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Postoptimal analysis of a vector minimax combinatorial problem

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Abstract

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.

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References

  1. V. F. Dem yanov and V. N. Malozemov, An Introduction to Minimax [in Russian], Nauka, Moscow (1972).

    Google Scholar 

  2. V. V. Fedorov, Numerical Maximin Methods [in Russian], Nauka, Moscow (1979).

    Google Scholar 

  3. D.-Z. Du and P. M. Pardalos (eds.), Minimax and Applications, Kluwer Acad. Publ., Dordrecht (1995).

    MATH  Google Scholar 

  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. I. V. Sergienko and V. P. Shilo, Discrete Optimization Problems. Challenges, Solution Methods, and Analysis [in Russian], Naukova Dumka, Kyiv (2003).

    Google Scholar 

  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).

    Article  MathSciNet  Google Scholar 

  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).

    Article  MathSciNet  MATH  Google Scholar 

  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).

    Article  MathSciNet  MATH  Google Scholar 

  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).

    Article  MathSciNet  MATH  Google Scholar 

  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).

    MathSciNet  MATH  Google Scholar 

  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).

    MathSciNet  Google Scholar 

  12. V. A. Emelichev, K. G. Kuzmin, and A. M. Leonovich, “Stability in combinatorial vector optimization problems,” Avtom. Telemekh., No. 2, 79–92 (2004).

  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).

    MathSciNet  Google Scholar 

  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).

    Article  MathSciNet  MATH  Google Scholar 

  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).

    Article  MathSciNet  MATH  Google Scholar 

  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.

  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).

    Article  MathSciNet  MATH  Google Scholar 

  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).

    Article  MathSciNet  MATH  Google Scholar 

  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).

    Article  MathSciNet  MATH  Google Scholar 

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Correspondence to V. A. Emelichev.

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Translated from Kibernetika i Sistemnyi Analiz, No. 3, pp. 95–108, May–June 2011.

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Emelichev, V.A., Korotkov, V.V. & Kuzmin, K.G. Postoptimal analysis of a vector minimax combinatorial problem. Cybern Syst Anal 47, 415–425 (2011). https://doi.org/10.1007/s10559-011-9323-5

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  • DOI: https://doi.org/10.1007/s10559-011-9323-5

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