I am interested in a variant of the orthogonal Procrustes problem
$$ \begin{array}{ll} \underset{X} {\text{minimize} } & \| X A - B \|_{\text F} \\ \text{subject to} & X^TX = I \\ & (X^T C)_{ii} \geq D_{ii} \end{array} $$
where $X,C,D\in\mathbb{R}^{3\times 3}$. $C$ is a rotation matrix. $D$ is diagonal and $0 < D_{ii} \leq 1$. X should be a rotation matrix.
Intuitively, I want to constrain the angle between the $i^{th}$ column of $X$ and the $i^{th}$ column of $C$.
If it helps, I can simplify $D=cI$ for some scalar $c$ with $0<c\leq 1$.
How do I solve this problem? Does it have a closed form solution?
