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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.

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  • $\begingroup$ Perhaps you can tel us exactly what $\phi$ is? $\endgroup$
    – Mark L. Stone
    Commented Dec 25, 2020 at 0:05

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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)

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).

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  • $\begingroup$ Just to confirm, $u_k \geq 0$ and $U_k \succeq 0$, right? $\endgroup$
    – Daniel Turizo
    Commented Dec 25, 2020 at 18:10
  • $\begingroup$ Yes.. ............. $\endgroup$
    – Mark L. Stone
    Commented Dec 25, 2020 at 19:42

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