  We propose a semismooth Newton algorithm for pathwise optimization (SNAP) for
the LASSO and Enet in sparse, high-dimensional linear regression. SNAP is
derived from a suitable formulation of the KKT conditions based on Newton
derivatives. It solves the semismooth KKT equations efficiently by actively and
continuously seeking the support of the regression coefficients along the
solution path with warm start. At each knot in the path, SNAP converges locally
superlinearly for the Enet criterion and achieves an optimal local convergence
rate for the LASSO criterion, i.e., SNAP converges in one step at the cost of
two matrix-vector multiplication per iteration. Under certain regularity
conditions on the design matrix and the minimum magnitude of the nonzero
elements of the target regression coefficients, we show that SNAP hits a
solution with the same signs as the regression coefficients and achieves a
sharp estimation error bound in finite steps with high probability. The
computational complexity of SNAP is shown to be the same as that of LARS and
coordinate descent algorithms per iteration. Simulation studies and real data
analysis support our theoretical results and demonstrate that SNAP is faster
and accurate than LARS and coordinate descent algorithms.
