  We present an active-set method for minimizing an objective that is the sum
of a convex quadratic and $\ell_1$ regularization term. Unlike two-phase
methods that combine a first-order active set identification step and a
subspace phase consisting of a \emph{cycle} of conjugate gradient (CG)
iterations, the method presented here has the flexibility of computing a
first-order proximal gradient step or a subspace CG step at each iteration. The
decision of which type of step to perform is based on the relative magnitudes
of some scaled components of the minimum norm subgradient of the objective
function. The paper establishes global rates of convergence, as well as work
complexity estimates for two variants of our approach, which we call the iiCG
method. Numerical results illustrating the behavior of the method on a variety
of test problems are presented.
