  We consider the proximal-gradient method for minimizing an objective function
that is the sum of a smooth function and a non-smooth convex function. A
feature that distinguishes our work from most in the literature is that we
assume that the associated proximal operator does not admit a closed-form
solution. To address this challenge, we study two adaptive and implementable
termination conditions that dictate how accurately the proximal-gradient
subproblem is solved. We prove that the number of iterations required for the
inexact proximal-gradient method to reach a $\tau > 0$ approximate first-order
stationary point is $\mathcal{O}(\tau^{-2})$, which matches the similar result
that holds when exact subproblem solutions are computed. Also, by focusing on
the overlapping group $\ell_1$ regularizer, we propose an algorithm for
approximately solving the proximal-gradient subproblem, and then prove that its
iterates identify (asymptotically) the support of an optimal solution. If one
imposes additional control over the accuracy to which each subproblem is
solved, we give an upper bound on the maximum number of iterations before the
support of an optimal solution is obtained.
