We are currently working on an engineering project involving convex optimization. The project relates to a low earth orbit spacecraft rendezvous. In the course of this project, we are required to convert an optimization problem given in the standard form to semidefinite programming or SDP form. Stating the problem below:
$$ \displaystyle \min_{\epsilon,\delta,X,Y,R,Q} \mu $$
such that the following matrix inequalities $$ \begin{bmatrix} \Omega & XE_1^T +Y^TG^T & Y^T & X \\ * & -\epsilon I & 0 & 0 \\ * & * & -R & 0 \\ * & * & * & -Q \\ \end{bmatrix} \leqq 0, $$
$$
\begin{bmatrix}
-\delta I & X_eo^T\\
* & -X
\end{bmatrix}
\leqq 0, $$
and
$$
\begin{bmatrix}
-X+v\delta I & 0 & 0 \\
* & -\mu I & Y \\
* & * & -vI \\
\end{bmatrix}
\leqq 0 $$
hold. This is the standard form we have.
We would like to convert this to SDP form given below.
$$ \displaystyle \min_{X, \epsilon, H^n} \langle X, M_0 \rangle_F \ \ \mathrm{s.t.}\ \ \ l_s \le \langle X, M_s \rangle_F \le u_s,\ s = 1 \dots p,$$ where the angular brackets denote Frobenius Inner Product and where $ M_0,M_1,\dots,M_p \ \in \ H^n $, and $(l_s,u_s)$ are pairs of extended real numbers such that $-\infty\le l_s\le u_s\le\infty$ for all $s=1,\ldots,p$.
Is there any way in which the above-mentioned conversion can be carried out?