A Semismooth Newton Method for Tikhonov Functionals with Sparsity Constraints

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.

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