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Superfast second-order methods for unconstrained convex optimization

Author

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  • NESTEROV Yurii,

    (Université catholique de Louvain, CORE, Belgium)

Abstract

In this paper, we present new second-order methods with converge rate O(k^{-4}), where k is the iteration counter. This is faster that athe existing lower bound for this type of schemes [1,2], which is O(k^{-7/2}). Our progress can be explained by a finer specification of the problem class. The main idea of this approach consists in implementation of the third-order scheme from [15] using the second-order oracle. At each iteration of our method, we solve a nontrivial auxiliary problem by a linearly convergent scheme based on the relati e non-degeneracy condition [3, 10]. During this process, the Hessian of the objective function is computed once, and the gradient is computed O(ln 1/epsilon) times, where epsilon is the desired accuracy of the solution for our problem.

Suggested Citation

  • 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).
    • Handle: RePEc:cor:louvco:2020007
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    File URL: https://sites.uclouvain.be/core/publications/coredp/coredp2020.html
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    Cited by:

    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).

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