# KKT conditions of problem with variational inequality constraint

I have an optimization problem with a variational inequality constraint: $$$$\begin{array}{ll} \min_x & f(x) \\ \mathrm{s.t.} & g_i(x) \leq 0, \quad i=1,\ldots,m \\ & h_j(x) = 0, \quad i=1,\ldots,n \\ & \phi(x,z) \geq 0, \quad \forall z \in \Omega_z \, , \end{array}$$$$ where $$\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.

• Perhaps you can tel us exactly what $\phi$ is? – Mark L. Stone Dec 25 '20 at 0:05

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)
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).
• Just to confirm, $u_k \geq 0$ and $U_k \succeq 0$, right? – Daniel Turizo Dec 25 '20 at 18:10