DMG Seminar by Prof Alexander Kruger

Title: Radius of Metric Regularity

By: Prof Alexander Kruger, Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam

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.

The infinite dimensional case is studied in [3]. The radius theorems for metric regularity and strong metric subregularity from [1,2] are improved.

I am going to summarise the developments in [1-3].

References

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. Gfrerer H., Kruger A.Y.: The radius of metric regularity revisited. Set-Valued Var. Anal. 31(3), 20 (2023)