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Discrete optimization by optimal control methods. II. The static traveling salesman problem

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Abstract

The static minisum traveling salesman problem is formulated as an optimal control problem. Two-sided algorithms based on the sufficient conditions for global optimality for solving this problem and a new algorithm for approximating the quality criterion from above to its optimal value are designed.

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References

  1. Sergeev S.I., Discrete Optimization by Optimal Control Methods. I, Avtom. Telemekh., 2006, no. 4, pp. 42–52.

  2. Melamed, I.I., Sergeev, S.I., and Sigal, I.Kh., The Traveling Salesman Problem. I–III, Avtom. Telemekh., 1989, no. 9, pp. 3–34; no. 10, pp. 3–29; no. 11, pp. 3–26.

  3. Lawler, E.L., Ed., Lenstra, J.K., Rinnoy Kan, A.H.G., and Shmoys, D.B., The Traveling Salesman Problem: A Guided Tour of Combinatorial Optimization, New York: Wiley, 1985.

    Google Scholar 

  4. Gutin, G., Ed. and Punnen, A.P., The Traveling Salesman Problem and Its Variations, London: Kluwer, 2002.

    Google Scholar 

  5. Rubinshtein, M.I. and Sergeev, S.I., Mathematical Models and Solution Methods for Minimization of Transport Costs in Production Systems, in Itogi nauki i tekhniki. Ser. Organizatsiya upravleniya transportom (Advances in Science and Technology: Transport Control Organization), Moscow: VINITI, 1992, vol. 12, pp. 3–90.

    Google Scholar 

  6. Pardalos, P.M., Ed. and Wolkowicz, H., Quadratic Assignment and Related Problems, New Jersey: Rutgers Univ., 1994.

    Google Scholar 

  7. Bayard, D.S., Reduced Complexity Dynamic Programming Based on Policy Iterations, J. Math. Anal. Appl., 1992, vol. 170, no. 1, pp. 75–103.

    Article  MATH  MathSciNet  Google Scholar 

  8. Gerasimov, V.A., Sequential Approximations for the Bellman Function, Avtom. Telemekh., 1987, no. 7, pp. 67–71.

  9. Krotov, V.F. and Sergeev, S.I., Algorithms for Linear and Linear Integer Programming Problems. I–IV, Avtom. Telemekh., 1980, no. 12, pp. 86–96; 1981, no. 1, pp. 86–96; no. 3, pp. 83–94; no. 4, pp. 103–112.

  10. Balas, E. and Christofides, N., A Restricted Lagrangian Approach to the Traveling Salesman Problem, Math. Program., 1981, no. 1, pp. 19–46.

  11. Sergeev, S.I., Computation Algorithms for the Traveling Salesman Problem. I, II, Avtom. Telemekh., 1994, no. 5, pp. 66–78; no. 6, pp. 106–115.

  12. Plotnikov, V.N. and Galkin, A.V., An Approximate Solution Method for the Traveling Salesman Problem, Ekonom. Mat. Metody, 1973, no. 6, pp. 1142–1146.

  13. Minina, T.R. and Perekrest, V.T., Approximation of Solutions of the Traveling Salesman Problem, Dokl. Akad. Nauk SSSR, 1975, vol. 220, no. 1, pp. 31–34.

    MathSciNet  Google Scholar 

  14. Lin, S., Computer Solution of the Traveling Salesman Problem, BSTJ, 1965, vol. 44, pp. 2245–2269.

    MATH  Google Scholar 

  15. Lin, S. and Kernighan, B.W., An Effective Heuristic Algorithm for the Traveling Salesman Problem, Oper. Res., 1973, vol. 21, no. 2, pp. 498–516.

    Article  MathSciNet  MATH  Google Scholar 

  16. Reiter, S. and Sherman, G., Discrete Optimizing, J. Soc. Industr. Appl. Math., 1965, vol. 13, pp. 846–889.

    MathSciNet  Google Scholar 

  17. Christofides, N., Mingozzi A., and Toth, P. State-Space Relaxation Procedures for Computation of Bounds to Routing Problems, Networks, 1981, vol. 11, no. 2, pp. 145–161.

    MathSciNet  MATH  Google Scholar 

  18. Zaretskii, L.S., The Traveling Salesman and Transport Problems: Solution via State Function Correction, Ekonom. Mat. Metody, 1979, vol. 15, no. 1, pp. 194–201.

    MATH  Google Scholar 

  19. Krotov, V.F., Solution and Optimization Algorithms for Systems of Equations. I–II, Izv. Akad. Nauk SSSR, Ser. Tekhn. Kibern., 1975, no. 5, pp. 3–15; no. 6, pp. 3–13.

  20. Sergeev, S.I. and Chernyshenko, A.V., An Algorithm for the Minimmax Traveling Salesman Problem. II, Avtom. Telemekh., 1995, no. 8, pp. 124–141.

  21. Sergeev, S.I., A Quadratic Assignment Problem. I, II, Avtom. Telemekh., 1999, no. 8, pp. 127–147; no. 9, pp. 137–143.

  22. Sergeev, S.I., A Refined Algorithm for the Quadratic Assignment Problem, Avtom. Telemekh., 2004, no. 11, pp. 57–71.

  23. Sergeev, S.I., Lower Bounds for the Three-Dimensional Assignment Problem. I, II, Tr. III Int. Conf. Identification of Systems and Control Problems (SICPRO’04), Moscow: Inst. Probl. Upravlen., 2004 (CD: ISBN-5-201-14966-9, pp. 1708–1726; pp. 1727–1740).

    Google Scholar 

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Authors and Affiliations

  1. Moscow State University of Economics, Statistics, and Informatics, Moscow, Russia

    S. I. Sergeev

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  1. S. I. Sergeev
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Original Russian Text © S.I. Sergeev, 2006, published in Avtomatika i Telemekhanika, 2006, No. 6, pp. 106–112.

This paper was recommended for publication by B.T. Polyak, a member of the Editorial Board

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Sergeev, S.I. Discrete optimization by optimal control methods. II. The static traveling salesman problem. Autom Remote Control 67, 927–932 (2006). https://doi.org/10.1134/S0005117906060075

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