About one discrete analog of Hausdorff semi-continuity of suitable mapping in a vector combinatorial problem with a parametric principle of optimality ("from Slater to lexicographic")

Authors

  • Vladimir A. Emelichev Belarussian State University, Minsk, Belarus
  • Andrey A. Platonov Belarussian State University, Minsk, Belarus

Keywords:

vector optimization, set of weak Slater optima, set of lexicographically optimal trajectories, quasistability, quasistability radius

Abstract

multicriteria linear combinatorial problem is considered, principle of optimality of which is defined by a partitioning of partial criteria onto groups with Slater preference relation within each group and the lexicographic preference relation between them. Quasistability of the problem is investigated. This type of stability is a discrete analog of Hausdorff lower semicontinuity of the many-valued mapping that defines the choice function. A formula of quasistability radius is derived for the case of metric \(l_\infty.\) Some conditions of quasistability are stated as corollaries.

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References

Tanino T. and Sawaragi Y., Stability of nondominated solutions in multicriteria decision-making, Journal of Optimization and Applications, 30, pp. 229-253, 1980, https://doi.org/10.1007/bf00934497

Sergienko I. V., Kozerackaja L. N. and Lebedeva T. T., Investigation of Stability and Parametric Analisys of Discrete Optimization Problems, Naukova Dumka, Kiev, 1995.

Sergienko I. V. and Shilo V. P., Discrete Optimization Problems, Naykova Dymka, Kiev, 2003.

Leontev V. K., Stability in linear discrete problems, Problems of Cybernetics. Nauka, Moskow, 35, pp. 169-185, 1979.

Emelichev V. A., Girlich E., Nikulin Yu. V. and Podkopaev D. P., Stability and regularization of vector problems of integer linear programming, Optimization, 51, pp. 645-676, 2002, https://doi.org/10.1080/0233193021000030760

Sotskov Yu. N., Leontev V. K. and Gordeev E. N., Some concepts of stability analysis in combinatorial optimization, Discrete Appl. Math., 58, pp. 169-190, 1995, https://doi.org/10.1016/0166-218x(93)e0126-j

Emelichev V. A. and Leonovich A. M., On stability conditions of vector l_{∞}-extreme combinatorial problem with Pareto principle of optimality, Rev. Anal. Numér. Théor. Approx., 32, pp. 31-37, 2003.

Emelichev V. A., Kravtsov M. K. and Podkopaev D. P., On the quasistability of trajectory problems of vector optimization, Math. Notes, 63, pp. 19-24, 1998, https://doi.org/10.1007/bf02316139

Emelichev, V. A. and Nikulin Yu. V.,On the stability and quasi-stability of a vector lexicographic quadric boolean programming problem, Rev. Anal. Numér. Théor. Approx., 30, pp. 36-46, 2001.

Emelichev V. A., Kuz'min K. G. and Leonovich A. M., Stability in the combinatorial vector optimization problems, Automation and Remote Control, 65, pp. 227-240, 2004, https://doi.org/10.1023/b:aurc.0000014719.45368.36

Emelichev, V. A., Kuz'min, K. G. and Nikulin Yu. V., Stability analysis of the Pareto optimal solutions for some vector boolean optimization problem, Optimization, 54, pp. 545-561, 2005, https://doi.org/10.1080/02331930500342708

Emelichev, V. A. and Kuzmin K. G.,The stability radius of efficient solution to a vector problem of boolean programming in the l₁ metric, Doclady mathematics, 71, pp. 266-268, 2005.

Emelichev V. A. and Stepanishina Yu. V., Multicriteria combinatorial linear problems: parametrization of the optimality principle and the stability of the effective solutions, Discrete Math. Appl., 11, pp. 435-444, 2001, https://doi.org/10.1515/dma.2001.11.5.435

Bukhtoyarov S. E. and Emelichev V. A., Parametrization of principle of optimality ("from Pareto to Slater") and stability of multicriterion trajectory problems, Diskretniy Analiz i Issled. Operaciy, ser. 2, 10, pp. 3-18, 2003.

Bukhtoyarov S. E., Emelichev V. A. and Stepanishina Yu. V., Stability of discrete vector problems with the parametric principle of optimality, Cybernetics and Systems Analysis, 39, pp. 604-614, 2003, https://doi.org/10.1023/b:casa.0000003509.09942.10

Emelichev V. A. and Bukhtoyarov S. E., Stability of generally efficient situation in finite cooperative games with parametric optimality principle ("from Pareto to Nash"), Computer Science Journal of Moldova, 11, pp. 316-323, 2003.

Bukhtoyarov S. E. and Emelichev V. A., On quasistability of vector trajectorial problem with the parametric principle of optimality, Izv. Vuzov, Matem., 1, pp. 25-30, 2004.

Bukhtoyarov S. E. and Emelichev V. A., On stability of an optimal situation infinite cooperative with a parametric concept of equilibrium (from lexicographic optimality to Nash equilibrium), Computer Science Journal of Moldova, 12, pp. 371-386, 2004.

Bukhtoyarov S. E., Emelichev V. A. and Kuz'min K. G., On quasistability radius of a vector combinatorial problem with parametric principle of optimality ("from Pareto to lexicographic"), ECCO XVIII, Minsk, Belarus, 1, p. 9, 2005.

Bukhtoyarov, S. E. and Emelichev, V. A.,On quasistability radius of a vector trajectorial problem with a principle of optimality generalizing Pareto and lexicographic principles, Computer Science Journal of Moldova, 13, pp. 47-58, 2005.

Emelicev V. A. and Kuzmin K. G.,Finite cooperative games with a parametric concept of equilibrium under uncertainty conditions, Journal of Computer and Systems Sciences International, 45, pp. 276-281, 2006, https://doi.org/10.1134/s1064230706020110

Emelichev V. A. and Berdysheva R. A., On stability and quasistability of trajectorial problem of sequential optimization, Dokl. Nazhion. Akad. Nauk Belarusi, 3, pp. 41-44, 1999.

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Published

2006-08-01

How to Cite

Emelichev, V. A., & Platonov, A. A. (2006). About one discrete analog of Hausdorff semi-continuity of suitable mapping in a vector combinatorial problem with a parametric principle of optimality ("from Slater to lexicographic"). Rev. Anal. Numér. Théor. Approx., 35(2), 131–139. Retrieved from https://ictp.acad.ro/jnaat/journal/article/view/2006-vol35-no2-art1

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