As we all know, KKT point and stationary point are well definied when the optimization variables are continuous in the problem.
Now, I want to know whether there exists some special points except for globally optimal solution in discrete optimization problems. For example, $$\begin{aligned}\min_{x,y}\quad& f_0(x,y), \\ \mathrm{s.t.}\quad&f_k(x,y)\leq 0,\forall k\\ &x\in\mathcal X,\\ & y\in\mathbb Z.\end{aligned}$$ where $\mathcal X$ is a compact and convex set. Now how to define the stationary point of this problem?
Thanks a lot for any help.