Abstract
Consideration was given to the vector (multicriteria) problem on the system of subsets of a finite set. In the case of linear subtests, a formula of the stability radius of efficient solution in the metric l 1 was obtained. For the vector problem with subtests of the form MINMAX MODUL, the necessary and sufficient stability conditions were established (retention or contraction of the Pareto set for “small” variations in the source data).
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REFERENCES
Hadamard, J., Sur les problems aux derivées partielles et leur signification partielle et leur signification physique, Bull.Princeton Univ., 1902.
Sergienko, I.V., Kozeratskaya, L.N., and Lebedeva, T.T., Issledovanie ustoichivosti i parametricheskii analiz diskretnykh optimizatsionnykh zadach (Discrete Optimization Problems: Stability Study and Parametric Analysis), Kiev: Naukova Dumka, 1995.
Kozeratskaya, L.N., Lebedeva, T.T., and Sergienko, I.V., Stability Study of Discrete Optimization Problems, Kibern.Sist.Anal., 1993, no. 3, pp. 78-93.
Sotskov, Yu.N., Leontev, V.K., and Gordeev, E.N., Some Concepts of Stability Analysis in Combinatorial Optimization, Discrete Appl.Math., 1995, vol. 58, no. 1, pp. 169-190.
Sotscov, Yu.N., Tanaev, V.S., and Werner, F., Stability Radius of an Optimal Schedule: A Survey and Recent Developments, in Industrial Applications of Combinatorial Optimization, Boston: Kluwer, 1998, vol. 16, pp. 72108.
Greenberg, N.J., An Annotated Bibliography for Post-solution Analysis in Mixed Integer and Combinatorial Optimization, in Advances in Computational and Stochastic Optimization, Logic Programming, and Heuristic Search, Woodruff, D.L., Ed., Boston: Kluwer, 1998, pp. 97-108.
Emelichev, V.A. and Podkopaev, D.P., Stability and Regularization of the Vector Problems Linear Integer Programming, Diskr.Anal.Issled.Operatsii, 2001, vol. 8, no. 1, pp. 47-69.
Emelichev, V.A., Girlich, E., Nikulin, Yu.N., and Podkopaev, D.P., Stability and Regularization of Vector Problem of Integer Linear Programming, Optimization, 2002, vol. 51, no. 4, pp. 645-676.
Belousov, E.G. and Andronov, V.G., Razreshimost' zadach polinomial'nogo programmirovaniya (Solvability of Polynomial Programming Problems), Moscow: Mosk. Gos. Univ., 1993.
Kolokolov, A.A. and Devyaterikova, M.V., Analysis of Stability of L-decomposition of the Sets in Finitedimensional Space, Diskr.Anal.Issled.Operatsii, 2000, vol. 7, no. 2, pp. 47-53.
Devyaterikova, M.V. and Kolokolov, A.A., Analysis of L-structure Stability of Convex Integer Programming Problems, in Oper.Res.Proc., 2000, Springer, pp. 49-54.
Podinovskii, V.V. and Nogin, V.D., Pareto-optimal'nye resheniya mnogokriterial'nykh zadach (Pareto-optimal Solutions of Multicriteria Problems), Moscow: Nauka, 1982.
Leonot'ev, V.K., Stability in Discrete Linear Problems, Probl.Kibern., Moscow: Nauka, 1979, no. 35.
Gordeev, E.N., Stability Study in the Optimization Problems on Matroids in the Metric l1, Kibern.Sist.Anal., 2001, no. 2, pp. 132-144.
Berdysheva, R.A. and Emelichev, V.A., On Stability Radius of the Lexicographic Optimum of the Trajectory Vector Problem,, Vestn.Belarus.Univ., 1998, ser. 1, no. 1, pp. 43-46.
Emelichev, V.A. and Nikulin, Yu.V., On Stability of the Effective Solution of the Integer Vector Problem of Linear Programming, Dokl.Belarus.Nat.Akad.Nauk, 2002, vol. 44, no. 4, pp. 26-28.
Emelichev, V.A. and Pokhil'ko, V.G., Sensitivity Analysis of the Effective Solutions of the Vector Problem of Minimization of Linear Forms on the Set of Substitutions, Discret.Mat., 2000, vol. 12, no. 3, pp. 37-48.
Emelichev, V.A. and Stepanishina, Yu.V., Multicriteria Combinatorial Linear Problems: Parametrization of the Optimality Principle and Stability of Effective Solutions, Discret.Mat., 2001, vol. 13, no. 4, pp. 43-51.
Emelichev, V.A. and Krichko, V.N., Stability Radius of Effective Solution of the Vector Quadratic Boolean Programming Problem, Zh.Vychisl.Mat.Mat.Fiz., 2001, vol. 41, no. 2, pp. 346-350.
Emelichev, V.A. and Kravtsov, M.K., On Stability of the Trajectory Problems of Vector Optimization, Kibern.Sist.Anal., 1995, no. 4, pp. 137-143.
Emelichev, E.A., Kravtsov, M.K., and Podkopaev, D.P., On Quasistability of the Trajectory Problems of Vector Optimization, Mat.Zametki, 1998, vol. 63, no. 1, pp. 21-27.
Smale, S., Global Analisis and Economics. V. Pareto Theory with Constraints, J.Math.Econ., 1974, vol. 1, pp. 213-221.
Kolmogorov, A.N. and Fomin, S.V., Elementy teorii funktsii i funktsional'nogo analiza (Elements of the Function Theory and Functional Analysis), Moscow: Nauka, 1979.
Emelichev, V.A. and Leonovich, A.M., Quasistability of the Vector l∞-extreme Combinatorial Problem with Pareto Principle of Optimality, Buletinul Acad.de St.a Republicii Moldova, Matematica, 2001, no. 1, pp. 44-50.
Emelichev, V.A. and Stepanishina, Yu.V., Quasistability of the Nonlinear Vector Trajectory Problem with the Pareto Optimality Principle, Izv.Vuzov, Mat., 2000, no. 12, pp. 27-32.
Emelichev, V.A. and Berdysheva, R.A., On Stability Measure of the Problem of Integer Vector Lexicographc Optimization, Izv.Belarus Akad.Nauk, Ser.Fiz.-Tech.Nauk, 1999, no. 4, pp. 119-124.
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Emelichev, V.A., Kuz'min, K.G. & Leonovich, A.M. Stability in the Combinatorial Vector Optimization Problems. Automation and Remote Control 65, 227–240 (2004). https://doi.org/10.1023/B:AURC.0000014719.45368.36
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DOI: https://doi.org/10.1023/B:AURC.0000014719.45368.36