Fréchet quasidifferential calculus with applications to metric regularity of continuous maps

Abstract

To cite this article: Amos Uderzo (2005) Fréchet quasidifferential calculus with applications to metric regularity of continuous maps, Optimization, 54:4-5, 469-493, A strong variant for the notion of quasidifferentiability of functions is considered in the light of recent achievements in variational analysis. Several characterization results, some calculus rules and examples are provided. Then a generic result for the corresponding subdifferentiability notion is established in general Banach spaces. Subsequently, such notion is employed in the study of generalized differential criteria for metric regularity. Necessary and sufficient conditions, along with a precise estimation of the norm of metric regularity, are obtained in the case of continuous maps between Banach spaces via a strong slope approach. All these results do not require Asplundity assumptions.

Metric Regularity of Quasidifferentiable Mappings and Optimality Conditions for Nonsmooth Mathematical Programming Problems

“…With the use of general results on metric regularity [22] we obtain new necessary and sufficient conditions for the metric regularity of multifunctions in terms of quasidifferentials of the distance function to this multifunction (see [15] for some results on the quasidifferentiability of this function). These conditions significantly generalize and improve some results from [36]. For example, our conditions, unlike the ones in [36], are invariant under the choice of quasidifferentials.…”

“…In this case it seems more reasonable to apply necessary and/or sufficient conditions for metric regularity in terms of quasidifferentials. Such conditions were studied by Uderzo in [36,37].…”

“…One of the main goals of this paper is to improve the main results of [36,37] and obtain simple conditions for metric regularity in terms of quasidifferentials. With the use of general results on metric regularity [22] we obtain new necessary and sufficient conditions for the metric regularity of multifunctions in terms of quasidifferentials of the distance function to this multifunction (see [15] for some results on the quasidifferentiability of this function).…”

Error bounds and metric subregularity

“…If (u, v) / ∈ B δ (x,ȳ), then, by (43), d((x, y), (x,ȳ)) > δ/2 and consequently, by (38), (36), (35), y)).…”

“…The first equality in (vii) can be found in numerous publications, cf. [2,3,25,30,34,35,38,40]. For the other equalities in (vii), see [12,Theorem 5].…”