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A parameterized Douglas–Rachford splitting algorithm for nonconvex optimization

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  • Bian, Fengmiao
  • Zhang, Xiaoqun

Abstract

In this paper, we study a parameterized Douglas–Rachford splitting method in Wang-Wang (2019)[5] for a class of nonconvex optimization problem. A new merit function is constructed to establish the convergence of the whole sequence generated by the parameterized Douglas–Rachford splitting method. As a by-product, this also provides convergence results of a special case of the adaptive Douglas–Rachford algorithm proposed by Dao and Phan (2019)[22] in nonconvex settings. We then apply the parameterized Douglas–Rachford splitting method to three important classes of nonconvex optimization problems arising in data science: sparsity constrained least squares problem, feasibility problem and low rank matrix completion. Numerical results validate the effectiveness of the parameterized Douglas–Rachford splitting method compared with some other classical methods.

Suggested Citation

  • Bian, Fengmiao & Zhang, Xiaoqun, 2021. "A parameterized Douglas–Rachford splitting algorithm for nonconvex optimization," Applied Mathematics and Computation, Elsevier, vol. 410(C).
    • Handle: RePEc:eee:apmaco:v:410:y:2021:i:c:s0096300321005142
    DOI: 10.1016/j.amc.2021.126425
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    References listed on IDEAS

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    1. Hédy Attouch & Jérôme Bolte & Patrick Redont & Antoine Soubeyran, 2010. "Proximal Alternating Minimization and Projection Methods for Nonconvex Problems: An Approach Based on the Kurdyka-Łojasiewicz Inequality," Mathematics of Operations Research, INFORMS, vol. 35(2), pages 438-457, May.
    2. Francisco J. Aragón Artacho & Rubén Campoy, 2018. "A new projection method for finding the closest point in the intersection of convex sets," Computational Optimization and Applications, Springer, vol. 69(1), pages 99-132, January.
    3. Dongying Wang & Xianfu Wang, 2019. "A parameterized Douglas–Rachford algorithm," Computational Optimization and Applications, Springer, vol. 73(3), pages 839-869, July.
    4. Guoyin Li & Tianxiang Liu & Ting Kei Pong, 2017. "Peaceman–Rachford splitting for a class of nonconvex optimization problems," Computational Optimization and Applications, Springer, vol. 68(2), pages 407-436, November.
    5. Francisco J. Aragón Artacho & Jonathan M. Borwein & Matthew K. Tam, 2014. "Recent Results on Douglas–Rachford Methods for Combinatorial Optimization Problems," Journal of Optimization Theory and Applications, Springer, vol. 163(1), pages 1-30, October.
    6. Hui Zou & Trevor Hastie, 2005. "Addendum: Regularization and variable selection via the elastic net," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 67(5), pages 768-768, November.
    7. Hui Zou & Trevor Hastie, 2005. "Regularization and variable selection via the elastic net," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 67(2), pages 301-320, April.
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