Suppose we want to minimize a linear objective inside an ellipsoid that is, 


$\min \_x l^Tx$


such that $(x - \mu)^TA(x - \mu) \leq \beta ^2$.


Here, A is PSD and $\mu$ is a fixed vector. Can this be written as a SDP ? 



Convex Optimization in an Ellipsoid
There is a closed-form expression for that value.
Indeed, a straight-forward computation yields
\begin{equation}
\begin{split}
\min\_{\langle A(x-\mu),x-\mu\rangle \le \beta^2} \langle l, x\rangle &= \min\_{\|v\|^2 \le \beta^2}\langle l, \mu + A^{-1/2}v\rangle = \langle l, \mu \rangle + \min\_{\|v\| \le \beta}\langle A^{-1/2}l,v\rangle
\\\\&
= \langle l, \mu\rangle - \beta \|A^{-1/2}l\|
= \langle l, \mu \rangle - \beta \sqrt{\langle A^{-1}l,l\rangle}
\end{split}
\end{equation}


