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Adaptive stability in combinatorial optimization problems

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Abstract

We consider a general approach to the construction of necessary, sufficient, and necessary and sufficient conditions that allow to ‘adapt’ a known optimal solution of an abstract combinatorial problem with a certain structure to a change in the initial data set for a fixed cost function ‘easily’ from the combinatorial point of view. We call this approach adaptive stability. Apparently, it is the first time that the approach is described for an abstract problem in a rigorous mathematical formalization.

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References

  1. A. M. Lyapunov, The General Problem of the Stability of Motion (Gostekhizdat, Moscow, 1950; Taylor and Francis, London, 1992).

    MATH  Google Scholar 

  2. N. N. Krasovskii, Some Problems in the Theory of Motion Stability (Fizmatgiz, Moscow, 1959) [in Russian].

    Google Scholar 

  3. V. K. Leont’ev, “Stability in linear discrete problems,” Problemy Kibernet., No. 35, 169–185 (1979).

    Google Scholar 

  4. E. N. Gordeev and V. K. Leont’ev, “A general approach to the study of the stability of solutions in discrete optimization problems,” Comp. Math. Math. Phys. 36(1), 53–58 (1996).

    MATH  MathSciNet  Google Scholar 

  5. V. A. Emelichev, K. G. Kuz’min, and A. M. Leonovich, “Stability in the combinatorial vector optimization problems,” Autom. Remote Control 65(2), 227–240 (2004).

    Article  MATH  MathSciNet  Google Scholar 

  6. V. A. Emelichev and K. G. Kuz’min, “Stability conditions for a multicriteria Boolean problem of minimizing projections of linear functions,” Trudy Inst. Mat. Mekh. UrO RAN 19(2), 125–133 (2013).

    MathSciNet  Google Scholar 

  7. V. Emelichev, E. Girlich, and O. Karelkina, “Postoptimal analysis of multicriteria combinatorial center location problem,” Bul. Acad. Stiinte Repub. Mold. Mat., No. 3, 13–29 (2009).

    Google Scholar 

  8. M. V. Devyaterikova and A. A. Kolokolov, “On the stability of some integer programming algorithms,” Russian Math. (Iz. VUZ) 47(12), 38–45 (2003).

    MATH  MathSciNet  Google Scholar 

  9. M. V. Devyaterikova, Stability Analysis for Problems and Algorithms of Integer Programming Based on Regular Partitions, Candidate’s Dissertation in Physics and Mathematics (Omsk, 2001).

    Google Scholar 

  10. I. V. Sergienko, L. N. Kozeratskaya, and T. T. Lebedeva, Stability and Parametric Analysis of Discrete Optimization Problems (Naukova Dumka, Kiev, 1995) [in Russian].

    Google Scholar 

  11. I. V. Sergienko and V. P. Shilo, Discrete Optimization Problems: Challenges, Solution Methods, and Research (Naukova Dumka, Kiev, 2003) [in Russian].

    Google Scholar 

  12. T. I. Sergienko, “On the stability with respect to constraints of a multicriteria problem of integer programming,” Dokl. Akad. Nauk USSR, No. 3, 79–81 (1989).

    Google Scholar 

  13. M. Libura, “Optimality conditions and sensitivity analysis for combinatorial optimization problems,” Control Cybernet. 25(6), 1165–1180 (1996).

    MATH  MathSciNet  Google Scholar 

  14. E. S. Poort, Aspects of Sensitivity Analysis for the Traveling Salesman Problem, PhD Dissertation (Groningen, 1997).

    Google Scholar 

  15. E. E. Ivanko, “Sufficient conditions for the optimal route stability in the traveling salesman problem in the case of adding a new vertex and deleting an existing vertex,” Vestn. Udmurt. Gos. Univ., No. 1, 46–56 (2010).

    Google Scholar 

  16. E. E. Ivanko, “A stability criterion for an optimal route in the traveling salesman problem under the addition of a new vertex,” Vestn. Udmurt. Gos. Univ., No. 1, 58–66 (2011).

    Google Scholar 

  17. E. E. Ivanko, “Sufficient stability conditions in the traveling salesman problem,” Trudy Inst. Mat. Mekh. UrO RAN 17(3), 155–168 (2011).

    Google Scholar 

  18. E. E. Ivanko, “A stability criterion for optimal solutions of a minimax problem about a partition into an arbitrary number of subsets under varying cardinality of the initial set,” Trudy Inst. Mat. Mekh. UrO RAN 18(4), 180–194 (2012).

    Google Scholar 

  19. E. E. Ivanko, “Optimal route stability in the traveling salesman problem under the addition and deletion of vertices,” in Discrete Optimization and Operations Research: Abstracts of the Russian Conference, Chemal, Russia, 2010 (Inst. Mat. SO RAN, Novosibirsk, 2010), p. 105.

    Google Scholar 

  20. E. E. Ivanko and A. M. Grigor’ev, “Domains of optimal route instability in the traveling salesman problem under the addition of a new vertex,” in Algorithmic Analysis of Ill-Posed Problems: Abstracts of the International Conference, Yekaterinburg, Russia, 2011 (Inst. Mat. Mekh. UrO RAN, Yekaterinburg, 2011), pp. 232–233.

    Google Scholar 

  21. E. E. Ivanko, “Stability in a combinatorial optimization problem as a polynomial ‘adaptability’ of optimal solutions under a perturbation of the set of initial data,” in Discrete Optimization and Operations Research: Abstracts of the International Conference, Novosibirsk, Russia, 2013 (Inst. Mat. SO RAN, Novosibirsk, 2013), p. 117.

    Google Scholar 

  22. E. E. Ivanko, Stability and Instability in Discrete Problems (RIO UrO RAN, Yekaterinburg, 2013) [in Russian].

    Google Scholar 

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Authors and Affiliations

  1. Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, ul. S. Kovalevskoi 16, Yekaterinburg, 620990, Russia

    E. E. Ivanko

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  1. E. E. Ivanko
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Correspondence to E. E. Ivanko.

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Original Russian Text © E.E. Ivanko, 2014, published in Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2014, Vol. 20, No. 1.

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Ivanko, E.E. Adaptive stability in combinatorial optimization problems. Proc. Steklov Inst. Math. 288 (Suppl 1), 79–87 (2015). https://doi.org/10.1134/S0081543815020091

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  • DOI: https://doi.org/10.1134/S0081543815020091

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