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

TL;DR

This paper introduces a regularized semi-smooth Newton method with projection steps for solving composite convex programs, bridging first- and second-order methods, and demonstrating superlinear convergence in numerical tests.

Contribution

It develops an adaptive semi-smooth Newton method leveraging fixed-point mappings and regularization, with proven global convergence and enhanced convergence speed.

Findings

Achieves superlinear or quadratic convergence in $\, ext{l}_1$-minimization.

Establishes global convergence to optimality.

Bridges first- and second-order methods for composite convex programs.

Abstract

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.