  In this paper we present an efficient active-set method for the solution of
convex quadratic programming problems with general piecewise-linear terms in
the objective, with applications to sparse approximations and
risk-minimization. The algorithm is derived by combining a proximal method of
multipliers (PMM) with a standard semismooth Newton method (SSN), and is shown
to be globally convergent under minimal assumptions. Further local linear (and
potentially superlinear) convergence is shown under standard additional
conditions. The major computational bottleneck of the proposed approach arises
from the solution of the associated SSN linear systems. These are solved using
a Krylov-subspace method, accelerated by certain novel general-purpose
preconditioners which are shown to be optimal with respect to the proximal
penalty parameters. The preconditioners are easy to store and invert, since
they exploit the structure of the nonsmooth terms appearing in the problem's
objective to significantly reduce their memory requirements. We showcase the
efficiency, robustness, and scalability of the proposed solver on a variety of
problems arising in risk-averse portfolio selection, $L^1$-regularized partial
differential equation constrained optimization, quantile regression, and binary
classification via linear support vector machines. We provide computational
evidence, on real-world datasets, to demonstrate the ability of the solver to
efficiently and competitively handle a diverse set of medium- and large-scale
optimization instances.
