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A unified convergence framework for nonmonotone inexact decomposition methods

Author

Listed:
  • Leonardo Galli

    (University of Florence)

  • Alessandro Galligari

    (University of Florence)

  • Marco Sciandrone

    (University of Florence)

Abstract

In this work we propose a general framework that provides a unified convergence analysis for nonmonotone decomposition algorithms. The main motivation to embed nonmonotone strategies within a decomposition approach lies in the fact that enforcing the reduction of the objective function could be unnecessarily expensive, taking into account that groups of variables are individually updated. We define different search directions and line searches satisfying the conditions required by the presented nonmonotone decomposition framework to obtain global convergence. We employ a set of large-scale network equilibrium problems as a computational example to show the advantages of a nonmonotone algorithm over its monotone counterpart. In conclusion, a new smart implementation for decomposition methods has been derived to solve numerical issues on large-scale partially separable functions.

Suggested Citation

  • Leonardo Galli & Alessandro Galligari & Marco Sciandrone, 2020. "A unified convergence framework for nonmonotone inexact decomposition methods," Computational Optimization and Applications, Springer, vol. 75(1), pages 113-144, January.
    • Handle: RePEc:spr:coopap:v:75:y:2020:i:1:d:10.1007_s10589-019-00150-5
    DOI: 10.1007/s10589-019-00150-5
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    References listed on IDEAS

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    1. C. J. Lin & S. Lucidi & L. Palagi & A. Risi & M. Sciandrone, 2009. "Decomposition Algorithm Model for Singly Linearly-Constrained Problems Subject to Lower and Upper Bounds," Journal of Optimization Theory and Applications, Springer, vol. 141(1), pages 107-126, April.
    2. Paul Tseng & Sangwoon Yun, 2010. "A coordinate gradient descent method for linearly constrained smooth optimization and support vector machines training," Computational Optimization and Applications, Springer, vol. 47(2), pages 179-206, October.
    3. Luca Zanni, 2006. "An Improved Gradient Projection-based Decomposition Technique for Support Vector Machines," Computational Management Science, Springer, vol. 3(2), pages 131-145, April.
    4. Cassioli, A. & Di Lorenzo, D. & Sciandrone, M., 2013. "On the convergence of inexact block coordinate descent methods for constrained optimization," European Journal of Operational Research, Elsevier, vol. 231(2), pages 274-281.
    5. Chengbo Li & Wotao Yin & Hong Jiang & Yin Zhang, 2013. "An efficient augmented Lagrangian method with applications to total variation minimization," Computational Optimization and Applications, Springer, vol. 56(3), pages 507-530, December.
    6. David Di Lorenzo & Alessandro Galligari & Marco Sciandrone, 2015. "A convergent and efficient decomposition method for the traffic assignment problem," Computational Optimization and Applications, Springer, vol. 60(1), pages 151-170, January.
    7. Chihwa Kao & Lung-fei Lee & Mark M. Pitt, 2001. "Simulated Maximum Likelihood Estimation of the Linear Expenditure System with Binding Non-Negativity Constraints," Annals of Economics and Finance, Society for AEF, vol. 2(1), pages 215-235, May.
    8. Leonardo Galli & Christian Kanzow & Marco Sciandrone, 2018. "A nonmonotone trust-region method for generalized Nash equilibrium and related problems with strong convergence properties," Computational Optimization and Applications, Springer, vol. 69(3), pages 629-652, April.
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