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A note on upper Lipschitz stability, error bounds, and critical multipliers for Lipschitz-continuous KKT systems

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Abstract

We prove a new local upper Lipschitz stability result and the associated local error bound for solutions of parametric Karush–Kuhn–Tucker systems corresponding to variational problems with Lipschitzian base mappings and constraints possessing Lipschitzian derivatives, and without any constraint qualifications. This property is equivalent to the appropriately extended to this nonsmooth setting notion of noncriticality of the Lagrange multiplier associated to the primal solution, which is weaker than second-order sufficiency. All this extends several results previously known only for optimization problems with twice differentiable data, or assuming some constraint qualifications. In addition, our results are obtained in the more general variational setting.

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Acknowledgments

The authors thank the anonymous referees and the Associate Editor for useful comments that led to an improved version of the original submission.

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Authors and Affiliations

  1. Uchebniy Korpus 2, OR Department, VMK Faculty, Moscow State University, MSU, Leninskiye Gory, 119991 , Moscow, Russia

    Alexey F. Izmailov & Alexey S. Kurennoy

  2. IMPA-Instituto de Matemática Pura e Aplicada, Estrada Dona Castorina 110, Jardim Botânico, Rio de Janeiro, RJ, 22460-320, Brazil

    Mikhail V. Solodov

Authors
  1. Alexey F. Izmailov
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  2. Alexey S. Kurennoy
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  3. Mikhail V. Solodov
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Correspondence to Mikhail V. Solodov.

Additional information

Research of the first two authors is supported by the Russian Foundation for Basic Research Grant 10-01-00251. The third author is supported in part by CNPq Grant 302637/2011-7, by PRONEX-Optimization, and by FAPERJ.

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Izmailov, A.F., Kurennoy, A.S. & Solodov, M.V. A note on upper Lipschitz stability, error bounds, and critical multipliers for Lipschitz-continuous KKT systems. Math. Program. 142, 591–604 (2013). https://doi.org/10.1007/s10107-012-0586-z

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  • DOI: https://doi.org/10.1007/s10107-012-0586-z

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