I am having trouble finding finding the KKT points of the following problem. $$ \begin{array}{ll} \min \max &{\tau^2} \\ \text{subject to} & \tau \leq 3 l_u ^2\\ & \tau \geq \gamma + l_u + 2\\ & \tau \geq \frac{3(\gamma - l_u)^2 + \|A\|^2}{C} \end{array} $$
where $l_u, \gamma, C, \|A\|^2$ are given constants.
The Lagrangian is $$ \mathcal{L}(\tau,\lambda, \mu, \omega)=\tau^2 + \lambda (\tau - 3 l_u ^2) + \mu (-\tau + \gamma + l_u + 2) + \omega (-\tau +\frac{3(\gamma - l_u)^2 + \|A\|^2}{C}) \quad \lambda, \mu, \omega \geq0, $$ KKT Conditions: $$ \begin{align} & L_{\tau}=2 \tau + \lambda - \mu -\omega =0 \\ & \lambda (\tau - 3 l_u ^2) = 0 \\ & \mu (-\tau + \gamma + l_u + 2) = 0 \\ & \omega (-\tau +\frac{3(\gamma - l_u)^2 + \|A\|^2}{C}) = 0 \\ & \tau \leq 3 l_u ^2\\ & \tau \geq \gamma + l_u + 2\\ & \tau \geq \frac{3(\gamma - l_u)^2 + \|A\|^2}{C} \\ & \lambda, \mu, \omega \geq0 \end{align} $$
- From first equation $\tau = \frac{\mu+\omega-\lambda}{2}$
- Then I plug in this into the second, third and fourth equations. But I did not manage to solve that.
Please advise.