683 research outputs found
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
Inertial Douglas-Rachford splitting for monotone inclusion problems
We propose an inertial Douglas-Rachford splitting algorithm for finding the
set of zeros of the sum of two maximally monotone operators in Hilbert spaces
and investigate its convergence properties. To this end we formulate first the
inertial version of the Krasnosel'ski\u{\i}--Mann algorithm for approximating
the set of fixed points of a nonexpansive operator, for which we also provide
an exhaustive convergence analysis. By using a product space approach we employ
these results to the solving of monotone inclusion problems involving linearly
composed and parallel-sum type operators and provide in this way iterative
schemes where each of the maximally monotone mappings is accessed separately
via its resolvent. We consider also the special instance of solving a
primal-dual pair of nonsmooth convex optimization problems and illustrate the
theoretical results via some numerical experiments in clustering and location
theory.Comment: arXiv admin note: text overlap with arXiv:1402.529
Douglas-Rachford Splitting: Complexity Estimates and Accelerated Variants
We propose a new approach for analyzing convergence of the Douglas-Rachford
splitting method for solving convex composite optimization problems. The
approach is based on a continuously differentiable function, the
Douglas-Rachford Envelope (DRE), whose stationary points correspond to the
solutions of the original (possibly nonsmooth) problem. By proving the
equivalence between the Douglas-Rachford splitting method and a scaled gradient
method applied to the DRE, results from smooth unconstrained optimization are
employed to analyze convergence properties of DRS, to tune the method and to
derive an accelerated version of it
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