. Nonlinear Sci. Numer. Simul. 21(1–3), 89–98 (2015)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Pardalos, P.M., Žilinskas, A., Žilinskas, J.: Non-convex Multi-objective Optimization. Springer, New York (2017)

    Google Scholar 

  7. 7.

    Ruetsch, G.: An interval algorithm for multi-objective optimization. Struct. Multidiscipl. Optim. 30(1), 27–37 (2005)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Fernández, J., Tóth, B.: Obtaining an outer approximation of the efficient set of nonlinear biobjective problems. J. Glob. Optim. 38(2), 315–331 (2007)

    MathSciNet  Article  Google Scholar 

  9. 9.

    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)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Kubica, B.J., Woźniak, A.: Tuning the interval algorithm for seeking Pareto sets of multi-criteria problems. In: Manninen, P., Öster, P. (eds.) International Workshop on Applied Parallel Computing, pp. 504–517. Springer, New York (2012)

  11. 11.

    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)

    MathSciNet  Article  Google Scholar 

  12. 12.

    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)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Niebling, J., Eichfelder, G.: A branch-and-bound based algorithm for nonconvex multiobjective optimization. SIAM J. Optim. 29(1), 794–821 (2019)

  14. 14.

    Araya, I., Campusano, J., Aliquintui, D.: Nonlinear biobjective optimization: improvements to interval branch & bound algorithms. J. Glob. Optim. 75(1), 91–110 (2019)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Goldsztejn, A., Domes, F., Chevalier, B.: First order rejection tests for multiple-objective optimization. J. Glob. Optim. 58(4), 653–672 (2014)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Hansen, E., Walster, G.W.: Global Optimization using Interval Analysis: Revised and Expanded, vol. 264. CRC Press, Boca Raton (2003)

    Google Scholar 

  17. 17.

    Jaulin, L., Kieffer, M., Didrit, O., Walter, E.: Applied Interval Analysis. Springer, Berlin (2001)

    Google Scholar 

  18. 18.

    Kutateladze, S.: Convex e-programming. Sov. Math. Dokl 20(2), 391–393 (1979)

    MATH  Google Scholar 

  19. 19.

    Kahan, W.: IEEE standard 754 for binary floating-point arithmetic. Lect. Notes Status IEEE 754(94720–1776), 11 (1996)

    Google Scholar 

  20. 20.

    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)

    MathSciNet  Article  Google Scholar 

  21. 21.

    Araya, I., Trombettoni, G., Neveu, B.: A contractor based on convex interval taylor. In: Beldiceanu, N., Jussien N., Pinson, É. (eds.) Integration of AI and OR Techniques in Contraint Programming for Combinatorial Optimzation Problems, pp. 1–16. Springer, New York (2012)

  22. 22.

    Hladík, M., Horáček, J.: Interval linear programming techniques in constraint programming and global optimization. In: Ceberio, M., Kreinovich, V. (eds.) Constraint Programming and Decision Making, pp. 47–59. Springer, New York (2014)

  23. 23.

    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)

    MathSciNet  Article  Google Scholar 

  24. 24.

    Ninin, J., Messine, F., Hansen, P.: A reliable affine relaxation method for global optimization. 4OR 13(3), 247–277 (2015)

    MathSciNet  Article  Google Scholar 

  25. 25.

    Jaulin, L.: Reliable minimax parameter estimation. Reliab. Comput. 7(3), 231–246 (2001)

    MathSciNet  Article  Google Scholar 

  26. 26.

    Spielman, D.A., Teng, S.-H.: Smoothed analysis of algorithms: why the simplex algorithm usually takes polynomial time. J. ACM (JACM) 51(3), 385–463 (2004)

    MathSciNet  Article  Google Scholar 

  27. 27.

    Chabert, G., Jaulin, L.: Contractor programming. Artif. Intell. 173, 1079–1100 (2009)

    MathSciNet  Article  Google Scholar 

  28. 28.

    Benhamou, F., Goualard, F., Granvilliers, L., Puget, J.-F.: Revising hull and box consistency. In: Int. Conf. on Logic Programming. Citeseer, Las Cruces (1999)

  29. 29.

    Trombettoni, G., Chabert, G.: Constructive interval disjunction. In: Bessière, C. (ed.) Principles and Practice of Constraint Programming (CP 2007), pp. 635–650. Springer, New York (2007)

  30. 30.

    Csendes, T., Ratz, D.: Subdivision direction selection in interval methods for global optimization. SIAM J. Numer. Anal. 34(3), 922–938 (1997)

    MathSciNet  Article  Google Scholar 

  31. 31.

    Moore, R.: Interval Analysis. Prentice-Hall, Englewood Cliffs (1966)

  32. 32.

    Araya, I., Neveu, B.: lSMEAR: a variable selection strategy for interval branch and bound solvers. J. Glob. Optim. 71(3), 483–500 (2018)

    MathSciNet  Article  Google Scholar 

  33. 33.

    Wilcoxon, F., Katti, S., Wilcox, R.A.: Critical values and probability levels for the Wilcoxon rank sum test and the Wilcoxon signed rank test. Sel. Tables Math. Stat. 1, 171–259 (1970)

    MATH  Google Scholar 

  34. 34.

    Dolan, E.D., Moré, J.J.: Benchmarking optimization software with performance profiles. Math. Progr. 91(2), 201–213 (2002)

    MathSciNet  Article  Google Scholar 

Download references


This work is supported by the Fondecyt Project 1200035.

Author information



Corresponding author

Correspondence to Ignacio Araya.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Araya, I., Aliquintui, D., Ardiles, F. et al. Nonlinear biobjective optimization: improving the upper envelope using feasible line segments. J Glob Optim 79, 503–520 (2021).

Download citation


  • Interval methods
  • Branch & bound
  • Multiobjective optimization