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The 2-Coordinate Descent Method for Solving Double-Sided Simplex Constrained Minimization Problems

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  • Amir Beck

    (Technion—Israel Institute of Technology)

Abstract

This paper considers the problem of minimizing a continuously differentiable function with a Lipschitz continuous gradient subject to a single linear equality constraint and additional bound constraints on the decision variables. We introduce and analyze several variants of a 2-coordinate descent method: a block descent method that performs an optimization step with respect to only two variables at each iteration. Based on two new optimality measures, we establish convergence to stationarity points for general nonconvex objective functions. In the convex case, when all the variables are lower bounded but not upper bounded, we show that the sequence of function values converges at a sublinear rate. Several illustrative numerical examples demonstrate the effectiveness of the method.

Suggested Citation

  • Amir Beck, 2014. "The 2-Coordinate Descent Method for Solving Double-Sided Simplex Constrained Minimization Problems," Journal of Optimization Theory and Applications, Springer, vol. 162(3), pages 892-919, September.
    • Handle: RePEc:spr:joptap:v:162:y:2014:i:3:d:10.1007_s10957-013-0491-5
    DOI: 10.1007/s10957-013-0491-5
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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. P. Tseng & S. Yun, 2009. "Block-Coordinate Gradient Descent Method for Linearly Constrained Nonsmooth Separable Optimization," Journal of Optimization Theory and Applications, Springer, vol. 140(3), pages 513-535, March.
    4. NESTEROV, Yurii, 2010. "Efficiency of coordinate descent methods on huge-scale optimization problems," LIDAM Discussion Papers CORE 2010002, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    5. 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.
    6. Giampaolo Liuzzi & Laura Palagi & Mauro Piacentini, 2010. "On the convergence of a Jacobi-type algorithm for Singly Linearly-Constrained Problems Subject to simple Bounds," DIS Technical Reports 2010-01, Department of Computer, Control and Management Engineering, Universita' degli Studi di Roma "La Sapienza".
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    Cited by:

    1. Cristofari, Andrea, 2023. "A decomposition method for lasso problems with zero-sum constraint," European Journal of Operational Research, Elsevier, vol. 306(1), pages 358-369.
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    3. Ion Necoara & Yurii Nesterov & François Glineur, 2017. "Random Block Coordinate Descent Methods for Linearly Constrained Optimization over Networks," Journal of Optimization Theory and Applications, Springer, vol. 173(1), pages 227-254, April.
    4. I. V. Konnov, 2016. "Selective bi-coordinate variations for resource allocation type problems," Computational Optimization and Applications, Springer, vol. 64(3), pages 821-842, July.

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