I have the following problem that I am looking advice on. Let $ \mathcal{F}$ be a convex subset of vector space $X$. The goal it to \begin{align*} \max_{x \in \mathcal{F}} f(x)\\ s.t. \ g(x) \le 0 \end{align*}
Also $f(x)$ is strictly concave and continuos over $\mathcal{F}$. However, the function $g(x)$ is continuos and concave.
So, the set formed by $\Omega= \mathcal{F} \cap \{x: g(x) \le 0 \}$ is not convex in general.
Clearly, this problem does not fall under umbrella of convex optimization.
Since the set $\Omega$ is compact $f(x)$ attains its maximum.
However, I am not sure how to find optimizing $x^*$.
Main Question: The frist step I thought is to pose Lagrangian (KKT). But my question is can I even pose a Lagrangian (KKT) as a necessary condition? For example does $\Omega$ have to be convex to pose KKT?
For, example are there any references (books or papers) you can point me to learn about something similar?
Any advice would be appreciated. Thank you