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Stability of Possibly Nonisolated Solutions of Constrained Equations, with Applications to Complementarity and Equilibrium Problems

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Abstract

We present a new covering theorem for a nonlinear mapping on a convex cone, under the assumptions weaker than the classical Robinson’s regularity condition. When the latter is violated, one cannot expect to cover the entire neighborhood of zero in the image space. Nevertheless, our covering theorem gives rise to natural conditions guaranteeing stability of a solution of a cone-constrained equation subject to wide classes of perturbations, and allowing for nonisolated solutions, and for systems with the same number of equations and variables. These features make these results applicable to various classes of variational problems, like nonlinear complementarity problems. We also consider the related stability issues for generalized Nash equilibrium problems.

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Authors and Affiliations

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

    A. V. Arutyunov & A. F. Izmailov

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

    A. V. Arutyunov & A. F. Izmailov

Authors
  1. A. V. Arutyunov
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  2. A. F. Izmailov
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Correspondence to A. F. Izmailov.

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Dedicated to the memory of Professor Jonathan Borwein.

Research of the first author is supported in part by the Russian Foundation for Basic Research Grant 17-01-00125, by the Ministry of Education and Science of the Russian Federation (Project 1.962.2017/4.6), by the RUDN University Program 5-100, and by the Volkswagen Foundation. The second author is supported by the Russian Science Foundation Grant 17-11-01168.

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Arutyunov, A.V., Izmailov, A.F. Stability of Possibly Nonisolated Solutions of Constrained Equations, with Applications to Complementarity and Equilibrium Problems. Set-Valued Var. Anal 26, 327–352 (2018). https://doi.org/10.1007/s11228-017-0459-y

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  • DOI: https://doi.org/10.1007/s11228-017-0459-y

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