I am trying to formulate following problem (with some constraint) as a semidefinite programming problem (SDP),
\begin{equation} \text{minimize } ~~ -a^T B^{-1} a \end{equation}
where $B$ is a positive definite matrix. I used auxiliary variable and Schur complement and tried to formulate it as following :
\begin{eqnarray} \text{minimize }~~ t \\ \text{s.t.} ~~~t \geq -a^T B^{-1} a \end{eqnarray}
then tried to to reformulate the constraint as a Linear Matrix inequality (LMI) using Schur complement, but I got stuck to following:
\begin{align*} \left[\begin{array}{cc} t & a' \\ a & -B^{-1} \end{array}\right] \overset{\Large\text{??}}\geq 0 \end{align*}
but the LMI is not positive definite because $-B^{-1}$ is not positive definite.
Is there another way /scape that I can formulate the original problem as SDP?