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Coupling the Gradient Method with a General Exterior Penalization Scheme for Convex Minimization

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  • Juan Peypouquet

    (Universidad Técnica Federico Santa María)

Abstract

In this paper, we propose and analyze an algorithm that couples the gradient method with a general exterior penalization scheme for constrained or hierarchical minimization of convex functions in Hilbert spaces. We prove that a proper but simple choice of the step sizes and penalization parameters guarantees the convergence of the algorithm to solutions for the optimization problem. We also establish robustness and stability results that account for numerical approximation errors, discuss implementation issues and provide examples in finite and infinite dimension.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:joptap:v:153:y:2012:i:1:d:10.1007_s10957-011-9936-x
    DOI: 10.1007/s10957-011-9936-x
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    References listed on IDEAS

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    1. M. H. Xu, 2007. "Proximal Alternating Directions Method for Structured Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 134(1), pages 107-117, July.
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    Cited by:

    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. Sebastian Banert & Radu Ioan Boţ, 2015. "Backward Penalty Schemes for Monotone Inclusion Problems," Journal of Optimization Theory and Applications, Springer, vol. 166(3), pages 930-948, September.
    3. 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.

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