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Fast and safe: accelerated gradient methods with optimality certificates and underestimate sequences

Author

Listed:
  • Majid Jahani

    (Lehigh University)

  • Naga Venkata C. Gudapati

    (Lehigh University
    University of Bologna)

  • Chenxin Ma

    (Lehigh University)

  • Rachael Tappenden

    (University of Canterbury)

  • Martin Takáč

    (Lehigh University)

Abstract

In this work we introduce the concept of an Underestimate Sequence (UES), which is motivated by Nesterov’s estimate sequence. Our definition of a UES utilizes three sequences, one of which is a lower bound (or under-estimator) of the objective function. The question of how to construct an appropriate sequence of lower bounds is addressed, and we present lower bounds for strongly convex smooth functions and for strongly convex composite functions, which adhere to the UES framework. Further, we propose several first order methods for minimizing strongly convex functions in both the smooth and composite cases. The algorithms, based on efficiently updating lower bounds on the objective functions, have natural stopping conditions that provide the user with a certificate of optimality. Convergence of all algorithms is guaranteed through the UES framework, and we show that all presented algorithms converge linearly, with the accelerated variants enjoying the optimal linear rate of convergence.

Suggested Citation

  • Majid Jahani & Naga Venkata C. Gudapati & Chenxin Ma & Rachael Tappenden & Martin Takáč, 2021. "Fast and safe: accelerated gradient methods with optimality certificates and underestimate sequences," Computational Optimization and Applications, Springer, vol. 79(2), pages 369-404, June.
    • Handle: RePEc:spr:coopap:v:79:y:2021:i:2:d:10.1007_s10589-021-00269-4
    DOI: 10.1007/s10589-021-00269-4
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    References listed on IDEAS

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    1. Kimon Fountoulakis & Rachael Tappenden, 2018. "A flexible coordinate descent method," Computational Optimization and Applications, Springer, vol. 70(2), pages 351-394, June.
    2. NESTEROV, Yurii, 2012. "Efficiency of coordinate descent methods on huge-scale optimization problems," LIDAM Reprints CORE 2511, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    3. NESTEROV, Yurii, 2013. "Gradient methods for minimizing composite functions," LIDAM Reprints CORE 2510, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    4. NESTEROV, Yu., 2007. "Gradient methods for minimizing composite objective function," LIDAM Discussion Papers CORE 2007076, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    5. NESTEROV, Yu., 2005. "Smooth minimization of non-smooth functions," LIDAM Reprints CORE 1819, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    6. Rachael Tappenden & Peter Richtárik & Jacek Gondzio, 2016. "Inexact Coordinate Descent: Complexity and Preconditioning," Journal of Optimization Theory and Applications, Springer, vol. 170(1), pages 144-176, July.
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