IDEAS home Printed from https://ideas.repec.org/a/spr/joptap/v153y2012i2d10.1007_s10957-011-9967-3.html
   My bibliography  Save this article

Dual Convergence for Penalty Algorithms in Convex Programming

Author

Listed:
  • Felipe Alvarez

    (Universidad de Chile)

  • Miguel Carrasco

    (Universidad de Los Andes)

  • Thierry Champion

    (Université du Sud Toulon-Var)

Abstract

Algorithms for convex programming, based on penalty methods, can be designed to have good primal convergence properties even without uniqueness of optimal solutions. Taking primal convergence for granted, in this paper we investigate the asymptotic behavior of an appropriate dual sequence obtained directly from primal iterates. First, under mild hypotheses, which include the standard Slater condition but neither strict complementarity nor second-order conditions, we show that this dual sequence is bounded and also, each cluster point belongs to the set of Karush–Kuhn–Tucker multipliers. Then we identify a general condition on the behavior of the generated primal objective values that ensures the full convergence of the dual sequence to a specific multiplier. This dual limit depends only on the particular penalty scheme used by the algorithm. Finally, we apply this approach to prove the first general dual convergence result of this kind for penalty-proximal algorithms in a nonlinear setting.

Suggested Citation

  • Felipe Alvarez & Miguel Carrasco & Thierry Champion, 2012. "Dual Convergence for Penalty Algorithms in Convex Programming," Journal of Optimization Theory and Applications, Springer, vol. 153(2), pages 388-407, May.
    • Handle: RePEc:spr:joptap:v:153:y:2012:i:2:d:10.1007_s10957-011-9967-3
    DOI: 10.1007/s10957-011-9967-3
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10957-011-9967-3
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s10957-011-9967-3?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. A. Kaplan & R. Tichatschke, 1998. "Proximal Methods in View of Interior-Point Strategies," Journal of Optimization Theory and Applications, Springer, vol. 98(2), pages 399-429, August.
    2. A. Auslender & R. Cominetti & M. Haddou, 1997. "Asymptotic Analysis for Penalty and Barrier Methods in Convex and Linear Programming," Mathematics of Operations Research, INFORMS, vol. 22(1), pages 43-62, February.
    3. R. Cominetti, 1997. "Coupling the Proximal Point Algorithm with Approximation Methods," Journal of Optimization Theory and Applications, Springer, vol. 95(3), pages 581-600, December.
    4. Felipe Alvarez & Miguel Carrasco & Karine Pichard, 2005. "Convergence of a Hybrid Projection-Proximal Point Algorithm Coupled with Approximation Methods in Convex Optimization," Mathematics of Operations Research, INFORMS, vol. 30(4), pages 966-984, November.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Felipe Alvarez & Miguel Carrasco & Karine Pichard, 2005. "Convergence of a Hybrid Projection-Proximal Point Algorithm Coupled with Approximation Methods in Convex Optimization," Mathematics of Operations Research, INFORMS, vol. 30(4), pages 966-984, November.
    2. Hedy Attouch & Alexandre Cabot & Zaki Chbani & Hassan Riahi, 2018. "Inertial Forward–Backward Algorithms with Perturbations: Application to Tikhonov Regularization," Journal of Optimization Theory and Applications, Springer, vol. 179(1), pages 1-36, October.
    3. Vu, Duc Thach Son & Ben Gharbia, Ibtihel & Haddou, Mounir & Tran, Quang Huy, 2021. "A new approach for solving nonlinear algebraic systems with complementarity conditions. Application to compositional multiphase equilibrium problems," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 190(C), pages 1243-1274.
    4. Julian Rasch & Antonin Chambolle, 2020. "Inexact first-order primal–dual algorithms," Computational Optimization and Applications, Springer, vol. 76(2), pages 381-430, June.
    5. Alfred Auslender & Miguel A. Goberna & Marco A. López, 2009. "Penalty and Smoothing Methods for Convex Semi-Infinite Programming," Mathematics of Operations Research, INFORMS, vol. 34(2), pages 303-319, May.
    6. Héctor Ramírez & David Sossa, 2017. "On the Central Paths in Symmetric Cone Programming," Journal of Optimization Theory and Applications, Springer, vol. 172(2), pages 649-668, February.
    7. Cominetti, Roberto & Dose, Valerio & Scarsini, Marco, 2024. "Monotonicity of equilibria in nonatomic congestion games," European Journal of Operational Research, Elsevier, vol. 316(2), pages 754-766.
    8. Angelia Nedić & Asuman Ozdaglar, 2008. "Separation of Nonconvex Sets with General Augmenting Functions," Mathematics of Operations Research, INFORMS, vol. 33(3), pages 587-605, August.
    9. Liwei Zhang & Jian Gu & Xiantao Xiao, 2011. "A class of nonlinear Lagrangians for nonconvex second order cone programming," Computational Optimization and Applications, Springer, vol. 49(1), pages 61-99, May.
    10. Ana Maria A. C. Rocha & M. Fernanda P. Costa & Edite M. G. P. Fernandes, 2017. "On a smoothed penalty-based algorithm for global optimization," Journal of Global Optimization, Springer, vol. 69(3), pages 561-585, November.
    11. Marcos Singer & Patricio Donoso & José Noguer, 2005. "Optimal Planning of a Multi-Station System with Sojourn Time Constraints," Annals of Operations Research, Springer, vol. 138(1), pages 203-222, September.
    12. X. Huang & J. Yao, 2013. "Characterizations of the nonemptiness and compactness for solution sets of convex set-valued optimization problems," Journal of Global Optimization, Springer, vol. 55(3), pages 611-626, March.
    13. Mounir Haddou & Patrick Maheux, 2014. "Smoothing Methods for Nonlinear Complementarity Problems," Journal of Optimization Theory and Applications, Springer, vol. 160(3), pages 711-729, March.
    14. Fu-Quan Xia & Nan-Jing Huang, 2011. "A Projection-Proximal Point Algorithm for Solving Generalized Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 150(1), pages 98-117, July.
    15. Alfred Auslender & Héctor C., 2006. "Penalty and Barrier Methods for Convex Semidefinite Programming," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 63(2), pages 195-219, May.
    16. Hedy Attouch & Szilárd Csaba László, 2024. "Convex optimization via inertial algorithms with vanishing Tikhonov regularization: fast convergence to the minimum norm solution," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 99(3), pages 307-347, June.
    17. X. X. Huang & X. Q. Yang, 2003. "A Unified Augmented Lagrangian Approach to Duality and Exact Penalization," Mathematics of Operations Research, INFORMS, vol. 28(3), pages 533-552, August.
    18. X. X. Huang & X. Q. Yang & K. L. Teo, 2004. "Characterizing Nonemptiness and Compactness of the Solution Set of a Convex Vector Optimization Problem with Cone Constraints and Applications," Journal of Optimization Theory and Applications, Springer, vol. 123(2), pages 391-407, November.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:joptap:v:153:y:2012:i:2:d:10.1007_s10957-011-9967-3. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.