The approximate subdifferential of composite functions

This paper deals with the approximate subdifferential chain rule in a Banach space. It establishes specific results when the real-valued function is locally Lipschitzian and the mapping is strongly compactly Lipschitzian.

Noncoincidence of Approximate and Limiting Subdifferentials of Integral Functionals

For a locally Lipschitz integral functional $I_f$ on $L^1(T,\mathbf{R}^n)$ associated with a measurable integrand f, the limiting subdifferential and the approximate subdifferential never coincide at a point $x_0$ where $f(t,\cdot)$ is not subdifferentially regular at $x_0(t)$ for a.e. $t\in T$. The coincidence of both subdifferentials occurs on a dense set of $L^1(T,\mathbf{R}^n)$ if and only if $f(t,\cdot)$ is convex for a.e. $t\in T$. Our results allow us to characterize Aubin's Lipschitz-like property as well as the convexity of multivalued mappings between $L^1$-spaces. New necessary optimality conditions for some Bolza problems are also obtained.

Convergence of subdifferentials and normal cones in locally uniformly convex Banach space

International audience; In this paper we study the behaviour of normal cones and subdifferentials with respect to two types of convergence of sets and functions: Mosco and Attouch–Wets convergences. Our analysis is devoted to proximal, Fréchet, and Mordukhovich limiting normal cones and subdifferentials. The results obtained can be seen as extensions of the Attouch theorem to the context of non-convex functions on locally uniformly convex Banach space. They also generalize, to sequences of subsmooth sets or functions, various results in the literature.

Differential properties of the Moreau envelope

International audience; In a vector space endowed with a uniformly Gâteaux differentiable norm, it is proved that the Moreau envelope enjoys many remarkable differential properties and that its subdifferential can be completely described through a certain approximate proximal mapping. This description shows in particular that the Moreau envelope is essentially directionally smooth. New differential properties are derived for the distance function associated with a closed set. Moreover, the analysis, when applied to the investigation of the convexity of Tchebyshev sets, allows us to recover several known results in the literature and to provide some new ones.

Metric regularity and subdifferential calculus in Banach spaces

In this paper we give verifiable conditions in terms of limiting Frechet subdifferentials ensuring the metric regularity of a multivalued functionF(x)=−g(x)+D. We apply our results to the study of the limiting Frechet subdifferential of a composite function defined on a Banach space.

A general metric regularity in asplund banach spaces

This paper establishes a simple and easily-applied criterion for determining whether a multivalued mapping is metrically regular relatively to a subset in the range space.

Coderivatives of multivalued mappings, locally compact cones and metric regularity

C 1,ω (·) -regularity and Lipschitz-like properties of subdifferential

Envelopes for sets and functions II: generalized polarity and conjugacy

International audience; Let X,Y be two nonempty sets, Φ an extended real-valued bivariate coupling function on X × Y and Γ a subset of X × Y. The present paper provides extensions to the well-known generalized Φ-conjugacy and Γ-polarity of diverse results of our previous work [2] related to φ-conjucacy and Λ-polarity, where Λ is a subset of a vector space E and φ is a function on E defining the particular coupling function (x,y)→φ(x−y) on E × E. A particular attention is devoted to the conjugacy functions (resp. polarity sets) which are mutually generating. Finally, for a superadditive conjugacy function Φ, we obtain a full description of the class of Φ-envelopes.

A note on Fréchet and approximate subdifferentials of composite functions

The aim of this note is to present in the reflexive Banach space setting a natural and simple proof of the formula of the approximate subdifferential of a composite function.

Metric regularity for strongly compactly Lipschitzian mappings

Chain rules for coderivatives of multivalued mappings in Banach spaces