What are the optimal solution and optimal value for the following semidefinite program
$$ \min_{ V } \{ \mbox{tr} (V\Sigma) : VV^T=I \}$$
where $\Sigma$ is a given positive semidefinite matrix, and both $V$ and $\Sigma$ are squared matrix?
Also, what are the optimal solution and optimal value for the following semidefinite program
$$ \min_{ V } \{ \sum_i v_i^T\Sigma v_i: VV^T=I \}$$
where $\Sigma$ is a given positive semidefinite matrix, both $V$ and $\Sigma$ are squared matrix, and $v_i$ is the $i$-th column of $V$?
Up to a change of orthonormal basis, we may assume that $\Sigma=diag(\lambda_i)$ where $\lambda_i>0$.
If $V=[v_{i,j}]$, then
$tr(V\Sigma)=\sum_i v_{i,i}\lambda_i\geq -\sum_i \lambda_i=tr((-I)\Sigma)$.
Conclusion. The $\min$ is reached for $V=-I$ and its value is $-tr(\Sigma)$.