I want to solve a Semidefinite Programming (SDP) problem with the solver mosek. The optimization problem is given by
$$ \begin{matrix} \min\limits_{\mathbf{X},\mathbf{x}} & \text{tr}(\mathbf{AX}) + \mathbf{b}^{\top}\mathbf{x} \\ s.t & \begin{bmatrix} \mathbf{X} & \mathbf{x}\\ \mathbf{x}^{\top} & 1 \\ \end{bmatrix} \succeq \mathbf{0}.& \\ \end{matrix} \tag{1} $$
However, mosek can solve an SDP problem in the form
$$ \begin{matrix} \min\limits_{\mathbf{Y},\mathbf{y}} & \text{tr}(\mathbf{Q}_0\mathbf{Y}) + \mathbf{c}_{0}^{\top}\mathbf{y} \\ s.t & \text{tr}(\mathbf{Q}_i\mathbf{Y}) + \mathbf{c}_i^{\top}\mathbf{y} \leq u_i, & i=1,2,\ldots\\ &\mathbf{Y} \succeq \mathbf{0}.& \\ \end{matrix} \tag{2} $$
I am wondering whether it is possible to write Problem $(1)$ in the form $(2)$.
Thanks in advance for any help!