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Affine-invariant contracting-point methods for Convex Optimization

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  • Doikov, Nikita

    (Université catholique de Louvain)

  • Nesterov, Yurii

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

Abstract

In this paper, we develop new affine-invariant algorithms for solving composite con- vex minimization problems with bounded domain. We present a general framework of Contracting-Point methods, which solve at each iteration an auxiliary subproblem re- stricting the smooth part of the objective function onto contraction of the initial domain. This framework provides us with a systematic way for developing optimization methods of different order, endowed with the global complexity bounds. We show that using an ap- propriate affine-invariant smoothness condition, it is possible to implement one iteration of the Contracting-Point method by one step of the pure tensor method of degree p ≥ 1. The resulting global rate of convergence in functional residual is then O(1/kp), where k is the iteration counter. It is important that all constants in our bounds are affine-invariant. For p = 1, our scheme recovers well-known Frank-Wolfe algorithm, providing it with a new interpretation by a general perspective of tensor methods. Finally, within our frame- work, we present efficient implementation and total complexity analysis of the inexact second-order scheme (p = 2), called Contracting Newton method. It can be seen as a proper implementation of the trust-region idea. Preliminary numerical results confirm its good practical performance both in the number of iterations, and in computational time.

Suggested Citation

  • 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).
    • Handle: RePEc:cor:louvco:2020029
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    References listed on IDEAS

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    1. DOIKOV Nikita, & NESTEROV Yurii,, 2019. "Contracting proximal methods for smooth convex optimization," LIDAM Discussion Papers CORE 2019027, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    2. Haihao Lu & Robert M. Freund & Yurii Nesterov, 2018. "Relatively smooth convex optimization by first-order methods, and applications," LIDAM Reprints CORE 2965, 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, December.
    4. Yurii Nesterov, 2018. "Smooth Convex Optimization," Springer Optimization and Its Applications, in: Lectures on Convex Optimization, edition 2, chapter 0, pages 59-137, Springer.
    5. 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.
    6. NESTEROV, Yurii, 2013. "Gradient methods for minimizing composite functions," LIDAM Reprints CORE 2510, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    7. 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).
    8. 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).
    9. NESTEROV Yurii,, 2020. "Inexact accelerated high-order proximal-point methods," LIDAM Discussion Papers CORE 2020008, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    10. NESTEROV Yurii,, 2020. "Superfast second-order methods for unconstrained convex optimization," LIDAM Discussion Papers CORE 2020007, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
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    Cited by:

    1. 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).

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    Keywords

    Convex Optimization; Frank-Wolfe algorithm; Newton method; Tensor Methods; Global Complexity Bounds;
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