What does it mean to call a constraint concave?

Question

I'm attempting to solve a constrained optimization problem via Kuhn-Tucker, and I'm being asked to write the necessary conditions for some point $x$ to be a solution.

The part of the K-T theorem stating the necessary conditions for a solution assume that the objective function $f$ is concave, and that the constraint functions $g_k$ are concave. My question is, what does it mean for a constraint to be concave? For example, one of the constraints in this problem is that $x\geq0$, but I only understand concavity to be describing a function taking singleton values over some set, not describing the set itself.

Answer 1

Here, your constraint function is $g(x)=x$, and your constraint is $g(x)\geq 0$. $g$ is a concave (but not strictly concave) function. So, you meet the KKT conditions, as long as $f$ is concave, and as long as your other constraint functions are concave.

Answer 2

As C. Windolf suggested, the constraint function here is $g(x) = x$ and, as you specified, the requirement is that $g(x)$ (or each $g_k(x)$ if the constraints are many) is concave.

A function is concave if and only if its hypograph is a convex set (see here for hypograph and concavity). Since the function is linear, its hypograph (the space lying below the function) is (weakly) convex. Very very roughly speaking: take two points in the set $\{(x, \mu) : \mu \leq g(x) = x\}$ and link them (take a convex combination of the two points). The combination will still belong (weakly) to the set, which therefore is (weakly) convex.

We can say that linear functions can be seen both as (weakly) convex and concave functions.