I am looking to solve the following problem. The first task is to comment on the convexity/concavity of the problem and then I need to compute minimizer of the problem.
$$\min f \left( \Psi \right) = \left( \displaystyle\sum_{i=1}^N y_{i}^T\Psi y_{i} \right) - \log | \Psi | \quad \text{subject to} \quad \Psi \geq 0$$
My attempt so far:
The first term is a sum of quadratic forms hence convex. $-\log|\Psi|$ is also a convex function in $\Psi$. hence, the problem seems convex to me.
In order to solve it, I have taken derivative w.r.t $\Psi$ and equated it to zero which gives me
$$\sum_{i=1}^N ( y_{i}y_{i}^T) - (\Psi^{-1})^T = 0$$
So using this, I get
$$\Psi^* =\left(( \sum_{i=1}^N \left( y_{i} y_{i}^T \right) \right)^{T})^{-1}$$
However, I'm not too confident about this approach. Any help in this regard would be highly appreciated.