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

A Regularized Semi-Smooth Newton Method With Projection Steps for Composite Convex Programs