A monotone scheme for sparsity optimization in lp with p Є (0,1]

Ghilli, Daria; Kunisch, Karl

doi:10.1016/j.ifacol.2017.08.102

Cited by 3 publications

(9 citation statements)

References 18 publications

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“…Many numerical techniques have been developed when Λ = I (e.g. in [20,27,29,30]) and attention has recently been given to the case of more general operators, here we mention e.g. [24,36,42] and we refer to the end of the introduction for further details.…”

Section: Introductionmentioning

confidence: 99%

“…Similar type of algorithms were proposed in [27] and [20] for problems as (1.1) in case of no matrix inside the p -term and in the infinite dimensional sequence spaces p , with p ∈ [0, 1]. In particular in [27] a primal-dual active set strategy has been studied for the case p = 0, its convergence was proved under a diagonal dominance condition and its performance was tested in three different situations.…”

Section: Introductionmentioning

confidence: 99%

“…In the case p ∈ (0, 1] a monotone convergent scheme was proposed but no numerical tests were made. Then inspired by [27], the two authors of the present paper developed in [20] a primal-dual active set strategy for p ∈ (0, 1] and tested its performance in diverse test cases. Note that in [20] the convergence of the primal-dual strategy was not investigated.…”

Section: Introductionmentioning

confidence: 99%

“…Then inspired by [27], the two authors of the present paper developed in [20] a primal-dual active set strategy for p ∈ (0, 1] and tested its performance in diverse test cases. Note that in [20] the convergence of the primal-dual strategy was not investigated. The monotone and primal-dual active set monotone algorithm studied in the present paper are inspired by the schemes proposed respectively in [27] and [20], but with the main novelties that now we treat the case of a regular matrix in the p -term and we provide diverse numerical tests for both the schemes.…”

Section: Introductionmentioning

confidence: 99%

“…Note that in [20] the convergence of the primal-dual strategy was not investigated. The monotone and primal-dual active set monotone algorithm studied in the present paper are inspired by the schemes proposed respectively in [27] and [20], but with the main novelties that now we treat the case of a regular matrix in the p -term and we provide diverse numerical tests for both the schemes. Moreover, we prove the convergence of the primal-dual active set strategy.…”

Section: Introductionmentioning

confidence: 99%

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

Ghilli

Kunisch

2018

Comput Optim Appl

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Nonsmooth nonconvex optimization problems involving the 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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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

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

Ghilli

Kunisch

2018

Comput Optim Appl

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On a Monotone Scheme for Nonconvex Nonsmooth Optimization with Applications to Fracture Mechanics

Ghilli

Kunisch

2019

J Optim Theory Appl

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A general class of nonconvex optimization problems is considered, where the penalty is the composition of a linear operator with a nonsmooth nonconvex mapping, which is concave on the positive real line. The necessary optimality condition of a regularized version of the original problem is solved by means of a monotonically convergent scheme. Such problems arise in continuum mechanics, as for instance cohesive fractures, where singular behaviour is usually modelled by nonsmooth nonconvex energies. The proposed algorithm is successfully tested for fracture mechanics problems. Its performance is also compared to two alternative algorithms for nonsmooth nonconvex optimization arising in optimal control and mathematical imaging.Nonsmooth nonconvex optimization and monotone algorithm and fracture mechanics and sparse recovery

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

Ghilli¹,

Kunisch²

2017

Preprint

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Nonsmooth nonconvex optimization problems involving the 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 developped and necessary optimality conditions for the original prolem 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 and active-set method and monotone algorithm and optimal control problems and image reconstruction and fracture mechanics.

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A monotone scheme for sparsity optimization in lp with p Є (0,1]

Cited by 3 publications

References 18 publications

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

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

On a Monotone Scheme for Nonconvex Nonsmooth Optimization with Applications to Fracture Mechanics

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

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