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A Characterization of Stability in Linear Programming

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  • Stephen M. Robinson

    (University of Wisconsin, Madison, Wisconsin)

Abstract

We prove that a necessary and sufficient condition for the primal and dual solution sets of a solvable, finite-dimensional linear programming problem to be stable under small but arbitrary perturbations in the data of the problem is that both of these sets be bounded. The distance from any pair of solutions of the perturbed problem to the solution sets of the original problem is then bounded by a constant multiple of the norm of the perturbations. These results extend earlier work of Williams.

Suggested Citation

  • Stephen M. Robinson, 1977. "A Characterization of Stability in Linear Programming," Operations Research, INFORMS, vol. 25(3), pages 435-447, June.
    • Handle: RePEc:inm:oropre:v:25:y:1977:i:3:p:435-447
    DOI: 10.1287/opre.25.3.435
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    Cited by:

    1. Marcel Klatt & Axel Munk & Yoav Zemel, 2022. "Limit laws for empirical optimal solutions in random linear programs," Annals of Operations Research, Springer, vol. 315(1), pages 251-278, August.
    2. Ning Zhang & Chang Fang, 2020. "Saddle point approximation approaches for two-stage robust optimization problems," Journal of Global Optimization, Springer, vol. 78(4), pages 651-670, December.
    3. M. J. Cánovas & R. Henrion & M. A. López & J. Parra, 2016. "Outer Limit of Subdifferentials and Calmness Moduli in Linear and Nonlinear Programming," Journal of Optimization Theory and Applications, Springer, vol. 169(3), pages 925-952, June.
    4. C. Filippi, 2004. "An Algorithm for Approximate Multiparametric Linear Programming," Journal of Optimization Theory and Applications, Springer, vol. 120(1), pages 73-95, January.
    5. Lamb, John D. & Tee, Kai-Hong, 2012. "Resampling DEA estimates of investment fund performance," European Journal of Operational Research, Elsevier, vol. 223(3), pages 834-841.
    6. Bulat Gafarov, 2019. "Simple subvector inference on sharp identified set in affine models," Papers 1904.00111, arXiv.org, revised Jul 2024.
    7. Dupacova, Jitka & Bertocchi, Marida, 2001. "From data to model and back to data: A bond portfolio management problem," European Journal of Operational Research, Elsevier, vol. 134(2), pages 261-278, October.
    8. Ludwig Kuntz & Stefan Scholtes, 2000. "Measuring the Robustness of Empirical Efficiency Valuations," Management Science, INFORMS, vol. 46(6), pages 807-823, June.
    9. Giorgio & Cesare, 2018. "A Tutorial on Sensitivity and Stability in Nonlinear Programming and Variational Inequalities under Differentiability Assumptions," DEM Working Papers Series 159, University of Pavia, Department of Economics and Management.
    10. Holger Scheel & Stefan Scholtes, 2003. "Continuity of DEA Efficiency Measures," Operations Research, INFORMS, vol. 51(1), pages 149-159, February.
    11. Mark Velednitsky, 2022. "Solving $$(k-1)$$ ( k - 1 ) -stable instances of k-terminal cut with isolating cuts," Journal of Combinatorial Optimization, Springer, vol. 43(2), pages 297-311, March.
    12. M. G. Fiestras-Janeiro & I. Garcia-Jurado & J. Puerto, 2000. "The Concept of Proper Solution in Linear Programming," Journal of Optimization Theory and Applications, Springer, vol. 106(3), pages 511-525, September.

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