On the Lexicographic Centre of Multiple Objective Optimization


We study the lexicographic centre of multiple objective optimization. Analysing the lexicographic-order properties yields the result that, if the multiple objective programming’s lexicographic centre is not empty, then it is a subset of all efficient solutions. It exists if the image set of multiple objective programming is bounded below and closed. The multiple objective linear programming’s lexicographic centre is nonempty if and only if there exists an efficient solution to the multiple objective linear programming. We propose a polynomial-time algorithm to determine whether there is an efficient solution to multiple objective linear programming, and we solve the multiple objective linear programming’s lexicographic centre by calculating at most the same number of dual linear programs as the number of objective functions and a system of linear inequalities.

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


  1. 1.

    Ogryczak, W.: On the lexicographic minimax approach to location problems. Eur. J. Oper. Res. 100, 566–585 (1997)

    Article  MATH  Google Scholar 

  2. 2.

    Luss, H.: On equitable resource allocation problems: a lexicographic minimax approach. Oper. Res. 47(3), 182–187 (1999)

    Article  Google Scholar 

  3. 3.

    Shmeidler, D.: The nucleolus of a characteristic function game. SIAM J. Appl. Math. 17, 1163–1170 (1969)

    Article  MathSciNet  Google Scholar 

  4. 4.

    Maschler, M., Peleg, B., Shapley, L.S.: Geometric properties of the kernel, nucleolus and related solution concepts. Math. Oper. Res. 4, 303–338 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  5. 5.

    Dragan, I.: A procedure for finding the nucleolus of a cooperative n person game. Zeitschrift für Oper. Res. 25(5), 119–131 (1981)

    MathSciNet  MATH  Google Scholar 

  6. 6.

    Benoît, J.-P.: The nucleolus is contested-garment-consistent: a direct proof. J. Econ. Theory 77, 192–196 (1997)

    Article  MATH  Google Scholar 

  7. 7.

    Potters, Jos A.M., Reijnierse, J.H., Ansing, M.: Computing the nucleolus by solving a prolonged simplex algorithm. Math. Oper. Res 21, 757–768 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  8. 8.

    Meertens, M., Potters, Jos A.M.: The nucleolus of trees with revenues. Math. Method Oper. Res. 64(2), 363–382 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  9. 9.

    Potters, Jos A.M., Reijnierse, H., Biswas, A.: The nucleolus of balanced simple flow networks. Game Econ. Behav. 54(1), 205–225 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  10. 10.

    Maschler, M., Potters, Jos A.M., Reijnierse, H.: The nucleolus of a standard tree game revisited: a study of its monotonicity and computational properties. Int. J. Game Theory 39(1–2), 89–104 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  11. 11.

    Marchi, E., Oviedo, J.A.: Lexicographic optimality in the multiple objective linear programming: the nucleolar solution. Eur. J. Oper. Res. 57(3), 355–359 (1992)

    Article  MATH  Google Scholar 

  12. 12.

    Kostreva, M.M., Ogryczak, W., Wierzbicki, A.: Equitable aggregations and multiple criteria analysis. Eur. J. Oper. Res. 158, 362–377 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  13. 13.

    Ogryczak, W., Sliwinski, T., Wierzbicki, A.: Fair resource allocation schemes and network dimensioning problems. J. Telecommun. Inf. Technol. 3, 34–42 (2003)

    Google Scholar 

  14. 14.

    Ogryczak, W., Śliwiński, T.: On direct methods for lexicographic min–max optimization. Lect. Notes Comput. Sci. 3982, 802–811 (2006)

    Article  Google Scholar 

  15. 15.

    Radunović, B., Le Boudec, J.-Y.: A unified framework for max–min and min–max fairness with applications. IEEE/ACM Trans. Netw. 15(5), 1073–1083 (2007)

    Article  Google Scholar 

  16. 16.

    Klein, R.S., Luss, H., Smith, D.R.: A lexicographic minimax algorithm for multiperiod resource allocation. Math. Program. 55, 213–234 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  17. 17.

    Tomaszewski, A.: A polynomial algorithm for solving a general max–min fairness problem. Eur. Trans. Telecommun. 16, 233–240 (2005)

    Article  Google Scholar 

  18. 18.

    Ogryczak, W., Pióro, M., Tomaszewski, A.: Telecommunications network design and max–min optimization problem. J. Telecommun. Inf. Technol. 3, 43–56 (2005)

    Google Scholar 

  19. 19.

    Ehrgott, M., Holder, A., Reese, J.: Beam selection in radiotherapy design. Linear Algebra Appl. 428, 1272–1312 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  20. 20.

    Pióro, M.: Fair routing and related optimization problems. In: Proceedings of the 15th International Conference on Advanced Computing and Communications, Guwahati, Assam, pp. 229–235. IEEE Computer Society (2007)

  21. 21.

    Pióro, M., Nilsson, P., Kubilinskas, E., Fodor, G.: On efficient max-min fair routing algorithms. In: Proceedings of the Eighth IEEE International Symposium on Computers and Communication (ISCC’03), Kemer - Antalya, Turkey, pp. 365–372 vol. 1 (2003)

  22. 22.

    Sun, M.: Some issues in measuring and reporting solution quality of interactive multiple objective programming procedures. Eur. J. Oper. Res. 162, 468–483 (2005)

    Article  MATH  Google Scholar 

  23. 23.

    Ogryczak, W., Wierzbicki, A.: On multi-criteria approaches to bandwidth allocation. Control Cybern. 33(3), 427–448 (2004)

    MathSciNet  MATH  Google Scholar 

  24. 24.

    Ogryczak, W., Wierzbicki, A., Milewskia, M.: A multi-criteria approach to fair and efficient bandwidth allocation. Omega 36, 451–463 (2008)

    Article  Google Scholar 

  25. 25.

    Hoang, D.T., Vitter, J.S.: Efficient Algorithms for MPEG Video Compression. Wiley, New York, NY (2001)

    Google Scholar 

  26. 26.

    Dragan, I.: A game theoretic approach for solving the multiobjective linear programming problems. Lib. Math. 30, 149–158 (2010)

    MathSciNet  MATH  Google Scholar 

  27. 27.

    Behringer, F.A.: A simplex based algorithm for the lexicographically extended linear maxmin problem. Eur. J. Oper. Res. 7, 274–283 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  28. 28.

    Pourkarimi, L., Zarepisheh, M.: A dual-based algorithm for solving lexicographic multiple objective programs. Eur. J. Oper. Res. 176, 1348–1356 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  29. 29.

    Klee, V., Minty, G.J.: How Good is the Simplex Algorithm? Inequalities-III. Academic Press, New York (1972)

    Google Scholar 

  30. 30.

    Friedmann, O., Hansen, T., Zwick, U.: Subexponential lower bounds for randomized pivoting rules for the simplex algorithm. In: STOC’11 Proceedings of the 43rd Annual ACM Symposium on Theory of Computing, pp. 283–292. ACM, San Jose, California, New York (2011)

  31. 31.

    Behringer, F.A.: Linear multiobjective maxmin optimization and some Pareto and lexmaxmin extensions. OR Spektrum 8, 25–32 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  32. 32.

    Khorram, E., Zarepisheh, M., Ghaznavi-ghosoni, B.A.: Sensitivity analysis on the priority of the objective functions in lexicographic multiple objective linear programs. Eur. J. Oper. Res. 207, 1162–1168 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  33. 33.

    Luenberger, D.G., Ye, Y.: Linear and Nonlinear Programming, 3rd edn. Springer, New York, NY (2010)

    Google Scholar 

Download references


We thank Dr. Irinel C. Dragan for his helpful comments and suggestions

Author information



Corresponding author

Correspondence to Shitao Yang.

Additional information

Communicated by Irinel C. Dragan.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Jiangao, Z., Yang, S. On the Lexicographic Centre of Multiple Objective Optimization. J Optim Theory Appl 168, 600–614 (2016). https://doi.org/10.1007/s10957-015-0810-0

Download citation


  • Multiple objective programming
  • Lexicographic order
  • Lexicographic centre
  • Efficient solution
  • Image set
  • \(\theta \)-image set

Mathematics Subject Classification

  • 49M29
  • 65K05
  • 90C05
  • 90C29
  • 90C47