Nesterov, Yurii ORCID: https://orcid.org/0000-0002-0542-8757
(2025)
Quartic Regularity.
Vietnam Journal of Mathematics
.
DOI 10.1007/s10013-024-00720-z
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Official URL: https://doi.org/10.1007/s10013-024-00720-z
Abstract
In this paper, we propose new linearly convergent second-order methods for minimizing convex quartic polynomials. This framework is applied for designing optimization schemes, which can solve general convex problems satisfying a new condition of quartic regularity. It assumes positive definiteness and boundedness of the fourth derivative of the objective function. For such problems, an appropriate quartic regularization of Damped Newton Method has global linear rate of convergence. We discuss several important consequences of this result. In particular, it can be used for constructing new second-order methods in the framework of high-order proximal-point schemes (Nesterov, Math. Program. 197, 1-26, 2023 and Nesterov, SIAM J. Optim. 31, 2807-2828, 2021). These methods have convergence rate (O) over tilde (k(-p)), where k is the iteration counter, p is equal to 3, 4, or 5, and tilde indicates the presence of logarithmic factors in the complexity bounds for the auxiliary problems, which are solved at each iteration of the schemes.
Item Type: | Article |
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Uncontrolled Keywords: | Composite convex minimization ; Second-order methods ; Global complexity bounds ; High-order proximal-point methods |
Divisions: | Corvinus Institute for Advanced Studies (CIAS) |
Subjects: | Mathematics, Econometrics |
Funders: | European Union’s Horizon 2020, Corvinus University of Budapest |
Projects: | 788368, Open Access funding |
DOI: | 10.1007/s10013-024-00720-z |
ID Code: | 11039 |
Deposited By: | MTMT SWORD |
Deposited On: | 27 Mar 2025 14:25 |
Last Modified: | 27 Mar 2025 14:25 |
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