Given an SDP problem with $m$ equality constraints and one Linear Matrix Inequality (LMI) in standard form:
$$ \begin{align} \min \quad & \mathbf{F}_0 \bullet \mathbf{Y} \\ \text{s.t.} \quad & \mathbf{F}_i \bullet \mathbf{Y} = c_i \quad (i = 1,\ldots,m), \quad \mathbf{Y} \succeq \mathbf{0}, \\ & f(\mathbf{Y}) \succeq 0 \\ \end{align} $$
Where $\mathbf{F}_j$ are known matrices and $\mathbf{Y}$ is the unknown matrix, how can we include the LMI into the formulation? I'm interested in expressing it in the SDPA format [Sec. 2.1 of this link]. In this format, we need to specify the matrices $\mathbf{F}_i$, as well as the constraint values $c_i$.
Since my problem above matches the dual in SDPA format (Eq. "SDP" in the previous link), except for the LMI, I am looking for a way to adapt it to inlcude an LMI.
I have thought of augmenting the matrix $\mathbf{Y}$ with a diagonal block corresponding to $f(\mathbf{Y})$: $$ \begin{align} \min \quad & \bar{\mathbf{F}}_0 \bullet \bar{\mathbf{Y}}, \quad \bar{\mathbf{Y}} = \begin{bmatrix} \mathbf{Y} & \mathbf{0} \\ \mathbf{0} & \mathbf{Z} \end{bmatrix} \\ \text{s.t.} \quad & \mathbf{F}_i \bullet \mathbf{Y} = c_i \\ & \mathbf{Z} - f(\mathbf{Y}) = \mathbf{0} \end{align} $$ Where the expression $\mathbf{Z} - f(\mathbf{Y}) = \mathbf{0}$ would need to be expanded into multiple constraints corresponding to the individual elements of the symmetric matrix $\mathbf{Z}$.
By doing this, I believe we can ensure that $f(\mathbf{Y}) \succeq \mathbf{0}$ holds. However, I am unsure if there are better alternatives.