Economics at your fingertips  

Avoiding bad steps in Frank-Wolfe variants

Francesco Rinaldi () and Damiano Zeffiro ()
Additional contact information
Francesco Rinaldi: Università di Padova
Damiano Zeffiro: Università di Padova

Computational Optimization and Applications, 2023, vol. 84, issue 1, No 9, 225-264

Abstract: Abstract The study of Frank-Wolfe (FW) variants is often complicated by the presence of different kinds of “good” and “bad” steps. In this article, we aim to simplify the convergence analysis of specific variants by getting rid of such a distinction between steps, and to improve existing rates by ensuring a non-trivial bound at each iteration. In order to do this, we define the Short Step Chain (SSC) procedure, which skips gradient computations in consecutive short steps until proper conditions are satisfied. This algorithmic tool allows us to give a unified analysis and converge rates in the general smooth non convex setting, as well as a linear convergence rate under a Kurdyka-Łojasiewicz (KL) property. While the KL setting has been widely studied for proximal gradient type methods, to our knowledge, it has never been analyzed before for the Frank-Wolfe variants considered in the paper. An angle condition, ensuring that the directions selected by the methods have the steepest slope possible up to a constant, is used to carry out our analysis. We prove that such a condition is satisfied, when considering minimization problems over a polytope, by the away step Frank-Wolfe (AFW), the pairwise Frank-Wolfe (PFW), and the Frank-Wolfe method with in face directions (FDFW).

Keywords: Nonconvex optimization; First-order optimization; Frank-Wolfe variants; Kurdyka-Łojasiewicz property; 46N10; 65K05; 90C06; 90C25; 90C30 (search for similar items in EconPapers)
Date: 2023
References: View references in EconPapers View complete reference list from CitEc
Citations: Track citations by RSS feed

Downloads: (external link) Abstract (text/html)
Access to the full text of the articles in this series is restricted.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link:

Ordering information: This journal article can be ordered from

DOI: 10.1007/s10589-022-00434-3

Access Statistics for this article

Computational Optimization and Applications is currently edited by William W. Hager

More articles in Computational Optimization and Applications from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().

Page updated 2023-11-13
Handle: RePEc:spr:coopap:v:84:y:2023:i:1:d:10.1007_s10589-022-00434-3