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Stability of Singular Solutions of Nonlinear Equations with Restricted Smoothness Assumptions

Author

Listed:
  • Andreas Fischer

    (Technische Universität Dresden)

  • Alexey F. Izmailov

    (Lomonosov Moscow State University (MSU))

  • Mario Jelitte

    (Technische Universität Dresden)

Abstract

This work is concerned with conditions ensuring stability of a given solution of a system of nonlinear equations with respect to large (not asymptotically thin) classes of right-hand side perturbations. Our main focus is on those solutions that are in a sense singular, and hence, their stability properties are not guaranteed by “standard” inverse function-type theorems. In the twice differentiable case, these issues have received some attention in the existing literature. Moreover, a few results in this direction are known in the case when the first derivative is merely B-differentiable. Here, we further elaborate on a similar setting, but the main attention is paid to the case of piecewise smooth equations. Specifically, we study the effect of singularity of a solution for some active smooth selection on the overall stability properties, and we provide sufficient conditions ensuring the needed stability properties in the cases when such smooth selections may exist. Finally, an application to a piecewise smooth reformulation of complementarity problems is given.

Suggested Citation

  • Andreas Fischer & Alexey F. Izmailov & Mario Jelitte, 2023. "Stability of Singular Solutions of Nonlinear Equations with Restricted Smoothness Assumptions," Journal of Optimization Theory and Applications, Springer, vol. 196(3), pages 1008-1035, March.
    • Handle: RePEc:spr:joptap:v:196:y:2023:i:3:d:10.1007_s10957-023-02159-7
    DOI: 10.1007/s10957-023-02159-7
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    References listed on IDEAS

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    1. HALKIN, Hubert, 1974. "Implicit functions and optimization problems without continuous differentiability of the data," LIDAM Reprints CORE 184, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    2. A. F. Izmailov & M. V. Solodov, 2002. "The Theory of 2-Regularity for Mappings with Lipschitzian Derivatives and its Applications to Optimality Conditions," Mathematics of Operations Research, INFORMS, vol. 27(3), pages 614-635, August.
    3. A. Fischer & A. F. Izmailov & M. Jelitte, 2021. "Newton-type methods near critical solutions of piecewise smooth nonlinear equations," Computational Optimization and Applications, Springer, vol. 80(2), pages 587-615, November.
    4. Aram V. Arutyunov & Alexey F. Izmailov, 2006. "Directional Stability Theorem and Directional Metric Regularity," Mathematics of Operations Research, INFORMS, vol. 31(3), pages 526-543, August.
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