The classical Orthogonal Procrustes Problem is
$$\begin{array}{ll} \text{minimize} & \|A\Omega-B\|_{F}\\ \text{subject to} & \Omega'\Omega=I\end{array}$$
where $A$ and $B$ are known matrices.
Suppose $A$ is the identity matrix. I would like to solve the less strict problem
$$\begin{array}{ll} \text{minimize} & \|\Omega-B\|_{F}\\ \text{subject to} & \Omega'\Omega = \mbox{diag}(d_1,d_2,d_3,\dots)\end{array}$$
i.e., I want to relax the orthonormality constraint to mere orthogonality. For my purposes it is only important that $\Omega' \Omega$ be diagonal. With a less strict constraint than orthonormality, my hope is that $\Omega$ can get closer to $B$. Any ideas on how to do this?