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Automation and Remote Control

, Volume 65, Issue 11, pp 1733–1746 | Cite as

Improved Lower Bounds for the Quadratic Assignment Problem

  • S. I. Sergeev
Article

Abstract

A model in the form of the Adams–Jonson model for the quadratic assignment problem is used. Three methods for improving the lower bounds based on subgradients computed by finite formula are designed. One of them is applied to improve the lower bound through continuous relaxation the Adams–Jonson model, which has thus far not been done.

Keywords

Mechanical Engineer Lower Bound System Theory Assignment Problem Quadratic Assignment Problem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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REFERENCES

  1. 1.
    Lowler, E., The Quadratic Assignment Problem, Manag. Sci., 1963, vol. 10, no. 7, pp. 586–598.Google Scholar
  2. 2.
    Sahni, S. and Gonzales, P., P-Complete Approximation Problems, J. Assoc. Comput. Math., 1976, vol. 23, no. 5, pp. 555–565.Google Scholar
  3. 3.
    Pierce, J.F. and Crowston, W.B., Tree-Search Algorithms for Quadratic Assignment Problems, Naval Res. Logist. Quarterly, 1971, vol. 8, no. 1, pp. 1–36.Google Scholar
  4. 4.
    Fink, G., Burkard, R.E., and Rendl, F., Quadratic Assignment Problems, Ann. Discr. Math., 1987, vol. 31, pp. 61–82.Google Scholar
  5. 5.
    Carraresi, P. and Malucelli, F., Quadratic Assignment Problems: A Review, Ricerca Oper., 1988, vol. 47, pp. 3–32.Google Scholar
  6. 6.
    Sergeev, S.I., Granitsy dlya kvadratichnoi zadachi naznacheniya. Modeli i metody ussledovaniya oper-atsii (Bounds for the Quadratic Assignment Problem: Models and Methods of Operations Research), Novosibirsk: Nauka, 1988.Google Scholar
  7. 7.
    Rubinshtein, M.I. and Sergeev, S.I., Optimal Location in Production and Transport Systems, Itogi Nauki Tekh., Ser. Tekh. Kibern., Moscow: VINITI, 1991, vol. 32, pp. 168–200.Google Scholar
  8. 8.
    Rubinshtein, M.I. and Sergeev, S.I., Minimization of Transport Cost in Production Systems: Mathematical Models and Methods, Itogi Nauki Tekh., Ser. Organiz. Upravl. Trans., Moscow: VINITI, 1992, vol. 12, pp. 3–90.Google Scholar
  9. 9.
    Quadratic Assignment and Related Problems, Pardalos, P.M. and Wolcowicz, H., Eds., New Jersey: Rutgers Univ., dy1994.Google Scholar
  10. 10.
    Cela, E., A Quadratic Assignment Problem: Theory and Algorithms, New York: Kluwer Academic, 1998.Google Scholar
  11. 11.
    Cristofildes, N., Mingozzi, A., and Toth, P., Contribution to the Quadratic Assignment Problem, Eur. J. Oper. Res., 1980, vol. 4, no. 4, pp. 243–247.Google Scholar
  12. 12.
    Adams, W.P. and Jonson, T.A., Improved Linear Programming-Based Lower Bounds for Quadratic Assignment Problems, Dimacs Ser. Discret. Math. Theor. Comput. Sci., 1994, vol. 16, pp. 43–77.Google Scholar
  13. 13.
    Fisher, M., The Lagrangian Relaxation Method for Solving Integer Programming Problems, Manag. Sci., 1981, vol. 28, no. 1, pp. 1–18.Google Scholar
  14. 14.
    Koopmans, N. and Beckmann, M., Assignment Problems and the Location of Economic Activities, Econometrica, 1957, vol. 25, no. 1, pp. 53–76.Google Scholar
  15. 15.
    The Travelling Salesman Problem. A Guided Tour of Combinatorial optimization, Lawler,E., Lenstra, J.K., Rinnoy Kan, A.H.G., and Shmoys, D.B., Eds., New York: Wiley, dy1985.Google Scholar
  16. 16.
    Melamed, I.I., Sergeev, S.I., and Sigal, I.Kh., The Travelling Salesman Problem: Exact Algorithms, Avtom. Telemekh., 1989, no. 10, pp. 3–29.Google Scholar
  17. 17.
    Carraresi, P. and Malucelli, F., A Reformulation Scheme and New Lower Bounds for The QAP, Dimacs Ser. Discret. Math. Theoretic. Comput. Sci., 1994, vol. 16, pp. 147–159.Google Scholar
  18. 18.
    Sergeev, S.I., Conditions for Optimality in Discrete Optimization Problems, Avtom. Telemekh., 1997, no. 3, pp. 3–19.Google Scholar
  19. 19.
    Sergeev, S.I., The Quadratic Assignment Problem. I, II, Avtom. Telemekh., 1999, no. 8, pp. 127–147; no. 9, pp. 137-143.Google Scholar
  20. 20.
    Assad, A. and Xu, W., On Lower Bounds for a Class of Quadratic 0.1 Programs, Oper. Res. Lett., 1985, vol. 4, no. 4, p. 175–180.Google Scholar
  21. 21.
    Resendl, M.G., Ramakrishnan, K.G., and Dresner, Z., Computing Lower Bounds for the Quadratic Assignment Problemwith an Interior Point Algorithm for Linear Programming, Oper. Res., 1995, vol. 43, no. 5, pp. 781–791.Google Scholar
  22. 22.
    Sergeev, S.I., A New Lower Bound for the Quadratic Assignment Problem, Zh. Vychisl. Mat. Mat. Fiz., 1987, vol. 27, no. 12, pp. 1802–1811.Google Scholar
  23. 23.
    Krotov, V.F. and Sergeev, S.I., Computation Algorithms for Solving Certain Linear and Linear Integer Programming Problems. I, II, Avtom. Telemekh., 1980, no. 12, pp. 86–96; dy1981, no. 1, pp. 8696.Google Scholar
  24. 24.
    24. Osnovy teorii optimal'nogo upravleniya (Elements of Optimal Control Theory), Krotov, V.F., Ed., Mscow: Vysshaya Shkola, dy1990.Google Scholar
  25. 25.
    Gavett, J.W. and Plyter, N.V., The Optimal Assignment of Facilities to Locations by Branch and Bound, Oper. Res., 1966, no. 2, pp. 260–283.Google Scholar
  26. 26.
    Murtagh, B.A., Jefferson, T.R., and Sornprasit, V., A Heuristic Procedure for Solving the Quadratic Assignment Problem, Eur. J. Oper. Res., 1982, vol. 9, no. 1, pp. 71–76.Google Scholar

Copyright information

© MAIK “Nauka/Interperiodica” 2004

Authors and Affiliations

  • S. I. Sergeev
    • 1
  1. 1.Moscow State University of Economics, Statistics, and InformaticsMoscowRussia

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