On the monotone and primal-dual active set schemes for ℓ^p-type problems, p ∈ (0,1]

2017/09/19 by Ghilli, Daria, Kunisch, Karl
#49K99 #49M05 #65K10 #Analysis of PDEs (math.AP) #FOS: Mathematics #Numerical Analysis (math.NA) #Optimization and Control (math.OC)

paper · doi:10.48550/arxiv.1709.06506

Abstract

Nonsmooth nonconvex optimization problems involving the ℓ^p quasi-norm, p ∈ (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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