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Generalized forward–backward splitting with penalization for monotone inclusion problems

Author

Listed:
  • Nimit Nimana

    (Khon Kaen University)

  • Narin Petrot

    (Naresuan University
    Naresuan University)

Abstract

We introduce a generalized forward–backward splitting method with penalty term for solving monotone inclusion problems involving the sum of a finite number of maximally monotone operators and the normal cone to the nonempty set of zeros of another maximally monotone operator. We show weak ergodic convergence of the generated sequence of iterates to a solution of the considered monotone inclusion problem, provided that the condition corresponding to the Fitzpatrick function of the operator describing the set of the normal cone is fulfilled. Under strong monotonicity of an operator, we show strong convergence of the iterates. Furthermore, we utilize the proposed method for minimizing a large-scale hierarchical minimization problem concerning the sum of differentiable and nondifferentiable convex functions subject to the set of minima of another differentiable convex function. We illustrate the functionality of the method through numerical experiments addressing constrained elastic net and generalized Heron location problems.

Suggested Citation

  • Nimit Nimana & Narin Petrot, 2019. "Generalized forward–backward splitting with penalization for monotone inclusion problems," Journal of Global Optimization, Springer, vol. 73(4), pages 825-847, April.
    • Handle: RePEc:spr:jglopt:v:73:y:2019:i:4:d:10.1007_s10898-018-00730-5
    DOI: 10.1007/s10898-018-00730-5
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    References listed on IDEAS

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    1. Nahla Noun & Juan Peypouquet, 2013. "Forward–Backward Penalty Scheme for Constrained Convex Minimization Without Inf-Compactness," Journal of Optimization Theory and Applications, Springer, vol. 158(3), pages 787-795, September.
    2. 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.
    3. 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.
    4. Juan Peypouquet, 2012. "Coupling the Gradient Method with a General Exterior Penalization Scheme for Convex Minimization," Journal of Optimization Theory and Applications, Springer, vol. 153(1), pages 123-138, April.
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    Cited by:

    1. Pornpimon Boriwan & Thanathorn Phoka & Narin Petrot, 2022. "The Lightly Robust Max-Ordering Solution Concept for Uncertain Multiobjective Optimization Problems: An Ambulance Location Problem with Unavailability," Sustainability, MDPI, vol. 14(12), pages 1-18, June.

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