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Mathematical Programming

, Volume 6, Issue 1, pp 62–88 | Cite as

Validation of subgradient optimization

  • Michael Held
  • Philip Wolfe
  • Harlan P. Crowder
Article
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Revised:

Abstract

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

Keywords

Exact Solution Mathematical Method Programming Problem Linear Programming Problem Subgradient Optimization 
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]
    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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