Mathematical Programming

, Volume 6, Issue 1, pp 62–88

Validation of subgradient optimization

  • Michael Held
  • Philip Wolfe
  • Harlan P. Crowder


The “relaxation” procedure introduced by Held and Karp for approximately solving a large linear programming problem related to the traveling-salesman problem is refined and studied experimentally on several classes of specially structured large-scale linear programming problems, and results on the use of the procedure for obtaining exact solutions are given. It is concluded that the method shows promise for large-scale linear programming


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  1. [1]
    J. Abadie and M. Sakarovitch, “Two methods of decomposition for linear programming”, in:Proceedings of the Princeton symposium on mathematical programming Ed. H.W. Kuhn (Princeton University Press, Princeton, N.J., 1970) pp 1–23.Google Scholar
  2. [2]
    S. Agmon, “The relaxation method for linear inequalities”,Canadian Journal of Mathematics 6 (1954) 382–392.Google Scholar
  3. [3]
    D.P. Bertsekas and S.K. Mitter, “Steepest descent for optimization problems with nondifferentiable cost functionals”, in:Proceedings of the 5 th annual Princeton conference on information sciences and systems, 1971.Google Scholar
  4. [4]
    G.B. Dantzig, D.R. Fulkerson and S.M. Johnson, “Solution of a large-scale traveling-salesman problem”,Operations Research 2 (1954) 393–410.Google Scholar
  5. [5]
    V.F. Dem'janov, “Seeking a minimax on a bounded set”,Soviet Mathematics Doklady 11 (1970) 517–521. [Translation of:Doklady Akademii Nauk SSSR 191 (1970).]Google Scholar
  6. [6]
    M.L. Fisher and J.F. Shapiro, “Constructive duality in integer programming”, Working Paper OR 008-72, Operations Research Center, Massachusetts Institute of Technology, Cambridge, Mass. (April, 1972).Google Scholar
  7. [7]
    L.R. Ford, Jr. and D.R. Fulkerson,Flows in networks (Princeton University Press, Princeton, N.J., 1962).Google Scholar
  8. [8]
    A.M. Geoffrion, “Elements of large-scale mathematical programming”,Management Science 16 (1970) 652–691.Google Scholar
  9. [9]
    R.C. Grinold, “Steepest ascent for large-scale linear programs”,SIAM Review 14 (1972) 447–464.Google Scholar
  10. [10]
    M. Held and R.M. Karp, “The traveling-salesman problem and minimum spanning trees”,Operations Research 18 (1970) 1138–1162.Google Scholar
  11. [11]
    M. Held and R.M. Karp, “The traveling-salesman problem and minimum spanning trees: part II”,Mathematical Programming 1 (1971) 6–25.Google Scholar
  12. [12]
    M. Held and R.M. Karp, “A dynamic programming approach to sequencing problems”,Journal of the Society for Industrial and Applied Mathematics 10 (1962) 196–210.Google Scholar
  13. [13]
    L.L. Karg and G.L. Thompson, “A heuristic approach to solving traveling-salesman problems”,Management Science 10 (1964) 225–248.Google Scholar
  14. [14]
    H.W. Kuhn and A.W. Tucker, “Nonlinear programming”, in:Proceedings of the second Berkeley symposium on mathematical statistics and probability Ed. J. Neyman (University of California Press, Berkeley, Calif., 1951) pp. 481–492.Google Scholar
  15. [15]
    L.S. Lasdon,Optimization theory for large systems (Macmillan, London, 1970).Google Scholar
  16. [16]
    R.E. Marsten and J.W. Blankenship, “Boxstep: a new strategy for Lagrangian decomposition”, Department of Industrial Engineering and Management Sciences, Northwestern University, Evanston, Ill. (March, 1973).Google Scholar
  17. [17]
    T. Motzkin and I.J. Schoenberg, “The relaxation method for linear inequalities”,Canadian Journal of Mathematics 6 (1954) 393–404.Google Scholar
  18. [18]
    J. von Neumann, “A certain zero-sum two-person game equivalent to the optimal assignment problem”, in:Contributions to the theory of games, Vol. II, Eds. H.W. Kuhn and A.W. Tucker, Annals of Mathematics Study No. 28 (Princeton University Press, Princeton, N.J., 1953).Google Scholar
  19. [19]
    W. Oettli, “An iterative method, having linear rate of convergence, for solving a pair of dual linear programs”,Mathematical Programming 3 (1972) 302–311.Google Scholar
  20. [20]
    B.T. Poljak, “A general method of solving extremum problems”, Soviet Mathematics Doklady 8 (1967) 593–597. [Translation ofDoklady Akademii Nauk SSSR 174 (1967).]Google Scholar
  21. [21]
    B.T. Poljak, “Minimization of unsmooth functionals”,U.S.S.R. Computational Mathematics and Mathematical Physics 14–29. [Translation of: Žurnal Vyčislitel'no\(\mathop i\limits^ \vee \) Matematiki i Matematičesko\(\mathop i\limits^ \vee \) Fiziki 9 (1969) 509–521.]Google Scholar
  22. [22]
    R.T. Rockafellar,Convex analysis (Princeton University Press, Princeton, N.J., 1970).Google Scholar
  23. [23]
    N.Z. Shor, “On the structure of algorithms for the numerical solution of optimal planning and design problems”, Dissertation, Cybernetics Institute, Academy of Sciences U.S.S.R. (1964).Google Scholar
  24. [24]
    P. Wolfe, M. Held and R.M. Karp, “Large-scale optimization and the relaxation method”, in:Proceedings of the 25 th national ACM meeting, Boston, Mass. (August 1972).Google Scholar

Copyright information

© The Mathematical Programming Society 1974

Authors and Affiliations

  • Michael Held
    • 1
  • Philip Wolfe
    • 2
  • Harlan P. Crowder
    • 2
  1. 1.IBM Systems Research InstituteNew YorkUSA
  2. 2.IBM Watson Research CenterYorktown HeightsUSA

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