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Adaptive Restart of the Optimized Gradient Method for Convex Optimization

Author

Listed:
  • Donghwan Kim

    (University of Michigan)

  • Jeffrey A. Fessler

    (University of Michigan)

Abstract

First-order methods with momentum, such as Nesterov’s fast gradient method, are very useful for convex optimization problems, but can exhibit undesirable oscillations yielding slow convergence rates for some applications. An adaptive restarting scheme can improve the convergence rate of the fast gradient method, when the parameter of a strongly convex cost function is unknown or when the iterates of the algorithm enter a locally strongly convex region. Recently, we introduced the optimized gradient method, a first-order algorithm that has an inexpensive per-iteration computational cost similar to that of the fast gradient method, yet has a worst-case cost function rate that is twice faster than that of the fast gradient method and that is optimal for large-dimensional smooth convex problems. Building upon the success of accelerating the fast gradient method using adaptive restart, this paper investigates similar heuristic acceleration of the optimized gradient method. We first derive a new first-order method that resembles the optimized gradient method for strongly convex quadratic problems with known function parameters, yielding a linear convergence rate that is faster than that of the analogous version of the fast gradient method. We then provide a heuristic analysis and numerical experiments that illustrate that adaptive restart can accelerate the convergence of the optimized gradient method. Numerical results also illustrate that adaptive restart is helpful for a proximal version of the optimized gradient method for nonsmooth composite convex functions.

Suggested Citation

  • Donghwan Kim & Jeffrey A. Fessler, 2018. "Adaptive Restart of the Optimized Gradient Method for Convex Optimization," Journal of Optimization Theory and Applications, Springer, vol. 178(1), pages 240-263, July.
  • Handle: RePEc:spr:joptap:v:178:y:2018:i:1:d:10.1007_s10957-018-1287-4
    DOI: 10.1007/s10957-018-1287-4
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    References listed on IDEAS

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    1. Patrick L. Combettes & Jean-Christophe Pesquet, 2011. "Proximal Splitting Methods in Signal Processing," Springer Optimization and Its Applications, in: Heinz H. Bauschke & Regina S. Burachik & Patrick L. Combettes & Veit Elser & D. Russell Luke & Henry (ed.), Fixed-Point Algorithms for Inverse Problems in Science and Engineering, chapter 0, pages 185-212, Springer.
    2. Renato D. C. Monteiro & Camilo Ortiz & Benar F. Svaiter, 2016. "An adaptive accelerated first-order method for convex optimization," Computational Optimization and Applications, Springer, vol. 64(1), pages 31-73, May.
    3. Donghwan Kim & Jeffrey A. Fessler, 2017. "On the Convergence Analysis of the Optimized Gradient Method," Journal of Optimization Theory and Applications, Springer, vol. 172(1), pages 187-205, January.
    4. NESTEROV, Yurii, 2013. "Gradient methods for minimizing composite functions," LIDAM Reprints CORE 2510, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    5. A. Chambolle & Ch. Dossal, 2015. "On the Convergence of the Iterates of the “Fast Iterative Shrinkage/Thresholding Algorithm”," Journal of Optimization Theory and Applications, Springer, vol. 166(3), pages 968-982, September.
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    Cited by:

    1. A. Scagliotti & P. Colli Franzone, 2022. "A piecewise conservative method for unconstrained convex optimization," Computational Optimization and Applications, Springer, vol. 81(1), pages 251-288, January.

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