Attempt to solve a combinatorial problem in the continuum by a method of extension-reduction

  • Emilio Spedicato
  • Giorgio Tagliabue
Numerical Methods
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3)


Combinatorial optimization problems in n variables are formulated as nonlinear programming problems in (n-1)2 variables and n2 constraints. Methods for solving the large unconstrained optimization problem generated are considered, with emphasis on conjugate-gradient algorithms based on the homogeneous model. The quadratic assignment problem is considered as an application example and results from the nonlinear programming approach are discussed.


Permutation Matrix Nonlinear Programming Problem Conjugate Gradient Algorithm Permutation Matrice Linear Search 
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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  1. 1.
    Nelder, J. A. and Nead, R.: A simplex method for function minimization, Comput. J., 7, 308–313, 1965Google Scholar
  2. 2.
    Powell, M. J. D.: An efficient method of finding the minimum of a function of several variables without calculating derivatives, Comput. J., 7, 155–162, 1964CrossRefGoogle Scholar
  3. 3.
    Fletcher, R. and Reeves, C. M.: Function minimization by conjugate gradients, Computer J., 7, 149–154, 1964CrossRefGoogle Scholar
  4. 4.
    Polak, E. and Ribiere, G.: Note sur le convergence de methodes des directions conjugees, University of California, Berkeley, Dept. of Electrical Engineering and Computer Sciences, working paper, 1969Google Scholar
  5. 5.
    Sorenson, H. W.: Conjugate Direction Procedures for Function Minimization, Journal of the Franklin Institute, 288, 421–441, 1969CrossRefGoogle Scholar
  6. 6.
    Fried, I.: N-step Conjugate Gradient Minimization Scheme for Nonquadratic Functions, AIAA Journal, 9, 2286–2287, 1971Google Scholar
  7. 7.
    Spedicato, E.: Un polialgoritmo per la minimizzazione di una funzione di più variabili, Atti del Convegno AICA su Tecniche di Simulazione e Algoritmi, Milano, Informatica, Numero speciale, 1972Google Scholar
  8. 8.
    Fielding, K.: Function minimization and linear search, Algorithm 387, Commun. of ACM, 13, 8, 1970Google Scholar
  9. 9.
    Spedicato, E.: Un polialgoritmo a gradiente coniuga to per la minimizzazione di funzioni nonlineari in molte variabili, Nota tecnica CISE-73.012, Milano, 1973Google Scholar
  10. 10.
    Spedicato, E.: CISE-Report to appearGoogle Scholar
  11. 11.
    Lawler, E. L.: The Quadratic Assignement Problem, Management Sci, 9, 586–599, 1963Google Scholar
  12. 12.
    Armour, G. C. and Buffa, E. S.: A Heuristic Algorithm and Simulation Approach to Relative Location of Facilities, Management Sci., 9, 294–309, 1963Google Scholar
  13. 13.
    Gilmore, P. C.: Optimal and Suboptimal Algorithms for the Quadratic Assignement, SIAM J., 10, 305–313, 1962Google Scholar
  14. 14.
    Hillier, F. S. and Connors, M. M.: Quadratic Assignment Problem Algorithms and the Location of Indivisible Facilities, Management Sci., 13, 42–57, 1966Google Scholar
  15. 15.
    Graves, G. W. and Whinston, A. B.: An Algorithm for the Quadratic Assignment Problem, Management Sci., 16, 453–471, 1970Google Scholar
  16. 16.
    Casanova, M. and Tagliabue, G.: CISE-Report to appearGoogle Scholar
  17. 17.
    Hansen, P.: Quadratic Zerc-One Programming by Implicit Enumeration, in Numerical Methods for non-linear Optimization, (F.A. Lootsma, ed.), Academic Press, 1972Google Scholar
  18. 18.
    Miele, A., Coggins, G. M. and Levy, A. V.: Updating rules for the penalty constant used in the penalty function method for mathematic programming problems, Aero-Astronautics Report n. 90, Rice University, Houston, 1972Google Scholar
  19. 19.
    Powell, M. J. D.: A Method for Nonlinear Constraints in Minimization Problems in Optimization, (R. Fletcher, ed.), Academic Press, 1969Google Scholar
  20. 20.
    Nugent, C. E., Vollmann, T. E. and Ruml J.: An experimental comparison of techniques for the assignment of facilities to locations, Operations Research, 16, 150–173, 1968.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1973

Authors and Affiliations

  • Emilio Spedicato
    • 1
  • Giorgio Tagliabue
    • 1
  1. 1.CISE, SegrateMilanoItaly

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