30 septiembre, 2022

Título: Radius theorems revisited

Ponente: Alexander Kruger (Federation University, Australia)

Organizador: Juan Parra

Fecha: Viernes de 30 septiembre de 2022 a las 13:00 horas

Lugar: Aula 0.2

Abstract: The topic of radius of “good behaviour” quantifying the “distance” of a given well-posed problem to the set of ill-posed problems of the same kind was explicitly initiated by Dontchev et al. in [1,2]. The authors studied the “good behaviour” of generalised equations characterised by regularity properties of set-valued mappings and established in finite dimensions exact formulas for the radii for the three fundamental properties: metric regularity, strong metric regularity and strong metric subregularity with respect to calm, Lipschitz continuous and linear perturbations in terms of the modulus of the respective property. In infinite dimensions, they obtained certain lower estimates for the radii. They also showed that for the property of (not strong) metric subregularity the analogues of the mentioned formulas and estimates fail. The radius of metric subregularity in finite dimensions was studied in [3], where new primal-dual derivatives of set-valued mappings and several moduli of subregularity were introduced and employed for characterising the radius with respect to the same and certain new classes of perturbations. The infinite dimensional case is studied in [4,5]. More moduli of subregularity and more classes of perturbations are considered. The radius theorems for metric regularity and strong metric subregularity from [2] are improved.
In the talk, I am going to summarise the developments in [1-5].
1. Dontchev, A.L., Lewis, A.S., Rockafellar, R.T.: The radius of metric regularity. Trans. Amer. Math. Soc.
355(2), 493–517 (2003).
2. Dontchev, A.L., Rockafellar, R.T.: Regularity and conditioning of solution mappings in variational
analysis. Set-Valued Anal. 12(1-2), 79–109 (2004).
3. Dontchev, A.L., Gfrerer H., Kruger A.Y., Outrata J.V.: The radius of metric subregularity. Set-Valued
Anal. 28(3), 451–473 (2020).
4. Gfrerer H., Kruger A.Y.: Radius theorems for subregularity in infinite dimensions. arXiv 2206.10347
5. Gfrerer H., Kruger A.Y.: The radius of metric regularity revisited. Preprint (2022).