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An Infinite Linear Program with a Duality Gap

Author

Listed:
  • R. J. Duffin

    (Carnegie Institute of Technology)

  • L. A. Karlovitz

    (Western Reserve University)

Abstract

The first part of this paper is concerned with the semi-infinite linear programs studied by Charnes, Cooper and Kortanek. A form of the Farkas lemma stated by Haar appears to apply to such programs and leads to a duality theorem. In this paper an example of a semi-infinite program is given which is consistent and which has a finite minimum. However the dual program is found to be inconsistent. With a variation of the example a situation is exhibited in which both the program and its dual are consistent and have finite extrema. In this case, however, the minimum of the former is not equal to the maximum of the latter. The existence of such a "duality gap" indicates that Haar's statement needs qualification. In the second part of the paper a correct form of Haar's theorem is stated and proved. The proof invokes the infinite programming theory of Duffin and Kretschmer. The last part of the paper develops a new duality theory for infinite programs which, as a special case, insures a weak form of duality for programs typified by the examples. This new duality theory is somewhat simpler than previous theories of infinite programming.

Suggested Citation

  • R. J. Duffin & L. A. Karlovitz, 1965. "An Infinite Linear Program with a Duality Gap," Management Science, INFORMS, vol. 12(1), pages 122-134, September.
    • Handle: RePEc:inm:ormnsc:v:12:y:1965:i:1:p:122-134
    DOI: 10.1287/mnsc.12.1.122
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    Cited by:

    1. M. A. Goberna & M. A. López, 2017. "Recent contributions to linear semi-infinite optimization," 4OR, Springer, vol. 15(3), pages 221-264, September.
    2. Qinghong Zhang, 2008. "Uniform LP duality for semidefinite and semi-infinite programming," Central European Journal of Operations Research, Springer;Slovak Society for Operations Research;Hungarian Operational Research Society;Czech Society for Operations Research;Österr. Gesellschaft für Operations Research (ÖGOR);Slovenian Society Informatika - Section for Operational Research;Croatian Operational Research Society, vol. 16(2), pages 205-213, June.
    3. M. A. Goberna & M. A. López, 2018. "Recent contributions to linear semi-infinite optimization: an update," Annals of Operations Research, Springer, vol. 271(1), pages 237-278, December.
    4. Amitabh Basu & Kipp Martin & Christopher Thomas Ryan, 2015. "Projection: A Unified Approach to Semi-Infinite Linear Programs and Duality in Convex Programming," Mathematics of Operations Research, INFORMS, vol. 40(1), pages 146-170, February.
    5. Ulaş Özen & Marco Slikker & Greys Sošić, 2022. "On the core of m$m$‐attribute games," Production and Operations Management, Production and Operations Management Society, vol. 31(4), pages 1770-1787, April.

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