Kybernetika 61 no. 5, 635-646, 2025

A note on the uniformity of strong subregularity around the reference point

Tomáš RoubalDOI: 10.14736/kyb-2025-5-0635

Abstract:

This paper investigates strong metric subregularity around the reference point as introduced by H. Gfrerer and J. V. Outrata in \cite{GfrererOutrata2022}. In the setting of Banach spaces, we analyse its stability under Lipschitz continuous perturbations and establish its uniformity over compact sets. Our results ensure that the property is preserved under small Lipschitz perturbations, which is crucial for maintaining robustness in variational analysis. Furthermore, we apply the developed theory to parametric inclusion problems. The analysis demonstrates that the uniformity of strong metric subregularity provides a theoretical foundation for addressing stability issues in parametrized optimization and control applications.

Keywords:

strong metric subregularity, Lipschitz continuity, uniformity, sum stability

Classification:

49J53, 49J52, 90C33

paper.pdf

References:

  1. S. Adly, R. Cibulka and H. Massias: Variational analysis and generalized equations in electronics. Set-Valued Var. Anal. 21 (2013), 333-358.   DOI:10.1007/s11228-013-0236-5
  2. M. Benko and P. Mehlitz: Calmness and calculus: Two basic patterns. Set-Valued Var. Anal. 30 (2021), 81-117.   DOI:10.1007/s11228-021-00589-x
  3. R. Cibulka, A. L. Dontchev and A. Y. Kruger: Strong metric subregularity of mappings in variational analysis and optimization. J. Math. Anal. Appl. 457 (2018), 2, 1247-1282.   DOI:10.1016/j.jmaa.2016.11.045
  4. R. Cibulka, J. Preininger and T. Roubal: On uniform regularity and strong regularity. Optimization 68 (2019), 2-3, 549-577.   DOI:10.1080/02331934.2018.1547383
  5. R. Cibulka and T. Roubal: Solution stability and path-following for a class of generalized equations. In: Control Systems and Mathematical Methods in Economics, vol. 687 of Lecture Notes in Econom. and Math. Systems. Springer, Cham, 2018, pp. 57-80.   DOI:10.1007/978-3-319-75169-6\_4
  6. A. L. Dontchev, M. I. Krastanov, R. T. Rockafellar and V. M. Veliov: An {E}uler-{N}ewton continuation method for tracking solution trajectories of parametric variational inequalities. SIAM J. Control Optim. 51 (2013), 3, 1823-1840.   DOI:10.1137/120876915
  7. A. L. Dontchev and R. T. Rockafellar: Implicit Functions and Solution Mappings. A View from Variational Analysis. (Second edition.) Springer, New York 2014.   CrossRef
  8. H. Gfrerer and J. Outrata: On a semismooth* Newton method for solving generalized equations. SIAM J. Optim. 31 (2021), 1, 489-517.   DOI:10.1137/19M1257408
  9. H. Gfrerer and J. Outrata: On (local) analysis of multifunctions via subspaces contained in graphs of generalized derivatives. J. Math. Anal. Appl. 508 (2022), 2, 125895.   DOI:10.1016/j.jmaa.2021.125895
  10. A. D. Ioffe: Variational Analysis of Regular Mappings. Theory and Applications. Springer Monographs in Mathematics, Springer, Cham 2017.   CrossRef
  11. H. V. Ngai, N. H. Tron and M. Théra: Metric regularity of the sum of multifunctions and applications. J. Optim. Theory Appl. 160 (2014), 2, 355-390.   DOI:10.1007/s10957-013-0385-6
  12. R. T. Rockafellar and R. J. B. Wets: Variational Analysis, vol. 317 of Grundlehren der Mathematischen Wissenschaften: Fundamental Principles of Mathematical Sciences. Springer-Verlag, Berlin 1998.   DOI:10.1007/978-3-642-02431-3