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

Author

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  • Daria Ghilli

    (Karl-Franzens-Universität)

  • Karl Kunisch

    (Karl-Franzens-Universität
    Johann Radon Institute for Computational and Applied Mathematics (RICAM), Austrian Academy of Sciences)

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.

Suggested Citation

  • Daria Ghilli & Karl Kunisch, 2019. "On monotone and primal-dual active set schemes for $$\ell ^p$$ ℓ p -type problems, $$p \in (0,1]$$ p ∈ ( 0 , 1 ]," Computational Optimization and Applications, Springer, vol. 72(1), pages 45-85, January.
    • Handle: RePEc:spr:coopap:v:72:y:2019:i:1:d:10.1007_s10589-018-0036-9
    DOI: 10.1007/s10589-018-0036-9
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    References listed on IDEAS

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    1. Xiaojun Chen & Weijun Zhou, 2014. "Convergence of the reweighted ℓ 1 minimization algorithm for ℓ 2 –ℓ p minimization," Computational Optimization and Applications, Springer, vol. 59(1), pages 47-61, October.
    2. Georg Stadler, 2009. "Elliptic optimal control problems with L 1 -control cost and applications for the placement of control devices," Computational Optimization and Applications, Springer, vol. 44(2), pages 159-181, November.
    3. Dante Kalise & Karl Kunisch & Zhiping Rao, 2017. "Infinite Horizon Sparse Optimal Control," Journal of Optimization Theory and Applications, Springer, vol. 172(2), pages 481-517, February.
    4. Kristian Bredies & Dirk A. Lorenz & Stefan Reiterer, 2015. "Minimization of Non-smooth, Non-convex Functionals by Iterative Thresholding," Journal of Optimization Theory and Applications, Springer, vol. 165(1), pages 78-112, April.
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