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

Abstract

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

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