I have an integer program to solve of the following form: \begin{align} \min_{x\in\mathbb{Z}^n}& \quad f(x)\\ \text{s.t} &\quad g(x)\leq 0, \end{align} where $g$ is such that "$g(x)\in\mathbb{R}^{m(x)}$", where quotation marks indicate a moderate abuse of notation.
What kind of optimization problem is this? Are there any optimization solvers that accept a black-box function $g(x)$ of which elements should be non-positive but such that the codomain size is a variable?