I'm reading this paper (EQ 11/15/16) on a way to relax QCQPs. They state the dual of their program, but I can't quite figure out how they got there.
We're minimizing over variables: $X$ (PSD $n\times n$), $Y\in\mathbb R^{n\times m}$ ($m\le n$). For some symmetric matrices $A_i$ and $B_i\in \mathbb R^{n\times m}$. The primal is: \begin{align*} &\text{minimize}_{X,Y}&& \langle A_0, X\rangle + 2\langle B_0, Y\rangle +c_0\\ &\text{Subject to}&& \langle A_i, X\rangle + 2\langle B_i, Y\rangle +c_i\\ &&&(e_k+e_\ell)^T X(e_k+e_\ell)\ge \Vert(e_i+e_j)^T Y\Vert\\ &&&(e_k-e_\ell)^T X(e_k-e_\ell)\ge \Vert(e_i-e_j)^T Y\Vert\\ \end{align*} where the $e_i$ form the standard basis for $\mathbb R^n$ and $k,\ell\in [n]$.
The dual is a maximization over $t_i\in \mathbb R_{\ge 0}$ and symmetric $m\times m$ matrices $T$:
\begin{align*} &\text{minimize}_{t, T} &&\text{tr}(T)\\ &\text{subject to}&&\begin{bmatrix} A_0&B_0\\ B_0^T&+-T+m^{-1}c_0I_m \end{bmatrix} + \sum_{i}t_i\begin{bmatrix} A_i&B_i\\ B^T_i&m^{-1}c_kI_m \end{bmatrix}\succeq 0\\ &&&[e_1,\dots, e_n]^T\left(A_0+\sum_i t_kA_k\right)[e_1,\dots, e_n]\in \mathbb D_n, \end{align*} where $\mathbb D_n$ is the set of $n\times n$ diagonally dominant matrices.
The PSD constraint makes sense to me, but I'm not sure how the last constraint is derived!
I'm just using the Lagrangian, but I don't understand why there's not a dependence on the $B_i$ matrices.
Thanks!