Cybernetics and Systems Analysis

, Volume 47, Issue 3, pp 415–425 | Cite as

Postoptimal analysis of a vector minimax combinatorial problem

  • V. A. Emelichev
  • V. V. Korotkov
  • K. G. Kuzmin


This paper considers a vector combinatorial problem with minimax criteria that provide the greatest uniformity of the parameters of efficient solutions. The necessary and sufficient conditions are obtained for five well-known types of stability of the problem against perturbations of parameters of the vector objective function.


vector minimax combinatorial problem range criteria stability with respect to vector criterion perturbations of initial data 


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  1. 1.
    V. F. Dem yanov and V. N. Malozemov, An Introduction to Minimax [in Russian], Nauka, Moscow (1972).Google Scholar
  2. 2.
    V. V. Fedorov, Numerical Maximin Methods [in Russian], Nauka, Moscow (1979).Google Scholar
  3. 3.
    D.-Z. Du and P. M. Pardalos (eds.), Minimax and Applications, Kluwer Acad. Publ., Dordrecht (1995).MATHGoogle Scholar
  4. 4.
    I. V. Sergienko, L. N. Kozeratskaya, and T. T. Lebedeva, Stability and Parametric Analyses of Discrete Optimization Problems [in Russian], Naukova Dumka, Kyiv (1995).Google Scholar
  5. 5.
    I. V. Sergienko and V. P. Shilo, Discrete Optimization Problems. Challenges, Solution Methods, and Analysis [in Russian], Naukova Dumka, Kyiv (2003).Google Scholar
  6. 6.
    L. N. Kozeratskaya, T. T. Lebedeva, and T. I. Sergienko, “Stability of discrete optimization problems,” Cybern. Syst. Analysis, 29, No. 3, 367–378 (1993).MathSciNetCrossRefGoogle Scholar
  7. 7.
    Yu. N. Sotskov, V. K. Leontev, and E. N. Gordeev, “Some concepts of stability analysis in combinatorial optimization,” Discrete Appl. Math., 58, No. 2, 169–190 (1995).MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    V. A. Emelichev, E. Girlich, Yu. V. Nikulin, and D. P. Podkopaev, “Stability and regularization of vector problems of integer linear programming,” Optimization, 51, No. 4, 645–676 (2002).MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    S. E. Bukhtoyarov, V. A. Emelichev, and Yu. V. Stepanishina, “Stability of discrete vector problems with the parametric principle of optimality,” Cybern. Syst. Analysis, 39, No. 4, 604–614 (2003).MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    M. Libura and Yu. Nikulin, “Stability and accuracy functions in multicriteria combinatorial optimization problem with Σ-MINMAX and Σ-MINMIN partial criteria,” Control and Cybernetics, 33, No. 3, 511–524 (2004).MathSciNetMATHGoogle Scholar
  11. 11.
    V. A. Emelichev, M. K. Kravtsov, and D. P. Podkopaev, “Quasistability of trajectory vector optimization problems,” Mat. Zametki, 63, Issue 1, 21–27 (1998).MathSciNetGoogle Scholar
  12. 12.
    V. A. Emelichev, K. G. Kuzmin, and A. M. Leonovich, “Stability in combinatorial vector optimization problems,” Avtom. Telemekh., No. 2, 79–92 (2004).Google Scholar
  13. 13.
    V. A. Emelichev and E. E. Gurevskii, “Stability kernel in a multicriterion combinatorial minimax problem,” Diskret. Analiz Issted. Oper., 15, No. 5, 6–19 (2008).MathSciNetGoogle Scholar
  14. 14.
    V. A. Emelichev and K. G. Kuz min, “Stability criteria in vector combinatorial bottleneck problems in terms of binary relations,” Cybern. Syst. Analysis, 44, No. 3, 397–405 (2008).MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    T. T. Lebedeva and T. I. Sergienko, “Different types of stability of vector integer optimization problem: General approach,” Cybern. Syst. Analysis, 44, No. 3, 429–433 (2008).MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    V. A. Emelichev, K. G. Kuz min, and V. V. Korotkov, “On stability of vector minimax problem,” in: Abstr. Intern. Conf. “Problems of decision making under uncertainties” (PDMU-2009), Skhidnytsia, Taras Shevchenko Nat. Univ. of Kyiv, Kyiv (2009), pp. 23–25.Google Scholar
  17. 17.
    T. T. Lebedeva and T. I. Sergienko, “Comparative analysis of different types of stability with respect to constraints of a vector integer-optimization problem,” Cybern. Syst. Analysis, 40, No. 1, 52–57 (2004).MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    T. T. Lebedeva, N. V. Semenova, and T. I. Sergienko, “Stability of vector problems of integer optimization: Relationship with the stability of sets of optimal and nonoptimal solutions,” Cybern. Syst. Analysis, 41, No. 4, 551–558 (2005).MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    T. T. Lebedeva and T. I. Sergienko, “Stability of a vector integer quadratic programming problem with respect to vector criterion and constraints,” Cybern. Syst. Analysis, 42, No. 5, 667–674 (2006).MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2011

Authors and Affiliations

  • V. A. Emelichev
    • 1
  • V. V. Korotkov
    • 1
  • K. G. Kuzmin
    • 1
  1. 1.Belarussian State UniversityMinskBelarus

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