Let $ \mathcal{P}$ be a convex problem with linear objective and quadratic constraints,
$$ \eqalign{
\mathcal{P}: & \min_{x,t} -t \\
& \text{s.t.} \\
{\color{blue}{\mu_k}}: & a^H_k x + a^T_k x^{*} - x^H D_k x + t p_k + r_k \leq 0, k = 1, \dots, K, \\
{\color{blue}{\lambda}}: & x^H C x - P \leq 0, \\
{\color{blue}{\nu}}: & diag(x) x^{*} - d 1 \preccurlyeq 0, \\
{\color{blue}{\beta}}: & -t \leq 0.
}
$$
$ p_k, r_k \geq 0 $,
$ x \in \mathbb{C}^N $,
$ C $ is a diagonal matrix with real non-negative elements,
$ D_k \in \mathbb{C}^{N \times N} $ is a symmetric positive semidefinite (rank-one)
$\mu_k, \lambda, \nu, \beta $ are the KKT multipliers
$ (\cdot)^H $ denotes Hermitian transpose
$ (\cdot)^{*} $ denotes the complex conjugate
$(\cdot)^T $ denotes the transpose.
Lagrangian:
$$
\eqalign
{
L(x,t,\mu_k,\lambda,\nu,\beta) = -t + \sum_k \mu_k (a^H_k x + a^T_k x^{*} - x^H D_k x + t p_k + r_k) + \lambda (x^H C x - P) + \nu^T (diag(x) x^{*} - d1) - t \beta
}
$$
Objective:
Find the dual of $ \mathcal{P} $ analytically.
My attempt:
I am stuck at this point by the following reasons. From the stationarity conditions, I would normally try to find $ x $ (and $ t $) and then replace them in the Lagrangian $ L(x,t,\mu_k,\lambda,\nu,\beta) $such that it only depends on the KKT multipliers $ \mu_k,\lambda,\nu,\beta $. However, from the stationarity conditions, I cannot find a way to get rid of $ x $ (as you can see). Also, the complementary slackness conditions do not seem to help at this point.
From stationarity conditions: $$ \eqalign { (\sum_k \mu_k D_k - \lambda C - diag(\nu))x = \sum_k \mu_k a_k \\ \sum_k \mu_k p_k = (1 + \beta) } $$