# FOC point vs Stationary point in local optimization

In this SIAM Review paper the authors are giving the following necessary condition for a point being a local maximum of a convex function:

Let $$F: \mathbb{R}^n \mapsto \mathbb{R}$$ be convex. If $$x$$ is a local maximum of $$F$$ over the nonempty compact set $$C$$, then \begin{align}\tag{FOC} \langle v - x, F'(x)\rangle \leq 0 \quad \forall v \in C. \end{align}

Moreover, the authors state that if $$F$$ is continuously differentiable, then any $$x$$ satisfying the above first-order condition will be referred to as a stationary point, and otherwise a point that satisfies the first-order condition.

What I do not understand is, what changes when a point is called a stationary point? It is still not guaranteed to be a local max, right? So what really changes when a point that satisfies the first-order conditions becomes a stationary point? What additional benefits do we get? (I am assuming this will be relevant when we analyze the convergence of the Frank-Wolfe algorithm or whatever we are using to find a local maximum, but I don't understand what changes with a point being stationary.)

• The premise here looks a little shaky. The quote says $C$ is nonempty and compact, but is it assumed to be convex? Also, the quote starts with $F$ being convex but then refers to $F'$ (presumably the gradient of $F$). Convex functions are not automatically differentiable. Is there an explicit assumption that $F'$ exists?
– prubin
Feb 13, 2022 at 16:56