Abstract

A definition of strong tangential transversality for a finite number of sets is proposed such that strong tangential transversality implies transversality. The main tool in the proofs are infinitesimal characterizations of transversality and subtransversality of a finite number of sets in the prime space which are of independent interest. A normal intersection property for Clarke normal cones, a sum rule for the Clarke subdifferential of a finite number of lower semicontinuous functions, and an abstract Lagrange multiplier rule are obtained as applications.

Author information

Contact details are reproduced from the original publication and may be historical.

Nadezhda K. Ribarska

(1) Faculty of Mathematics and Informatics, Sofia University, Sofia, Bulgaria
(2) Inst. of Mathematics and Informatics, Bulgarian Academy of Sciences, Sofia, Bulgaria

ribarska@fmi.uni-sofia.bg

Keywords

  • Sub(transversality)
  • primal space characterizations
  • strong tangential transversality
  • normal intersection properties
  • sum rule
  • Lagrange multiplier rule

Mathematics Subject Classification

49K27, 49J53, 46N10

References

  1. S. Apostolov, M. Bivas: Characterizations of metric (sub)regularity via (sub)transversality, J. Convex Analysis 32/4 (2025).https://doi.org/10.68381/jca32065Google Scholar
  2. S. Apostolov, M. Bivas, N. Ribarska: Characterizations of some transversality-type properties, Set-Valued Var. Analysis 30 (2022) 1041–1060.https://doi.org/10.1007/s11228-022-00633-4Google Scholar
  3. M. Bivas, M. Krastanov, N. Ribarska: On tangential transversality, J. Math. Analysis Appl. 481/1 (2020), art.no. 123445, 21p.https://doi.org/10.1016/j.jmaa.2019.123445Google Scholar
  4. M. Bivas, M. Krastanov, N. Ribarska: On strong tangential transversality, J. Math. Analysis Appl. 490/1 (2020), art.no. 124235, 16p.https://doi.org/10.1016/j.jmaa.2020.124235Google Scholar
  5. M. Bivas, N. Ribarska, M. Valkov: Properties of uniform tangent sets and Lagrange multiplier rule, C. R. Acad. Bulgare Sci. 71/7 (2018) 875–884.https://doi.org/10.7546/CRABS.2018.07.02Google Scholar
  6. H. T. Bui, N. D. Cuong, A. Y. Kruger: Geometric and metric characterizations of transversality properties, Vietnam J. Math. 48/2 (2020) 277–297.https://doi.org/10.1007/s10013-020-00388-1Google Scholar
  7. N. D. Cuong, A. Y. Kruger: Transversality properties: Primal sufficient conditions, Set-Valued Var. Analysis 29 (2021) 221–256.https://doi.org/10.1007/s11228-020-00545-1Google Scholar
  8. D. Drusvyatskiy, A. D. Ioffe, A. S. Lewis: Transversality and alternating projections for nonconvex sets, Found. Comput. Math. 15/6 (2015) 1637–1651.https://doi.org/10.1007/s10208-015-9279-3Google Scholar
  9. A. Ioffe: Variational Analysis of Regular Mappings. Theory and Applications, Springer Monographs in Mathematics, Springer, Berlin (2017).https://doi.org/10.1007/978-3-319-64277-2Google Scholar
  10. A. Jourani, L. Thibault: Extensions of subdifferential calculus rules in Banach spaces, Canad. J. Math. 48/4 (1996) 834–848.https://doi.org/10.4153/CJM-1996-042-1Google Scholar
  11. M. I. Krastanov, N. K. Ribarska: Nonseparation of sets and optimality conditions, SIAM J. Control Optimization 55/3 (2017) 1598–1618.https://doi.org/10.1137/16M1074965Google Scholar
  12. N. Ribarska, M. Tasheva: Transversality versus strong tangential transversality, J. Math. Analysis Appl. 524/2 (2023), art.no. 127215, 13p.https://doi.org/10.1016/j.jmaa.2023.127215Google Scholar

Suggested citation

N. K. Ribarska, M. Tasheva. “Transversality and Strong Tangential Transversality of a Finite Number of Sets.” Journal of Convex Analysis 32 (2025), No. 3, 767–788. https://doi.org/10.68381/jca32038

Download BibTeXPublished by Heldermann Verlag.

Published by Heldermann Verlag, 2025. Rights now held by Banach Press LLC.