Is there a closed form solution to this constrained optimization problem:

\begin{equation} \min_{R \in SO(3),\, \mathbf t \in \mathbb R^3} = \sum_{i = 1}^N \| M_i(R \mathbf p_i + \mathbf t) \|^2, \end{equation}

where $M_i$ are given $3 \times 3$ matrices and $\mathbf p_i \in \mathbb R^3$ are given vectors.

To put it in words, I am interested in the rigid transform, given by a $3 \times 3$ rotation matrix $R$ and a translation vector $\mathbf t \in \mathbb R^3$, which minimize the above expression.

It is also fine to minimize the norms, instead of the squared norms, if it makes it any easier.