On Douglas–Rachford operators that fail to be proximal mappings

On Douglas–Rachford operators that fail to be proximal mappings The problem of finding a zero of the sum of two maximally monotone operators is of central importance in optimization. One successful method to find such a zero is the Douglas–Rachford algorithm which iterates a firmly nonexpansive operator constructed from the resolvents of the given monotone operators. In the context of finding minimizers of convex functions, the resolvents are actually proximal mappings. Interestingly, as pointed out by Eckstein in 1989, the Douglas–Rachford operator itself may fail to be a proximal mapping. We consider the class of symmetric linear relations that are maximally monotone and prove the striking result that the Douglas–Rachford operator is generically not a proximal mapping. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Mathematical Programming Springer Journals

On Douglas–Rachford operators that fail to be proximal mappings

Loading next page...
 
/lp/springer_journal/on-douglas-rachford-operators-that-fail-to-be-proximal-mappings-bevrLJzQTq
Publisher
Springer Berlin Heidelberg
Copyright
Copyright © 2016 by Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society
Subject
Mathematics; Calculus of Variations and Optimal Control; Optimization; Mathematics of Computing; Numerical Analysis; Combinatorics; Theoretical, Mathematical and Computational Physics; Mathematical Methods in Physics
ISSN
0025-5610
eISSN
1436-4646
D.O.I.
10.1007/s10107-016-1076-5
Publisher site
See Article on Publisher Site

Abstract

The problem of finding a zero of the sum of two maximally monotone operators is of central importance in optimization. One successful method to find such a zero is the Douglas–Rachford algorithm which iterates a firmly nonexpansive operator constructed from the resolvents of the given monotone operators. In the context of finding minimizers of convex functions, the resolvents are actually proximal mappings. Interestingly, as pointed out by Eckstein in 1989, the Douglas–Rachford operator itself may fail to be a proximal mapping. We consider the class of symmetric linear relations that are maximally monotone and prove the striking result that the Douglas–Rachford operator is generically not a proximal mapping.

Journal

Mathematical ProgrammingSpringer Journals

Published: Oct 12, 2016

References

You’re reading a free preview. Subscribe to read the entire article.


DeepDyve is your
personal research library

It’s your single place to instantly
discover and read the research
that matters to you.

Enjoy affordable access to
over 12 million articles from more than
10,000 peer-reviewed journals.

All for just $49/month

Explore the DeepDyve Library

Unlimited reading

Read as many articles as you need. Full articles with original layout, charts and figures. Read online, from anywhere.

Stay up to date

Keep up with your field with Personalized Recommendations and Follow Journals to get automatic updates.

Organize your research

It’s easy to organize your research with our built-in tools.

Your journals are on DeepDyve

Read from thousands of the leading scholarly journals from SpringerNature, Elsevier, Wiley-Blackwell, Oxford University Press and more.

All the latest content is available, no embargo periods.

See the journals in your area

DeepDyve Freelancer

DeepDyve Pro

Price
FREE
$49/month

$360/year
Save searches from Google Scholar, PubMed
Create lists to organize your research
Export lists, citations
Access to DeepDyve database
Abstract access only
Unlimited access to over
18 million full-text articles
Print
20 pages/month
PDF Discount
20% off