Abstract:

We consider a class of nonsmooth convex optimization problems where
the objective function is the composition of a strongly convex differentiable function
with a linear mapping, regularized by the sum of both 1-norm and 2-norm of the
optimization variables. This class of problems arise naturally from applications in
sparse group Lasso, which is a popular technique for variable selection. An effective
approach to solve such problems is by the Proximal Gradient Method (PGM). In this
paper we prove a local error bound around the optimal solution set for this problem
and use it to establish the linear convergence of the PGM method without assuming
strong convexity of the overall objective function.

Key words: Proximal gradient method , Error bound , Linear convergence , Sparse
group Lasso