四次规律性。

IF 0.7 Q2 MATHEMATICS
Vietnam Journal of Mathematics Pub Date : 2025-01-01 Epub Date: 2025-03-12 DOI:10.1007/s10013-024-00720-z
Yurii Nesterov
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引用次数: 0

摘要

在本文中,我们提出了一种新的二阶线性收敛最小化凸四次多项式的方法。将该框架用于求解满足四次正则性新条件的一般凸问题的优化方案设计。它假定目标函数的四阶导数具有正确定性和有界性。对于这类问题,适当的四次正则化阻尼牛顿法具有全局线性收敛速度。我们讨论这个结果的几个重要后果。特别是,它可以用于在高阶近点格式的框架中构造新的二阶方法(Nesterov, Math。[j] .中国计算机工程学报。1997,1 - 26,2023;Nesterov, SIAM . optitij . 31, 2807- 2828,2021]。这些方法的收敛速率为O ~ (k - p),其中k为迭代计数器,p等于3,4或5,波浪表示辅助问题的复杂性界中存在对数因子,这些辅助问题在方案的每次迭代中得到解决。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Quartic Regularity.

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 ~ ( 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.

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来源期刊
CiteScore
1.90
自引率
0.00%
发文量
52
期刊介绍: Vietnam Journal of Mathematics was originally founded in 1973 by the Vietnam Academy of Science and Technology and the Vietnam Mathematical Society. Published by Springer from 1997 to 2005 and since 2013, this quarterly journal is open to contributions from researchers from all over the world, where all submitted articles are peer-reviewed by experts worldwide. It aims to publish high-quality original research papers and review articles in all active areas of pure and applied mathematics.
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