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

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.Comment: 25 pages, 4 figure

On efficiency of nonmonotone Armijo-type line searches

Monotonicity and nonmonotonicity play a key role in studying the global convergence and the efficiency of iterative schemes employed in the field of nonlinear optimization, where globally convergent and computationally efficient schemes are explored. This paper addresses some features of descent schemes and the motivation behind nonmonotone strategies and investigates the efficiency of an Armijo-type line search equipped with some popular nonmonotone terms. More specifically, we propose two novel nonmonotone terms, combine them into Armijo's rule and establish the global convergence of sequences generated by these schemes. Furthermore, we report extensive numerical results and comparisons indicating the performance of the nonmonotone Armijo-type line searches using the most popular search directions for solving unconstrained optimization problems. Finally, we exploit the considered nonmonotone schemes to solve an important inverse problem arising in signal and image processing