On the monotone and primal-dual active set schemes for $\ell^p$-type problems, $p \in (0,1]$
This work provides computational methods for a class of nonconvex optimization problems, but the novelty is incremental as it combines existing techniques.
The paper develops a monotonically convergent scheme and a primal-dual active set algorithm for nonsmooth nonconvex optimization problems involving the ℓ^p quasi-norm (p∈(0,1]). Numerical tests on optimal control, fracture mechanics, and microscopy image reconstruction demonstrate the algorithms' performance.
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.