  Regularized least-squares approaches have been successfully applied to linear
system identification. Recent approaches use quadratic penalty terms on the
unknown impulse response defined by stable spline kernels, which control model
space complexity by leveraging regularity and bounded-input bounded-output
stability. This paper extends linear system identification to a wide class of
nonsmooth stable spline estimators, where regularization functionals and data
misfits can be selected from a rich set of piecewise linear-quadratic (PLQ)
penalties. This class includes the 1-norm, Huber, and Vapnik, in addition to
the least-squares penalty.
  By representing penalties through their conjugates, the modeler can specify
any piecewise linear-quadratic penalty for misfit and regularizer, as well as
inequality constraints on the response. The interior-point solver we implement
(IPsolve) is locally quadratically convergent, with $O(\min(m,n)^2(m+n))$
arithmetic operations per iteration, where $n$ the number of unknown impulse
response coefficients and $m$ the number of observed output measurements.
IPsolve is competitive with available alternatives for system identification.
This is shown by a comparison with TFOCS, libSVM, and the FISTA algorithm. The
code is open source (https://github.com/saravkin/IPsolve).
  The impact of the approach for system identification is illustrated with
numerical experiments featuring robust formulations for contaminated data,
relaxation systems, nonnegativity and unimodality constraints on the impulse
response, and sparsity promoting regularization. Incorporating constraints
yields particularly significant improvements.
