We discuss minimization of a smooth function to which is added a separable regularization function that induces structure in the solution. A block-coordinate relaxation approach with proximal linearized subproblems yields convergence to critical points, while identification of the optimal manifold (under a nondegeneracy condition) allows acceleration techniques to be applied on a reduced space. The work is motivated by experience with an algorithm for regularized logistic regression, and computational results for the algorithm on problems of this type are presented.
SIAM Journal on Optimization 22 (2012), pp. 159-186. PDF available here http://pages.cs.wisc.edu/~swright/papers/80856.pdf