Optimization problem with constraint is another optimization problem

Question

I would like to know how to solve a problem like this: $$\min f(x) s.t.\; g(x)\leq 0$$. However, we do not have explicit form of $g(x)$. What we know is that $g(x)$ is another solution of an optimization problem, say $g(x)=\inf_{\beta} h(x,\beta)$. Here, $h(x,\beta)$ is known, but assuming that we cannot solve the optimization analytically. I would like to know what kind of such optimization called and ways to solve this kind of problem.