I was stuck in computing the dual problem in a paper by Yinyu Ye which raises an SDP relaxation such that
$$ \min_{Z \in \mathcal{K}} \left\{ h(Z) := \sqrt{\sum_{(i, j) \in \mathcal{E}} \gamma_{ij}^2 (g_{ij}^T Z g_{ij} - d_{ij}^2)^2 } \right\} \quad \mathcal{K} = \left\{Z: Z = \begin{pmatrix} Y & X^T \\ X & I_d \end{pmatrix} \succeq 0 \right\} $$
The paper gives the dual problem in the form
$$ \max \left\{ \begin{aligned} & \sum_{(i, j) \in \mathcal{E}} \lambda_{ij} d_{ij}^2 + \sum_{(i, j) \in \mathcal{I}} \lambda_{ij} \delta_{ij} : \\ &\sum_{(i, j) \in \mathcal{E}} \lambda_{ij} g_{ij} g_{ij}^T + \sum_{(i,j) \in \mathcal{I}} \lambda_{ij} \frac{e_ie_j^T + e_je_i^T}{2} \preceq 0,~|\lambda_{ij}| \leq \gamma_{ij}, \sum_{(i, j) \in \mathcal{E}} \frac{\lambda_{ij}^2}{\gamma_{ij}^2} \leq 1 \end{aligned} \right\} $$
where index set $\mathcal{I}$ corresponds to the indices of block $I_d$ in $Z$. And $\delta$ is the Kronecker delta function. May I ask how to formulate the dual problem? The paper simply skips the intermediate steps and gives the dual.