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A piecewise conservative method for unconstrained convex optimization

Author

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  • A. Scagliotti

    (Scuola Internazionale Superiore di Studi Avanzati)

  • P. Colli Franzone

    (Università di Pavia)

Abstract

We consider a continuous-time optimization method based on a dynamical system, where a massive particle starting at rest moves in the conservative force field generated by the objective function, without any kind of friction. We formulate a restart criterion based on the mean dissipation of the kinetic energy, and we prove a global convergence result for strongly-convex functions. Using the Symplectic Euler discretization scheme, we obtain an iterative optimization algorithm. We have considered a discrete mean dissipation restart scheme, but we have also introduced a new restart procedure based on ensuring at each iteration a decrease of the objective function greater than the one achieved by a step of the classical gradient method. For the discrete conservative algorithm, this last restart criterion is capable of guaranteeing a qualitative convergence result. We apply the same restart scheme to the Nesterov Accelerated Gradient (NAG-C), and we use this restarted NAG-C as benchmark in the numerical experiments. In the smooth convex problems considered, our method shows a faster convergence rate than the restarted NAG-C. We propose an extension of our discrete conservative algorithm to composite optimization: in the numerical tests involving non-strongly convex functions with $$\ell ^1$$ ℓ 1 -regularization, it has better performances than the well known efficient Fast Iterative Shrinkage-Thresholding Algorithm, accelerated with an adaptive restart scheme.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:coopap:v:81:y:2022:i:1:d:10.1007_s10589-021-00332-0
    DOI: 10.1007/s10589-021-00332-0
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    References listed on IDEAS

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    1. 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.
    2. NESTEROV, Yurii, 2013. "Gradient methods for minimizing composite functions," LIDAM Reprints CORE 2510, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    3. Yurii Nesterov, 2018. "Lectures on Convex Optimization," Springer Optimization and Its Applications, Springer, edition 2, number 978-3-319-91578-4, September.
    4. Olivier Fercoq & Zheng Qu, 2020. "Restarting the accelerated coordinate descent method with a rough strong convexity estimate," Computational Optimization and Applications, Springer, vol. 75(1), pages 63-91, January.
    5. 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.
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