  We study a generalized framework for structured sparsity. It extends the
well-known methods of Lasso and Group Lasso by incorporating additional
constraints on the variables as part of a convex optimization problem. This
framework provides a straightforward way of favouring prescribed sparsity
patterns, such as orderings, contiguous regions and overlapping groups, among
others. Existing optimization methods are limited to specific constraint sets
and tend to not scale well with sample size and dimensionality. We propose a
novel first order proximal method, which builds upon results on fixed points
and successive approximations. The algorithm can be applied to a general class
of conic and norm constraints sets and relies on a proximity operator
subproblem which can be computed explicitly. Experiments on different
regression problems demonstrate the efficiency of the optimization algorithm
and its scalability with the size of the problem. They also demonstrate state
of the art statistical performance, which improves over Lasso and StructOMP.
