Cubic Regularization Methods with Second-Order Complexity Guarantee Based on a New Subproblem Reformulation

doi:10.1007/s40305-022-00398-5

Journal of the Operations Research Society of China ›› 2022, Vol. 10 ›› Issue (3): 471-506.doi: 10.1007/s40305-022-00398-5

收稿日期:2021-06-26 修回日期:2021-12-16 出版日期:2022-09-30 发布日期:2022-09-06

Cubic Regularization Methods with Second-Order Complexity Guarantee Based on a New Subproblem Reformulation

Ru-Jun Jiang¹, Zhi-Shuo Zhou¹, Zi-Rui Zhou²

1. School of Data Science, Fudan University, Shanghai 200433, China;
2. Huawei Technologies Canada, Burnaby, BC, Canada

Received:2021-06-26 Revised:2021-12-16 Online:2022-09-30 Published:2022-09-06
Contact: Ru-Jun Jiang,Zhi-Shuo Zhou,Zi-Rui Zhou E-mail:rjjiang@fudan.edu.cn;zhouzs18@fudan.edu.cn;zirui.zhou@huawei.com
Supported by:
The first author is supported in part by the National Natural Foundation of China (Nos.11801087 and 12171100).

摘要/Abstract

Abstract: Thecubicregularization (CR) algorithmhasattractedalotofattentionsintheliterature in recent years.We propose a new reformulation of the cubic regularization subproblem.The reformulation is an unconstrained convex problem that requires computing the minimum eigenvalue of the Hessian.Then,based on this reformulation,we derive a variant of the (non-adaptive) CR provided a known Lipschitz constant for the Hessian and a variant of adaptive regularization with cubics (ARC).We show that the iteration complexity of our variants matches the best-known bounds for unconstrained minimization algorithms using first-and second-order information.Moreover,we show that the operation complexity of both of our variants also matches the state-of-the-art bounds in the literature.Numerical experiments on test problems from CUTEst collection show that the ARC based on our new subproblem reformulation is comparable to the existing algorithms.

Key words: Cubic regularization subproblem, First-order methods, Constrained convex optimization, Complexity analysis

中图分类号:

. [J]. Journal of the Operations Research Society of China, 2022, 10(3): 471-506.

Ru-Jun Jiang, Zhi-Shuo Zhou, Zi-Rui Zhou. Cubic Regularization Methods with Second-Order Complexity Guarantee Based on a New Subproblem Reformulation[J]. Journal of the Operations Research Society of China, 2022, 10(3): 471-506.

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Cubic Regularization Methods with Second-Order Complexity Guarantee Based on a New Subproblem Reformulation

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