Is there something similar to Lagrange multipliers, but to get critical points of an $f$ on $g \leq m$ instead of on $g = m$?

Question

I have $$f,g:\mathbb{R^n} \rightarrow \mathbb{R}, m \in \mathbb{R}$$

Under any conditions of 'goodness' of the functions, is there a way to get the critical points of $f$ on $D = \{ g \leq m \}$

(Suppose $g \geq 0$)

Lagrange multipliers gives us a method for when there is an equality in $D$.

Something more: what I really want is to get the maximum over the integer points of $D$, but this is a good start.

I don't really know if this problem is solvable. There are some tools for when $g$ is linear, though.