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Critical solutions of nonlinear equations: stability issues

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Abstract

It is known that when the set of Lagrange multipliers associated with a stationary point of a constrained optimization problem is not a singleton, this set may contain so-called critical multipliers. This special subset of Lagrange multipliers defines, to a great extent, stability pattern of the solution in question subject to parametric perturbations. Criticality of a Lagrange multiplier can be equivalently characterized by the absence of the local Lipschitzian error bound in terms of the natural residual of the optimality system. In this work, taking the view of criticality as that associated to the error bound, we extend the concept to general nonlinear equations (not necessarily with primal–dual optimality structure). Among other things, we show that while singular noncritical solutions of nonlinear equations can be expected to be stable only subject to some poor “asymptotically thin” classes of perturbations, critical solutions can be stable under rich classes of perturbations. This fact is quite remarkable, considering that in the case of nonisolated solutions, critical solutions usually form a thin subset within all the solutions. We also note that the results for general equations lead to some new insights into the properties of critical Lagrange multipliers (i.e., solutions of equations with primal–dual structure).

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Acknowledgments

The authors thank the two anonymous referees for their comments, which improved the original presentation.

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Authors and Affiliations

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

    A. F. Izmailov

  2. RUDN University, Miklukho-Maklaya Str. 6, Moscow, Russia, 117198

    A. F. Izmailov

  3. Department of Mathematics, Physics and Computer Sciences, Derzhavin Tambov State University, TSU, Internationalnaya 33, Tambov, Russia, 392000

    A. S. Kurennoy

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

    M. V. Solodov

Authors
  1. A. F. Izmailov
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  2. A. S. Kurennoy
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  3. M. V. Solodov
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Correspondence to M. V. Solodov.

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Dedicated to Professor R. Tyrrell Rockafellar.

This research is supported in part by the Russian Foundation for Basic Research Grant 14-01-00113, by the Russian Science Foundation Grant 15-11-10021, by the Grant of the Russian Federation President for the state support of leading scientific schools NSh-8215.2016.1, by Volkswagen Foundation, by CNPq Grants PVE 401119/2014-9 and 303724/2015-3, and by FAPERJ.

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Izmailov, A.F., Kurennoy, A.S. & Solodov, M.V. Critical solutions of nonlinear equations: stability issues. Math. Program. 168, 475–507 (2018). https://doi.org/10.1007/s10107-016-1047-x

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  • DOI: https://doi.org/10.1007/s10107-016-1047-x

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