If by chance, the $C_k$'s are manifolds, then the following refs are relevant:

• Bauschke et al., "Optimal rates of convergence of matrices with applications" http://arxiv.org/pdf/1407.0671v1.pdf
• Liang et al., "Activity Identification and Local Linear Convergence of Douglas-Rachford/ADMM under Partial Smoothness" http://arxiv.org/abs/1412.6858

If by chance the $C_k$'s are manifolds, then the following refs are relevant:

• Bauschke et al., "Optimal rates of convergence of matrices with applications" http://arxiv.org/pdf/1407.0671v1.pdf
• Liang et al., "Activity Identification and Local Linear Convergence of Douglas-Rachford/ADMM under Partial Smoothness" http://arxiv.org/abs/1412.6858