Mathematical Programming

, Volume 10, Issue 1, pp 354–366 | Cite as

Some theoretical implications of local optimization

  • Sam L. Savage
Article
First Online:
Received:
Revised:

Abstract

This paper presents some theoretical results concerning the effectiveness of an approximate technique, known as local optimization, as applied to a wide class of problems.

First, conditions are described under which the technique ensures exact solutions. Then, in regard to cases in which these conditions cannot be met in practice, a method is presented for estimating the probability that the approximate (locally optimal) solution delivered by such a technique is in fact the exact (globally optimal) solution.

This probability may be viewed as a possible criterion of effectiveness in the design of neighborhoods for specific local optimization algorithms.

Keywords

Exact Solution Optimization Algorithm Theoretical Result Mathematical Method Local 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.
This is a preview of subscription content, log in to check access.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    M. Bellmore and G. Nemhauser, “The traveling salesman problem: A survey”,Operations Research 16 (1968) 538–558.MathSciNetCrossRefMATHGoogle Scholar
  2. [2]
    F. Bock, “An algorithm for solving ‘traveling-salesman’ and related network optimization problems”, 14th National Meeting of ORSA (1958).Google Scholar
  3. [3]
    G. Croes, “A method for solving traveling-salesman problems”,Operations Research 6 (1958) 792–812.MathSciNetCrossRefGoogle Scholar
  4. [4]
    S. Gupta, “Probability integral of multivariate normal and multivariatet”,Annals of Mathematical Statistics 34 (1963) 805.Google Scholar
  5. [5]
    T.C. Hu,Integer programming and network flows (Addison Wesley, Reading, Mass., 1967).Google Scholar
  6. [6]
    R.M. Karp, “Reducibility among combinatorial problems”, in:Complexity of computer computations (Plenum Press, New York, 1972).Google Scholar
  7. [7]
    M. Krone, “Heuristic programming applied to scheduling problems”, Dissertation, Princeton University, Princeton, N.J. (1970).Google Scholar
  8. [8]
    J. Kruskal, “On the shortest spanning subtree of a graph and the traveling salesman problem”,Proceedings of the American Mathematical Society 7 (1956) 48–50.MathSciNetCrossRefMATHGoogle Scholar
  9. [9]
    H. Kuhn and A. Tucker,Linear inequalities and related systems (Princeton University Press, Princeton, N.J., 1956) p. 19.MATHGoogle Scholar
  10. [10]
    S. Lin, “Computer solutions of the traveling salesman problem”,Bell System Technical Journal 44 (1965) 2245–2269.MathSciNetCrossRefMATHGoogle Scholar
  11. [11]
    S. Reiter and G. Sherman, “Discrete optimizing”,Journal of the Society for Industrial Applications in Mathematics 13 (1965) 864–889.MathSciNetCrossRefMATHGoogle Scholar
  12. [12]
    D.J. Rosenkrantz, R.E. Stearns and P.M. Lewis, “Approximate algorithms for the traveling salesman problem”, 15th Annual Symposium on switching and automata theory of the IEEE, 1974.Google Scholar
  13. [13]
    S.L. Savage, “The solution of discrete linear optimization problems by neighborhood search techniques”, Dissertation, Yale University, New Haven, Conn. (1973).Google Scholar
  14. [14]
    S.L. Savage, “Statistical indicators of optimality”, 14th Annual Symposium on switching and automata theory of the IEEE, University of Iowa, 1973.Google Scholar
  15. [15]
    S.L. Savage, P. Weiner and M.J. Krone, “Convergent local search”, Research Rept. No. 14, Yale University, New Haven, Conn. (1972).Google Scholar
  16. [16]
    N. Simonnard,Linear programming (Prentice-Hall, Englewood Cliffs, N.J., 1966).MATHGoogle Scholar
  17. [17]
    K. Steiglitz, P. Weiner and D. Kleitman, “The design of minimal cost survivable networks”,IEEE Transactions on Circuit Theory, CT-16 4 (1969) 455–460.CrossRefGoogle Scholar
  18. [18]
    J.D. Ullman, “The performance of a memory allocation algorithm”,4th symposium on the theory of computation (Princeton University Computer Science Laboratory, Princeton, N.J., 1972) pp. 143–150.Google Scholar
  19. [19]
    P. Weiner, S. Savage and A. Bagchi, “Neighborhood search algorithms for finding optimal traveling salesman tours must be inefficient”,Proceedings of the fifth annual ACM symposium on the theory of computing, Austin, Texas, 1973.Google Scholar

Copyright information

© The Mathematical Programming Society 1976

Authors and Affiliations

  • Sam L. Savage
    • 1
  1. 1.University of ChicagoChicagoUSA

Personalised recommendations