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

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Publisher
Springer Journals
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

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