209 research outputs found
A first-order stochastic primal-dual algorithm with correction step
We investigate the convergence properties of a stochastic primal-dual
splitting algorithm for solving structured monotone inclusions involving the
sum of a cocoercive operator and a composite monotone operator. The proposed
method is the stochastic extension to monotone inclusions of a proximal method
studied in {\em Y. Drori, S. Sabach, and M. Teboulle, A simple algorithm for a
class of nonsmooth convex-concave saddle-point problems, 2015} and {\em I.
Loris and C. Verhoeven, On a generalization of the iterative soft-thresholding
algorithm for the case of non-separable penalty, 2011} for saddle point
problems. It consists in a forward step determined by the stochastic evaluation
of the cocoercive operator, a backward step in the dual variables involving the
resolvent of the monotone operator, and an additional forward step using the
stochastic evaluation of the cocoercive introduced in the first step. We prove
weak almost sure convergence of the iterates by showing that the primal-dual
sequence generated by the method is stochastic quasi Fej\'er-monotone with
respect to the set of zeros of the considered primal and dual inclusions.
Additional results on ergodic convergence in expectation are considered for the
special case of saddle point models
Variable Metric Forward-Backward Splitting with Applications to Monotone Inclusions in Duality
We propose a variable metric forward-backward splitting algorithm and prove
its convergence in real Hilbert spaces. We then use this framework to derive
primal-dual splitting algorithms for solving various classes of monotone
inclusions in duality. Some of these algorithms are new even when specialized
to the fixed metric case. Various applications are discussed
Solving monotone inclusions involving parallel sums of linearly composed maximally monotone operators
The aim of this article is to present two different primal-dual methods for
solving structured monotone inclusions involving parallel sums of compositions
of maximally monotone operators with linear bounded operators. By employing
some elaborated splitting techniques, all of the operators occurring in the
problem formulation are processed individually via forward or backward steps.
The treatment of parallel sums of linearly composed maximally monotone
operators is motivated by applications in imaging which involve first- and
second-order total variation functionals, to which a special attention is
given.Comment: 25 page
A forward-backward view of some primal-dual optimization methods in image recovery
A wide array of image recovery problems can be abstracted into the problem of
minimizing a sum of composite convex functions in a Hilbert space. To solve
such problems, primal-dual proximal approaches have been developed which
provide efficient solutions to large-scale optimization problems. The objective
of this paper is to show that a number of existing algorithms can be derived
from a general form of the forward-backward algorithm applied in a suitable
product space. Our approach also allows us to develop useful extensions of
existing algorithms by introducing a variable metric. An illustration to image
restoration is provided
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