  The goal of this paper is to study approaches to bridge the gap between
first-order and second-order type methods for composite convex programs. Our
key observations are: i) Many well-known operator splitting methods, such as
forward-backward splitting (FBS) and Douglas-Rachford splitting (DRS), actually
define a fixed-point mapping; ii) The optimal solutions of the composite convex
program and the solutions of a system of nonlinear equations derived from the
fixed-point mapping are equivalent. Solving this kind of system of nonlinear
equations enables us to develop second-order type methods. Although these
nonlinear equations may be non-differentiable, they are often semi-smooth and
their generalized Jacobian matrix is positive semidefinite due to monotonicity.
By combining with a regularization approach and a known hyperplane projection
technique, we propose an adaptive semi-smooth Newton method and establish its
convergence to global optimality. Preliminary numerical results on
$\ell_1$-minimization problems demonstrate that our second-order type
algorithms are able to achieve superlinear or quadratic convergence.
