3
$\begingroup$

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.

$\endgroup$
1
  • $\begingroup$ Perhaps you can tel us exactly what $\phi$ is? $\endgroup$ – Mark L. Stone Dec 25 '20 at 0:05
2
$\begingroup$

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).

$\endgroup$
2
  • $\begingroup$ Just to confirm, $u_k \geq 0$ and $U_k \succeq 0$, right? $\endgroup$ – Daniel Turizo Dec 25 '20 at 18:10
  • $\begingroup$ Yes.. ............. $\endgroup$ – Mark L. Stone Dec 25 '20 at 19:42

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.