# How to define a stationary point of the MINLP problem?

As we all know, KKT point and stationary point are well defined when the optimization variables are continuous in the problem.

Now, I want to know whether there exist some special points except for the globally optimal solution in discrete optimization problems. For example, suppose $$f$$'s are differentiable with respect to $$x$$ for any given $$y$$. \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. Except for the globally optimal solution, what else do some special points (e.g. stationary point) exist? And how to define these points?

Thanks a lot for any help.

Your question is not clear. You are interested in special points in discrete optimization spaces. But you describe a mixed-variable problem involving both continuous variables $$x$$ and integer variables $$y$$.
For a given $$y$$, the theory of calculus of variations applies to the continuous subproblem on $$x$$. But I don't think you are interested in such a trivial answer.