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.