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Lipschitz Modulus of the Optimal Value in Linear Programming

Author

Listed:
  • María Jesús Gisbert

    (Miguel Hernández University of Elche)

  • María Josefa Cánovas

    (Miguel Hernández University of Elche)

  • Juan Parra

    (Miguel Hernández University of Elche)

  • Fco. Javier Toledo

    (Miguel Hernández University of Elche)

Abstract

The present paper is devoted to the computation of the Lipschitz modulus of the optimal value function restricted to its domain in linear programming under different types of perturbations. In the first stage, we study separately perturbations of the right-hand side of the constraints and perturbations of the coefficients of the objective function. Secondly, we deal with canonical perturbations, i.e., right-hand side perturbations together with linear perturbations of the objective. We advance that an exact formula for the Lipschitz modulus in the context of right-hand side perturbations is provided, and lower and upper estimates for the corresponding moduli are also established in the other two perturbation frameworks. In both cases, the corresponding upper estimates are shown to provide the exact moduli when the nominal (original) optimal set is bounded. A key strategy here consists in taking advantage of the background on calmness in linear programming and providing the aimed Lipschitz modulus through the computation of a uniform calmness constant.

Suggested Citation

  • María Jesús Gisbert & María Josefa Cánovas & Juan Parra & Fco. Javier Toledo, 2019. "Lipschitz Modulus of the Optimal Value in Linear Programming," Journal of Optimization Theory and Applications, Springer, vol. 182(1), pages 133-152, July.
    • Handle: RePEc:spr:joptap:v:182:y:2019:i:1:d:10.1007_s10957-018-01456-w
    DOI: 10.1007/s10957-018-01456-w
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    References listed on IDEAS

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    1. Thomas Saaty & Saul Gass, 1954. "Parametric Objective Function (Part 1)," Operations Research, INFORMS, vol. 2(3), pages 316-319, August.
    2. S. I. Gass & Thomas L. Saaty, 1955. "Parametric Objective Function (Part 2)---Generalization," Operations Research, INFORMS, vol. 3(4), pages 395-401, November.
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