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Lipschitz Modulus of the Optimal Value in Linear Programming

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.

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Acknowledgements

This research has been partially supported by project MTM2014-59179-C2-2-P and its associated grant BES-2015-073220, both from MINECO, Spain and FEDER, “Una manera de hacer Europa”, European Union. The authors wish to thank the referees for their suggestions and comments, which have improved the original version of the paper.

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Correspondence to Juan Parra.

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Gisbert, M.J., Cánovas, M.J., Parra, J. et al. Lipschitz Modulus of the Optimal Value in Linear Programming. J Optim Theory Appl 182, 133–152 (2019). https://doi.org/10.1007/s10957-018-01456-w

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  • DOI: https://doi.org/10.1007/s10957-018-01456-w

Keywords

  • Lipschitz modulus
  • Optimal value
  • Linear programming
  • Variational analysis
  • Calmness

Mathematics Subject Classification

  • 90C31
  • 49J53
  • 49K40
  • 90C05