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Optimization Methods for Fully Composite Problems

Author

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  • Doikov, Nikita

    (ICTEAM)

  • Nesterov, Yurii

    (Université catholique de Louvain, LIDAM/CORE, Belgium)

Abstract

In this paper, we propose a new Fully Composite Formulation of convex optimization problems. It includes, as a particular case, the problems with functional constraints, max-type minimization problems, and problems of Composite Minimization, where the objective can have simple nondifferential components. We treat all these formulations in a unified way, highlighting the existence of very natural optimization schemes of different order. We prove the global convergence rates for our methods under the most general conditions. Assuming that the upper-level component of our objective function is subhomogeneous, we develop efficient modification of the basic Fully Composite first-order and second-order Methods, and propose their accelerated variants.

Suggested Citation

  • Doikov, Nikita & Nesterov, Yurii, 2021. "Optimization Methods for Fully Composite Problems," LIDAM Discussion Papers CORE 2021001, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    • Handle: RePEc:cor:louvco:2021001
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    References listed on IDEAS

    as
    1. Doikov, Nikita & Nesterov, Yurii, 2020. "Affine-invariant contracting-point methods for Convex Optimization," LIDAM Discussion Papers CORE 2020029, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    2. Geovani N. GRAPIGLIA & Yurii NESTEROV, 2017. "Regularized Newton methods for minimizing functions with Hölder continuous Hessians," LIDAM Reprints CORE 2846, 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. Nikita Doikov & Yurii Nesterov, 2021. "Minimizing Uniformly Convex Functions by Cubic Regularization of Newton Method," Journal of Optimization Theory and Applications, Springer, vol. 189(1), pages 317-339, April.
    5. Yurii Nesterov, 2018. "Complexity bounds for primal-dual methods minimizing the model of objective function," LIDAM Reprints CORE 2992, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    6. Rodomanov, Anton & Nesterov, Yurii, 2020. "Smoothness Parameter of Power of Euclidean Norm," LIDAM Reprints CORE 3099, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    7. Yurii Nesterov, 2018. "Lectures on Convex Optimization," Springer Optimization and Its Applications, Springer, edition 2, number 978-3-319-91578-4, September.
    8. Yurii Nesterov, 2018. "The Primal-Dual Model of an Objective Function," Springer Optimization and Its Applications, in: Lectures on Convex Optimization, edition 2, chapter 0, pages 423-487, Springer.
    9. Anton Rodomanov & Yurii Nesterov, 2020. "Smoothness Parameter of Power of Euclidean Norm," Journal of Optimization Theory and Applications, Springer, vol. 185(2), pages 303-326, May.
    10. DOIKOV, Nikita, & NESTEROV Yurii,, 2020. "Convex optimization based on global lower second-order models," LIDAM Discussion Papers CORE 2020023, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
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