The standard SDP formulation is given as : \begin{equation} \begin{aligned} \min_{X\in H^{n}} \quad& \langle X,M_{0} \rangle\\ \textrm{s.t.} \quad& l_{s} \leq \langle X, M_{s} \rangle \leq u_{s}, \quad & s=1,...,p,\\ \quad & X \succcurlyeq 0 \end{aligned} \end{equation}
where $M_{0}, M_{1}, ..., M_{p} \in \mathbb{H}^{n} $ and
\begin{align*} (l_{s}, u_{s}) \in (\{-\infty\} \cup \mathbb{R} ) \times (\mathbb{R} \cup \{+\infty\}) \end{align*}
for every $s=1,...,p$
Here, $\langle \rangle$ indicates the frobenius inner product.
The SDP relaxation of the TSP problem can be written (as per this paper) as : \begin{equation} \begin{aligned} \min_{X} \quad& -\frac{1}{2}\langle D,X \rangle + \frac{\alpha}{2}\sum_{i=1}^{n}\sum_{j=1}^{n}d_{ij}\\ \textrm{s.t.} \quad& \langle A_{i},X \rangle = 2 + \alpha - \beta \quad & i=1,...,n\\ \quad& \langle B_{i}, X\rangle = 2(n\alpha - \beta) \quad & i=1,...,n \\ \quad & 2(\alpha - 1) \leq \langle C_{ij},X \rangle \leq 2\alpha \quad & i,j = 1,...,n\\ \quad & X \succcurlyeq 0 \end{aligned} \end{equation}
Where $D$ is the Distance Matrix of the graph,
$A_{i} \in S_{n}$ and $a_{ii} = 1$ with all other entries $0$,
$B_{i} \in S_{n}$ and $b_{ii} = 2$ with all other entries in row $i$ and column $i$ are $1$ and all other entries is $0$
$C_{ij}$ has $1$ in the positions $c_{ij}$ and $c_{ji}$ with all other entries $0$ and
$n$ is the number of vertices in the graph
Is it possible to convert the SDP relaxation of the TSP problem into the standard SDP formulation?