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A forward-backward dynamical approach for nonsmooth problems with block structure coupled by a smooth function

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  • Boţ, Radu Ioan
  • Kanzler, Laura

Abstract

In this paper we aim to minimize the sum of two nonsmooth (possibly also nonconvex) functions in separate variables connected by a smooth coupling function. To tackle this problem we choose a continuous forward-backward approach and introduce a dynamical system which is formulated by means of the partial gradients of the smooth coupling function and the proximal point operator of the two nonsmooth functions. Moreover, we consider variable rates of implicity of the resulting system. We discuss the existence and uniqueness of a solution and carry out the asymptotic analysis of its convergence behaviour to a critical point of the optimization problem, when a regularization of the objective function fulfills the Kurdyka-Łojasiewicz property. We further provide convergence rates for the solution trajectory in terms of the Łojasiewicz exponent. We conclude this work with numerical simulations which confirm and validate the analytical results.

Suggested Citation

  • Boţ, Radu Ioan & Kanzler, Laura, 2021. "A forward-backward dynamical approach for nonsmooth problems with block structure coupled by a smooth function," Applied Mathematics and Computation, Elsevier, vol. 394(C).
    • Handle: RePEc:eee:apmaco:v:394:y:2021:i:c:s009630032030775x
    DOI: 10.1016/j.amc.2020.125822
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    References listed on IDEAS

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    1. Radu Ioan Boţ & Ernö Robert Csetnek & Szilárd Csaba László, 2016. "An inertial forward–backward algorithm for the minimization of the sum of two nonconvex functions," EURO Journal on Computational Optimization, Springer;EURO - The Association of European Operational Research Societies, vol. 4(1), pages 3-25, February.
    2. B. Abbas & H. Attouch & Benar F. Svaiter, 2014. "Newton-Like Dynamics and Forward-Backward Methods for Structured Monotone Inclusions in Hilbert Spaces," Journal of Optimization Theory and Applications, Springer, vol. 161(2), pages 331-360, May.
    3. J. Bolte, 2003. "Continuous Gradient Projection Method in Hilbert Spaces," Journal of Optimization Theory and Applications, Springer, vol. 119(2), pages 235-259, November.
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