Inclusion of Subdifferentials, Linear Well-Conditioning, and Steepest Descent Equation - Cabot, A.; Thibault, L. (SIAM journal on optimization, 2013)
Given a normed space $X$, we study the following problem: characterize the functions $f:X→\mathbb{R}\cup\{+∞\}$ such that for some $\overline{x} \in {\rm dom}\kern 0.12em f$ and some neighborhood $U$ of $\overline{x}$, $\partial f(x) \subset Q\ \text{for all }x\in U.$ 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 positively 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 developed for an inequality on the directional derivative in place of the above subdifferential inclusion. [PUBLICATION ABSTRACT]