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

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.