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

Daria Ghilli () and Karl Kunisch ()
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Daria Ghilli: Karl-Franzens-Universität
Karl Kunisch: Karl-Franzens-Universität

Computational Optimization and Applications, 2019, vol. 72, issue 1, No 2, 45-85

Abstract: Abstract Nonsmooth nonconvex optimization problems involving the $$\ell ^p$$ ℓ p quasi-norm, $$p \in (0, 1]$$ 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.

Keywords: Nonsmooth nonconvex optimization; Active-set method; Monotone algorithm; Optimal control problems; Image reconstruction; Fracture mechanics; 49K99; 49M05; 65K10 (search for similar items in EconPapers)
Date: 2019
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DOI: 10.1007/s10589-018-0036-9

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