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

Author

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  • Daria Ghilli

    (Karl-Franzens University)

  • Karl Kunisch

    (Karl-Franzens University
    Johann Radon Institute)

Abstract

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.

Suggested Citation

  • Daria Ghilli & Karl Kunisch, 2019. "On a Monotone Scheme for Nonconvex Nonsmooth Optimization with Applications to Fracture Mechanics," Journal of Optimization Theory and Applications, Springer, vol. 183(2), pages 609-641, November.
    • Handle: RePEc:spr:joptap:v:183:y:2019:i:2:d:10.1007_s10957-019-01545-4
    DOI: 10.1007/s10957-019-01545-4
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    References listed on IDEAS

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    1. Fan J. & Li R., 2001. "Variable Selection via Nonconcave Penalized Likelihood and its Oracle Properties," Journal of the American Statistical Association, American Statistical Association, vol. 96, pages 1348-1360, December.
    2. 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.
    3. 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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