Abstract
The paper considers a vector (multiobjective) path problem with minimax partial criteria. Solving this problem means finding a Pareto set. Binary relations on a set of paths are used to formulate the necessary and sufficient conditions for five types of problem stability against perturbations of the parameters of a vector criterion. Some sufficient stability conditions in terms of Pareto, Smale, and Slater sets are obtained as corollaries.
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References
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).
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).
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).
I. V. Sergienko, L. N. Kozeratskaya, and T. T. Lebedeva, Stability and Parametric Analyses of Discrete Optimization Problems [in Russian], Naukova Dumka, Kyiv (1995).
I. V. Sergienko and V. P. Shilo, Discrete Optimization Problems. Challenges, Solution Techniques, and Investigations [in Russian], Naukova Dumka, Kyiv (2003).
M. A. Aizerman and F. T. Alekserov, Theory of Choice, North Holland, Amsterdam (1995).
B. A. Berezovskii, V. I. Borzenko, and L. N. Kempner, Binary Relations in Multiobjective Optimization [in Russian], Nauka, Moscow (1981).
V. V. Podinovskii and V. D. Nogin, Pareto-Optimal Solutions of Multiobjective Problems [in Russian], Nauka, Moscow (1982).
L. A. Sholomov, “Studying relations in criterion spaces and synthesis of multiple-choice operators,” in: Mathematical Problems in Cybernetics [in Russian], Issue 5, 109–143 (1994).
V. A. Emelichev and A. V. Pashkevich, “On the parametrization of the optimality principle in a criterion space,” Diskret. Analiz Issled. Oper., Ser. 2, 9, No. 1, 21–32 (2002).
V. K. Leont’ev and E. N. Gordeev, “Qualitative investigation of path problems,” Cybernetics, 23, No. 5, 636–646 (1986).
E. N. Gordeev and V. K. Leont’ev, “General approach to the stability analysis of solutions in discrete optimization problems,” Zh. Vych. Mat. Mat. Fiz., 36, No. 1, 66–72 (1996).
V. A. Emelichev, M. K. Kravtsov, and D. P. Podkopaev, “On quasistability of path problems of vector optimization,” Mat. Zametki, 63, Issue 1, 21–27 (1998).
E. N. Gordeev, “Stability analysis in optimization problems on matroids in the metric l 1,” Cybern. Syst. Analysis, 37, No. 2, 251–259 (2001).
S. E. Bukhtoyarov, V. A. Emelichev, and Yu. V. Stepanishina, “Stability of vector discrete problems with the parametric principle of optimality,” Cybern. Syst. Analysis, 39, No. 4, 604–614 (2003).
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Translated from Kibernetika i Sistemnyi Analiz, No. 3, pp. 103–111, May–June 2008.
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Emelichev, V.A., Kuz’min, K.G. Stability criteria in vector combinatorial bottleneck problems in terms of binary relations. Cybern Syst Anal 44, 397–404 (2008). https://doi.org/10.1007/s10559-008-9001-4
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DOI: https://doi.org/10.1007/s10559-008-9001-4