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Nonsmooth nonconvex optimization problems involving the $\ell^p$ quasi-norm,$p \in (0, 1]$, of a linear map are considered. A monotonically convergentscheme for a regularized version of the original problem is developed andnecessary optimality conditions for the original problem in the form of acomplementary system amenable for computation are given. Then an algorithm forsolving the above mentioned necessary optimality conditions is proposed. It isbased on a combination of the monotone scheme and a primal-dual active setstrategy. The performance of the two algorithms is studied by means of a seriesof numerical tests in different cases, including optimal control problems,fracture mechanics and microscopy image reconstruction.

On the monotone and primal-dual active set schemes for $\ell^p$-type problems, $p \in (0,1]$