Inexact first-order primal–dual algorithms

Julian Rasch () and Antonin Chambolle ()
Additional contact information
Julian Rasch: Westfälische Wilhelms-Universität (WWU) Münster
Antonin Chambolle: CMAP

Computational Optimization and Applications, 2020, vol. 76, issue 2, No 4, 430 pages

Abstract: Abstract We investigate the convergence of a recently popular class of first-order primal–dual algorithms for saddle point problems under the presence of errors in the proximal maps and gradients. We study several types of errors and show that, provided a sufficient decay of these errors, the same convergence rates as for the error-free algorithm can be established. More precisely, we prove the (optimal) $$O\left( 1/N\right)$$O1/N convergence to a saddle point in finite dimensions for the class of non-smooth problems considered in this paper, and prove a $$O\left( 1/N^2\right)$$O1/N2 or even linear $$O\left( \theta ^N\right)$$OθN convergence rate if either the primal or dual objective respectively both are strongly convex. Moreover we show that also under a slower decay of errors we can establish rates, however slower and directly depending on the decay of the errors. We demonstrate the performance and practical use of the algorithms on the example of nested algorithms and show how they can be used to split the global objective more efficiently.

Keywords: First-order primal–dual algorithm; Inexact proximal operator; Nested algorithms; Convex optimization (search for similar items in EconPapers)
Date: 2020
References: View references in EconPapers View complete reference list from CitEc
Citations:

Downloads: (external link)
http://link.springer.com/10.1007/s10589-020-00186-y 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: https://EconPapers.repec.org/RePEc:spr:coopap:v:76:y:2020:i:2:d:10.1007_s10589-020-00186-y

Ordering information: This journal article can be ordered from
http://www.springer.com/math/journal/10589

DOI: 10.1007/s10589-020-00186-y

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 ().