A semismooth Newton method for Tikhonov functionals with sparsity constraints

Minimization problems in ℓ 2 for Tikhonov functionals with sparsity constraints are considered. Sparsity of the solution is ensured by a weighted ℓ 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.