I have the following optimization problem
$$\min_{M \in SO(3), \mathbf{t} \in \mathbb{R}^3, \mathbf{v} \in \mathbb{R}^3} \sum_i^n \lVert M (\mathbf{p}_i + R_i \mathbf{v}) + \mathbf{t} - \mathbf{q}_i \rVert^2 $$
that seeks the rotation $M$ and the translation $\mathbf{t}$ that best align, in a least-squares sense, the two sets of points $\mathcal{Q}=\{\mathbf{q}_1, \mathbf{q}_2, ...,\mathbf{q}_n\}$ and $\mathcal{Z}=\{\mathbf{z}_1, \mathbf{z}_2, ...,\mathbf{z}_n\}$, with $\mathbf{z}_i = \mathbf{p}_i + R_i \mathbf{v}$, $\forall{i}$. Vectors $\mathbf{p}_i, \mathbf{q}_i \in \mathbb{R}^3$ and rotation matrices $R_i \in SO(3)$ are all known and constant.
This formulation resembles the Generalized Wahba Problem [1,2], except for the additional translation vector $\mathbf{v}$ that is added to $\mathbf{p}_i$, after being rotated by $R_i$. As it stands, I am able to find an optimal solution via an alternating optimization algorithm: at each iteration $k$, $\mathbf{v}_k^*$ is estimated while $\mathbf{t}$ and $M$ are kept constant at $\mathbf{t}_{k-1}^*$ and $M_{k-1}^*$ respectively; then $\mathbf{t}_k^*$ and $M_k^*$ are estimated while $\mathbf{v} = \mathbf{v}_k^*$.
However, I wanted to formulate the dual problem to further analyse the optimality of the primal one. In this context, [3,4] were both good reads but I did not manage to come up with any formulation of the dual problem, since the additional translation $\mathbf{v}$ really messed up my approaches.
Any help or hint on how to find it is greatly appreciated! Thank you very much!
[1] Malcolm D. Shuster, The Generalized
Wahba Problem, The Journal of the Astronautical Sciences, Vol. 54, No. 2, April–June 2006, pp. 245–259.
[2] Olga Sorkine-Hornung and Michael Rabinovich, Least-Squares Rigid Motion Using SVD, 2016, Technical Note.
[3] Henry Wolkowicz, A note on lack of strong duality for quadratic problems
with orthogonal constraints, European Journal of Operational Research, Vol. 143, No. 2, 2002, pp. 356–364.
[4] Jesus Briales and Javier Gonzalez-Jimenez, Convex Global 3D Registration with Lagrangian Duality, Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR), 2017, pp. 4960-4969 .