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Combining Lagrangian decomposition and excessive gap smoothing technique for solving large-scale separable convex optimization problems

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  • Quoc Tran Dinh
  • Carlo Savorgnan
  • Moritz Diehl

Abstract

A new algorithm for solving large-scale convex optimization problems with a separable objective function is proposed. The basic idea is to combine three techniques: Lagrangian dual decomposition, excessive gap and smoothing. The main advantage of this algorithm is that it automatically and simultaneously updates the smoothness parameters which significantly improves its performance. The convergence of the algorithm is proved under weak conditions imposed on the original problem. The rate of convergence is $O(\frac {1}{k})$ , where k is the iteration counter. In the second part of the paper, the proposed algorithm is coupled with a dual scheme to construct a switching variant in a dual decomposition framework. We discuss implementation issues and make a theoretical comparison. Numerical examples confirm the theoretical results. Copyright Springer Science+Business Media New York 2013

Suggested Citation

  • Quoc Tran Dinh & Carlo Savorgnan & Moritz Diehl, 2013. "Combining Lagrangian decomposition and excessive gap smoothing technique for solving large-scale separable convex optimization problems," Computational Optimization and Applications, Springer, vol. 55(1), pages 75-111, May.
    • Handle: RePEc:spr:coopap:v:55:y:2013:i:1:p:75-111
    DOI: 10.1007/s10589-012-9515-6
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    References listed on IDEAS

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    1. B. S. He & H. Yang & S. L. Wang, 2000. "Alternating Direction Method with Self-Adaptive Penalty Parameters for Monotone Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 106(2), pages 337-356, August.
    2. Robert F. Love & Svend A. Kraemer, 1973. "A Dual Decomposition Method for Minimizing Transportation Costs in Multifacility Location Problems," Transportation Science, INFORMS, vol. 7(4), pages 297-316, November.
    3. NESTEROV, Yu., 2005. "Excessive gap technique in nonsmooth convex minimization," LIDAM Reprints CORE 1818, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    4. NESTEROV, Yu., 2005. "Smooth minimization of non-smooth functions," LIDAM Reprints CORE 1819, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
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    Cited by:

    1. Jueyou Li & Guo Chen & Zhaoyang Dong & Zhiyou Wu, 2016. "A fast dual proximal-gradient method for separable convex optimization with linear coupled constraints," Computational Optimization and Applications, Springer, vol. 64(3), pages 671-697, July.
    2. Nima Rabiei & Jose Muñoz, 2015. "AAR-based decomposition algorithm for non-linear convex optimisation," Computational Optimization and Applications, Springer, vol. 62(3), pages 761-786, December.
    3. Jueyou Li & Zhiyou Wu & Changzhi Wu & Qiang Long & Xiangyu Wang, 2016. "An Inexact Dual Fast Gradient-Projection Method for Separable Convex Optimization with Linear Coupled Constraints," Journal of Optimization Theory and Applications, Springer, vol. 168(1), pages 153-171, January.
    4. Jean-Hubert Hours & Colin N. Jones, 2017. "An Alternating Trust Region Algorithm for Distributed Linearly Constrained Nonlinear Programs, Application to the Optimal Power Flow Problem," Journal of Optimization Theory and Applications, Springer, vol. 173(3), pages 844-877, June.
    5. Quoc Tran-Dinh, 2017. "Adaptive smoothing algorithms for nonsmooth composite convex minimization," Computational Optimization and Applications, Springer, vol. 66(3), pages 425-451, April.
    6. Frank E. Curtis & Arvind U. Raghunathan, 2017. "Solving nearly-separable quadratic optimization problems as nonsmooth equations," Computational Optimization and Applications, Springer, vol. 67(2), pages 317-360, June.
    7. Liusheng Hou & Hongjin He & Junfeng Yang, 2016. "A partially parallel splitting method for multiple-block separable convex programming with applications to robust PCA," Computational Optimization and Applications, Springer, vol. 63(1), pages 273-303, January.

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