Inclusion of Subdifferentials, Linear Well-Conditioning, and Steepest Descent Equation

• Published in: SIAM journal on optimization Vol. 23; no. 1; pp. 552 - 575
• Main Authors: Cabot, A., Thibault, L.
• Format: Journal Article
• Language: English
• Published: Philadelphia Society for Industrial and Applied Mathematics 01.01.2013
• Subjects: Banach spaces; Convex analysis; Derivatives; Descent; Dynamical systems; Inclusions; Inequalities; Inequality; Joints; Mathematical analysis; Mathematical functions; Optimization; Translations
• ISSN: 1052-6234; 1095-7189
• DOI: 10.1137/120886212

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 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]