Context & Motivation: This is a preliminary question that I wanted to answer in order to generalize the nonconvex projected gradient descent method in Chapter 3 of the book Non-convex Optimization for Machine Learning to the nonsmooth case. Specifically, I wanted to know if the classic subgradient could be analyzed in the nonconvex setting using the same principles in the convex case, i.e., by looking at the size of the subgradients of the objective function in the nonconvex setting.
Question: Let $\Omega \subseteq {\mathbb R}^n$ be a possibly nonconvex open set, and suppose $X$ is a compact subset of the interior of $\Omega$, i.e., $X\subseteq {\rm int}\ \Omega$. Moreover, let $f:\Omega \mapsto {\mathbb R}$ be such that for every $x\in\Omega$ there exists an open ball $\cal B$ around $x$ such that $f$ is convex on $\cal B$.
Does there exist a constant $L > 0$ such that $\|v\| \leq L$ for every $v\in\partial f(x)$ and $x\in X$ (where $\partial f(x)$ is the convex subdifferential of $f$ at $x$)? In other words, are the subdifferentials of $f$ uniformly bounded when restricted to a compact subset?
Too long for a comment.
Here is the idea of the proof.
Suppose $\varphi$ a convex locally lipschitzian function that has a closed support hyperplane (which by definition, is a continuous linear form). Given a unit vector $u,$ and $x$ a centre of the ball, you can restrict the function $\varphi$ on the line segment inside this ball that passess through $x$ in direction $u.$ $\varphi$ restricted to this line segment is a convex function from an interval to the reals, which therefore has lateral derivatives (a.k.a. side derivatives), both of which are non-decreasing. Since $\varphi$ is lipschitzian, these lateral derivatives are bounded by the lipchitzian control-constant, a fortiori, the bounds are independent of $u.$ In finite dimensions, your hyperplane is necessarily given as a row matrix, this will show that all entries in this row matrix are bounded by constants independent of $u$ and therefore, the associated linear transformation will also have bounded movement (i.e. the support hyperplane can only tilt a little on every direction). QED