Abstract
An approach is proposed to the analysis of stability with respect to perturbations of the initial data of the vector integer optimization problem when using different (Pareto, Slater, and Smale) optimality principles. The properties of optimal and non-optimal solutions are investigated from the point of view of the stability of their membership in the sets of optimal and non-optimal solutions, respectively.
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I. V. Sergienko, L. N. Kozeratskaya, and T. T. Lebedeva, Stability and Parametric Analyses of Discrete Optimization Problems [in Russian], Naukova Dumka, Kyiv (1995).
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).
N. V. Semenova, “Stability with respect to constraints of vector integer optimization problems with convex quadratic constraint functions,” Teoriya Optym. Rishen’, V. M. Glushkov Inst. of Cybernetics NANU, Kyiv, No. 6, 131–138 (2007).
T. I. Sergienko, “Stability with respect to vector criterion and constraints of integer problems of searching for Slater and Smale optimal solutions,” Komp. Matematika, V. M. Glushkov Inst. of Cybernetics NANU, Kyiv, No. 1, 145–151 (2008).
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).
T. T. Lebedeva and T. I. Sergienko, “Stability conditions of vector integer problems of searching for Smale optimal solutions,” Komp. Matematika, V. M. Glushkov Inst. of Cybernetics NANU, Kyiv, No. 2, 156–163 (2010).
T. T. Lebedeva and T. I. Sergienko, “Stability conditions with respect to vector criterion and constraints of multicriteria integer optimization problems,” Dop. NANU, No. 4, 37–40 (2011).
T. T. Lebedeva, N. V. Semenova, and T. I. Sergienko, “Stability analysis of vector discrete optimization problems with different principles of optimality,” Dop. NANU, No. 11, 34–39 (2012).
V. A. Emelichev and K. G. Kuzmin, “Stability radius of a vector integer linear programming problem: Case of a regular norm in the space of criteria,” Cybern. Syst. Analysis, 46, No. 1, 72–79 (2010).
V. A. Emelichev and V. V. Korotkov, “Stability radius of a vector investment problem with Savage’s minimax risk criteria,” Cybern. Syst. Analysis, 48, No. 3, 378–386 (2012).
V. A. Emelichev and V. V. Korotkov, “Stability radius of the efficient solution of the bottleneck vector quadratic Boolean problem,” Diskret. Analiz Issled. Oper., 18, No. 6, 3–16 (2011).
V. A. Emelichev, O. V. Karelkina, and K. G. Kuzmin, “Qualitative stability analysis of multicriteria combinatorial minimin problem,” Control and Cybernetics, 41, No. 1, 57–79 (2012).
V. A Emelichev and V. V. Korotkov, “Stability of the lexicographic solution of the vector minimax quadratic Boolean problem,” Tr. Inst. Matematiki NAN Belarusi, 19, No. 2, 26–36 (2011).
V. V. Podinovskii and V. D. Nogin, Pareto Optimal Solutions of Multicriteria Problems [in Russian], Nauka, Moscow (1982).
S. Smale, “Global analysis and economics, V. Pareto theory with constraints,” J. Math. Econ., No. 1, 213–221 (1974).
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The study was sponsored by the State Fund for Fundamental Researches of Ukraine (Project F 54.1/039).
Translated from Kibernetika i Sistemnyi Analiz, No. 2, March–April, 2014, pp. 75–82.
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Lebedeva, T.T., Semenova, N.V. & Sergienko, T.I. Qualitative Characteristics of the Stability Vector Discrete Optimization Problems with Different Optimality Principles. Cybern Syst Anal 50, 228–233 (2014). https://doi.org/10.1007/s10559-014-9609-5
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DOI: https://doi.org/10.1007/s10559-014-9609-5