  In this paper we present an active-set method for the solution of
$\ell_1$-regularized convex quadratic optimization problems. It is derived by
combining a proximal method of multipliers (PMM) strategy with a standard
semismooth Newton method (SSN). The resulting linear systems are solved using a
Krylov-subspace method, accelerated by certain general-purpose preconditioners
which are shown to be optimal with respect to the proximal parameters.
Practical efficiency is further improved by warm-starting the algorithm using a
proximal alternating direction method of multipliers. We show that the outer
PMM achieves global convergence under mere feasibility assumptions. Under
additional standard assumptions, the PMM scheme achieves global linear and
local superlinear convergence. The SSN scheme is locally superlinearly
convergent, assuming that its associated linear systems are solved accurately
enough, and globally convergent under certain additional regularity
assumptions. We provide numerical evidence to demonstrate the effectiveness of
the approach by comparing it against OSQP and IP-PMM (an ADMM and a regularized
IPM solver, respectively) on several elastic-net linear regression and
$L^1$-regularized PDE-constrained optimization problems.
