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Mathematical Programming

, Volume 7, Issue 1, pp 223–235 | Cite as

On the existence of optimal solutions to integer and mixed-integer programming problems

  • R. R. Meyer
Article

Abstract

The purpose of this paper is to present sufficient conditions for the existence of optimal solutions to integer and mixed-integer programming problems in the absence of upper bounds on the integer variables. It is shown that (in addition to feasibility and boundedness of the objective function) (1) in the pure integer case a sufficient condition is that all of the constraints (other than non-negativity and integrality of the variables) beequalities, and (2) that in the mixed-integer caserationality of the constraint coefficients is sufficient. Some computational implications of these results are also given.

Keywords

Objective Function Mathematical Method Programming Problem Integer Variable Integer Case 
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]
    G.B. Dantzig,Linear programming and extensions (Princeton University Press, Princeton, N.J., 1963).Google Scholar
  2. [2]
    S. Halfin, “Arbitrarily complex corner polyhedra are dense inR n”,SIAM Journal on Applied Mathematics 23 (1972) 157–163.Google Scholar
  3. [3]
    R.G. Jeroslow, “On the unlimited number of faces in integer hulls of linear problems with two constraints”, Tech. Rept. No. 67, Department of Operations Research, Cornell University, Ithaca, N.Y. (April, 1969).Google Scholar
  4. [4]
    R.G. Jeroslow, “Comments on integer hulls of two linear constraints”,Operations Research 19 (1971) 1061–1069.Google Scholar
  5. [5]
    R.T. Rockafellar,Convex analysis (Princeton University Press, Princeton, N.J., 1970).Google Scholar
  6. [6]
    D.S. Rubin, “On the unlimited number of faces in integer hulls of linear programs with a single constraint”,Operations Research 18 (1970) 940–946.Google Scholar
  7. [7]
    D.S. Rubin, “The neighboring vertex cut and other cuts derived with Gomory's asymptotic algorithm”, Ph.D. Thesis, The University of Chicago, 1970.Google Scholar
  8. [8]
    H.M. Stark,An introduction to number theory (Markham Publishing Company, Chicago, 1970).Google Scholar
  9. [9]
    L.A. Wolsey, “Group theoretic results in mixed integer programming”,Operations Research 19 (1971) 1691–1697.Google Scholar

Copyright information

© The Mathematical Programming Society 1974

Authors and Affiliations

  • R. R. Meyer
    • 1
  1. 1.University of WisconsinMadisonUSA

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