I have an optimization problem with a *variational inequality constraint*:
$$
\begin{equation}
\begin{array}{ll}
\min_x & f(x) \\
\mathrm{s.t.} & g_i(x) \leq 0, \quad i=1,\ldots,m \\
& h_j(x) = 0, \quad i=1,\ldots,n \\
& \phi(x,z) \geq 0, \quad \forall z \in \Omega_z \, ,
\end{array}
\end{equation}
$$
where $\Omega_z$ defines a feasible set for vector $z$. The previous problem is identical to a standard constrained optimization problem, except for the variational inequality constraint. My question is: are there any "KKT" conditions for this type of problem, similar to the standard KKT necessary conditions?

Thanks beforehand.

EDIT: $\phi(x,z) = z^T M(x)z$ where $M(x)$ is symmetric, and $\Omega_z = \left\{{z \, | \, z \neq 0}\right\}$. The variational inequality basically requires $M(x)$ to be positive semidefinite.