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On Representations of Semi-Infinite Programs which Have No Duality Gaps

Author

Listed:
  • A. Charnes

    (Northwestern University)

  • W. W. Cooper

    (Carnegie Institute of Technology)

  • K. Kortanek

    (University of Chicago and Northwestern University)

Abstract

Duality gaps which may occur in semi-infinite programs are shown to be interpretable as a phenomenon of an improper representation of the constraint set, u T P i \geqq c i , i \varepsilon I. Thus, any semi-infinite system of linear inequalities has a canonically closed equivalent (with interior points) which has no duality gap. With respect to the original system of inequalities, duality gaps may be closed by adjoining additional linear inequalities to the original system. Also, for consistent, but not necessarily canonically closed programs, a partial regularization of original data removes duality gaps that may occur. In contrast, a new "weakly consistent" duality theorem without duality gap may have a value determined by an inequality which is strictly redundant with respect to the constraint set defined by the total inequality system.

Suggested Citation

  • A. Charnes & W. W. Cooper & K. Kortanek, 1965. "On Representations of Semi-Infinite Programs which Have No Duality Gaps," Management Science, INFORMS, vol. 12(1), pages 113-121, September.
    • Handle: RePEc:inm:ormnsc:v:12:y:1965:i:1:p:113-121
    DOI: 10.1287/mnsc.12.1.113
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    Citations

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    Cited by:

    1. Bel? Jerez, 2000. "General Equilibrium with Asymmetric Information: a Dual Approach," UFAE and IAE Working Papers 510.02, Unitat de Fonaments de l'Anàlisi Econòmica (UAB) and Institut d'Anàlisi Econòmica (CSIC).
    2. Qinghong Zhang, 2017. "Strong Duality and Dual Pricing Properties in Semi-Infinite Linear Programming: A non-Fourier–Motzkin Elimination Approach," Journal of Optimization Theory and Applications, Springer, vol. 175(3), pages 702-717, December.
    3. M. A. Goberna & M. A. López, 2017. "Recent contributions to linear semi-infinite optimization," 4OR, Springer, vol. 15(3), pages 221-264, September.
    4. M. Goberna & M. Todorov & V. Vera de Serio, 2012. "On stable uniqueness in linear semi-infinite optimization," Journal of Global Optimization, Springer, vol. 53(2), pages 347-361, June.
    5. 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.
    6. M. A. Goberna & V. Jornet & R. Puente & M. I. Todorov, 1999. "Analytical Linear Inequality Systems and Optimization," Journal of Optimization Theory and Applications, Springer, vol. 103(1), pages 95-119, October.
    7. András Prékopa & Anh Ninh & Gabriela Alexe, 2016. "On the relationship between the discrete and continuous bounding moment problems and their numerical solutions," Annals of Operations Research, Springer, vol. 238(1), pages 521-575, March.
    8. Glover, Fred & Sueyoshi, Toshiyuki, 2009. "Contributions of Professor William W. Cooper in Operations Research and Management Science," European Journal of Operational Research, Elsevier, vol. 197(1), pages 1-16, August.
    9. 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.
    10. Xiao-Bing Li & Suliman Al-Homidan & Qamrul Hasan Ansari & Jen-Chih Yao, 2020. "Robust Farkas-Minkowski Constraint Qualification for Convex Inequality System Under Data Uncertainty," Journal of Optimization Theory and Applications, Springer, vol. 185(3), pages 785-802, June.
    11. Jerez, Belen, 2003. "A dual characterization of incentive efficiency," Journal of Economic Theory, Elsevier, vol. 112(1), pages 1-34, September.
    12. András Prékopa & Anh Ninh & Gabriela Alexe, 2016. "On the relationship between the discrete and continuous bounding moment problems and their numerical solutions," Annals of Operations Research, Springer, vol. 238(1), pages 521-575, March.

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