A Semismooth Newton Method for Tikhonov Functionals with Sparsity Constraints

TL;DR

This paper introduces a semismooth Newton method for solving Tikhonov minimization problems with sparsity constraints, demonstrating local superlinear convergence and favorable numerical performance.

Contribution

It develops a semismooth Newton algorithm tailored for sparsity-constrained Tikhonov functionals, with proven convergence properties and practical efficiency.

Findings

The method converges superlinearly locally.

Numerical results show improved performance over existing methods.

The optimality condition is slantly differentiable, enabling Newton-type methods.

Abstract

Minimization problems in $\ell^2$ for Tikhonov functionals with sparsity constraints are considered. Sparsity of the solution is ensured by a weighted $\ell^1$ penalty term. The necessary and sufficient condition for optimality is shown to be slantly differentiable (Newton differentiable), hence a semismooth Newton method is applicable. Local superlinear convergence of this method is proved. Numerical examples are provided which show that our method compares favorably with existing approaches.