12 research outputs found
Attouch-Th\'era duality revisited: paramonotonicity and operator splitting
The problem of finding the zeros of the sum of two maximally monotone
operators is of fundamental importance in optimization and variational
analysis. In this paper, we systematically study Attouch-Th\'era duality for
this problem. We provide new results related to Passty's parallel sum, to
Eckstein and Svaiter's extended solution set, and to Combettes' fixed point
description of the set of primal solutions. Furthermore, paramonotonicity is
revealed to be a key property because it allows for the recovery of all primal
solutions given just one arbitrary dual solution. As an application, we
generalize the best approximation results by Bauschke, Combettes and Luke [J.
Approx. Theory 141 (2006), 63-69] from normal cone operators to paramonotone
operators. Our results are illustrated through numerous examples
Generalized Forward-Backward Splitting
This paper introduces the generalized forward-backward splitting algorithm
for minimizing convex functions of the form , where
has a Lipschitz-continuous gradient and the 's are simple in the sense
that their Moreau proximity operators are easy to compute. While the
forward-backward algorithm cannot deal with more than non-smooth
function, our method generalizes it to the case of arbitrary . Our method
makes an explicit use of the regularity of in the forward step, and the
proximity operators of the 's are applied in parallel in the backward
step. This allows the generalized forward backward to efficiently address an
important class of convex problems. We prove its convergence in infinite
dimension, and its robustness to errors on the computation of the proximity
operators and of the gradient of . Examples on inverse problems in imaging
demonstrate the advantage of the proposed methods in comparison to other
splitting algorithms.Comment: 24 pages, 4 figure