Optimization Online


Tensor Methods for Finding Approximate Stationary Points of Convex Functions

Geovani Grapiglia (grapiglia***at***ufpr.br)
Yurii Nesterov (yurii.nesterov***at***uclouvain.be)

Abstract: In this paper we consider the problem of finding \epsilon-approximate stationary points of convex functions that are p-times differentiable with \nu-Hölder continuous pth derivatives. We present tensor methods with and without acceleration. Specifically, we show that the non-accelerated schemes take at most O(\epsilon^{-1/(p+\nu-1)}) iterations to reduce the norm of the gradient of the objective below a given \epsilon\in (0,1). For accelerated tensor schemes we establish improved complexity bounds of O(\epsilon^{-(p+\nu)/[(p+\nu-1)(p+\nu+1)]}) and O(|\log(\epsilon)|\epsilon^{-1/(p+\nu)}), when the Hölder parameter \nu\in [0,1] is known. For the case in which \nu is unknown, we obtain a bound of O(\epsilon^{-(p+1)/[(p+\nu-1)(p+2)]}) for a universal accelerated scheme. Finally, we also obtain a lower complexity bound of O(\epsilon^{-2/[3(p+\nu)-2]}) for finding \epsilon-approximate stationary points using p-order tensor methods.

Keywords: unconstrained minimization, high-order methods, tensor methods, Hölder condition, worst-case complexity

Category 1: Convex and Nonsmooth Optimization (Convex Optimization )

Citation: Optimization Methods and Software (2020), DOI: 10.1080/10556788.2020.1818082

Download: [PDF]

Entry Submitted: 07/13/2019
Entry Accepted: 07/13/2019
Entry Last Modified: 05/22/2021

Modify/Update this entry

  Visitors Authors More about us Links
  Subscribe, Unsubscribe
Digest Archive
Search, Browse the Repository


Coordinator's Board
Classification Scheme
Give us feedback
Optimization Journals, Sites, Societies
Mathematical Optimization Society