How to transform this problem, $$\min_{x_1, x_2} x_1 + x_2$$
$$\text{subject to} \\ \|P_\Omega(X-Z)\|_{F}^2 \leq x_1\\ \lambda\| X \|_{\text{tr}} \leq x_2$$
to the standard SDP problem, \begin{equation*} \begin{array}{ll} \min_{x \in \mathbb{R}^p} & c^T x \\ \text{subject to} & x_1 A_1 + \cdots + x_p A_p \preceq B, \end{array} \end{equation*}
Given that solving $\| X \|_{\text{tr}}$ is equivalent to the optimization problem defined as, $$\max_{Y \in \mathbb{R}^{m \times n}} \text{trace}(X^T Y) \\ $$ $$\text{subject to} \left[ \begin{array}{cc} I_m & Y \\ Y^T & I_n \end{array} \right]\succeq 0$$
I know that $\text{trace}(X^T Y) = \langle X,Y\rangle$, so solving $\| X \|_{\text{tr}}$, so it actually looks very much like the conic problem, but how to transform that into standard SDP, where $c$ and $x$ are vectors?