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.



KKT conditions of problem with variational inequality constraint
This is (in general) a Nonlinear Semidefinite Programming problem.


The KKT optimality conditions for it (other than flipping the sign for $g\_i(x))$ are stated in (12)-(14) of [NAG Library Routine Document
e04svf (handle\_solve\_pennon)](https://www.nag.com/numeric/fl/nagdoc_latest/html/e04/e04svf.html)


Edit: Just to clarify, (12-(14) are the ternination condiitions for that solver. In the actual optimality conditions, all the $\epsilon$ 's would be zero (rendering some of these as equalities).


