I would like to compute the dual of the following problem by using the KKT conditions. However, due to form of the first constraint I am not able to obtain the dual. The problem is the following \begin{eqnarray} \min_{x,a} a &\\ \text{s.t.}: &\\ & -x^H P^H_1 (cI + P_2 x x^H P^H_2 + P_3 x x^H P^H_3)^{-1} P_1 x - a \leq 0 \\ & x^H D x \leq r \\ & |x_n|^2 \leq s, n = 1,\dots,N \\ & a \leq 0 \end{eqnarray} where $ c, s, r > 0$, $ a \in \mathbb{R} $, $ x \in \mathbb{C}^{N} $, $ P_1, P_2, P_3 \in \mathbb{C}^{M \times N} $
I have proved that the matrix $ (cI + P_2 x x^H P^H_2 + P_3 x x^H P^H_3)^{-1} $ is positive definite (PD). Then, it follows that the matrix $ - x^H P^H_1 (cI + P_2 x x^H P^H_2 + P_3 x x^H P^H_3)^{-1} P_1 x $ is negative definite. $ D $ is a diagonal positive semidefinite (PSD). Thus, the first constraint is non-convex, the second and third constraints are convex whereas the last one is linear. Any help will be welcomed.
** Update:
If it helps, I have the gradient of $ z = x^H P^H_1 (cI + P_2 x x^H P^H_2 + P_3 x x^H P^H_3)^{-1} $ w.r.t. $ x $ $$ \nabla_w z = x^H P^H_1 Y^{-1} P_1 - ( x^H P^H_2 Y^{-1} P_1 x x^H P^H_1 Y^{-1} P_2 + x^H P^H_3 Y^{-1} P_1 x x^H P^H_1 Y^{-1} P_3 ) $$ where $ Y = (cI + P_2 x x^H P^H_2 + P_3 x x^H P^H_3) $