A globally convergent and locally quadratically convergent modified B-semismooth Newton method for $\ell_1$-penalized minimization

TL;DR

This paper introduces a modified semismooth Newton method for -penalized minimization problems that guarantees global convergence without extra assumptions and achieves local quadratic convergence, demonstrated through image deblurring and regression examples.

Contribution

The paper generalizes a previous Newton method to nonlinear problems, proving global convergence without extra conditions and establishing local quadratic convergence under a technical assumption.

Findings

Method achieves global convergence without additional assumptions.

Full Newton steps are accepted eventually, leading to quadratic convergence.

Numerical tests show effective performance in image deblurring and regression.

Abstract

We consider the efficient minimization of a nonlinear, strictly convex functional with $\ell_1$-penalty term. Such minimization problems appear in a wide range of applications like Tikhonov regularization of (non)linear inverse problems with sparsity constraints. In (2015 Inverse Problems (31) 025005), a globalized Bouligand-semismooth Newton method was presented for $\ell_1$-Tikhonov regularization of linear inverse problems. Nevertheless, a technical assumption on the accumulation point of the sequence of iterates was necessary to prove global convergence. Here, we generalize this method to general nonlinear problems and present a modified semismooth Newton method for which global convergence is proven without any additional requirements. Moreover, under a technical assumption, full Newton steps are eventually accepted and locally quadratic convergence is achieved. Numerical examples from image deblurring and robust regression demonstrate the performance of the method.