Séance Séminaire

Séminaire ACSIOM

Inclusion de sous-différentiels, conditionnement linéaire et équation de la plus grande pente

Given a normed space $X$, we study the following problem: characterize the functions $f:X\to\mathbb{R}\cup${$+\infty$} such that for some $\overline{x} \in {{\rm dom}\kern 0.12em} f$ and some neighbourhood $U$ of $\overline{x}$ $$\partial f(x) \subset Q \quad \text{for all }x\in U. $$ The operator $\partial$ denotes an abstract subdifferential and $Q$ is a fixed subset of $X^*$ (the topological dual space of $X$). When $X$ is a Banach space and $f$ is lower semicontinuous, we provide a characterization via the support function of $Q$, based on the Zagrodny mean value theorem. If $Q=\partial f(\overline{x})$, the lower semicontinuous functions $f$ verifying the above subdifferential inclusion are homogeneous, up to some translation. Connections with the class of linearly well-conditioned functions are explored and an application is given to the steepest descent equation. A parallel study is developped for an inequality on the directional derivative in place of the above subdifferential inclusion.