303 research outputs found
A stochastic inertial forward-backward splitting algorithm for multivariate monotone inclusions
We propose an inertial forward-backward splitting algorithm to compute the
zero of a sum of two monotone operators allowing for stochastic errors in the
computation of the operators. More precisely, we establish almost sure
convergence in real Hilbert spaces of the sequence of iterates to an optimal
solution. Then, based on this analysis, we introduce two new classes of
stochastic inertial primal-dual splitting methods for solving structured
systems of composite monotone inclusions and prove their convergence. Our
results extend to the stochastic and inertial setting various types of
structured monotone inclusion problems and corresponding algorithmic solutions.
Application to minimization problems is discussed
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
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
Almost sure convergence of the forward-backward-forward splitting algorithm
In this paper, we propose a stochastic forward-backward-forward splitting
algorithm and prove its almost sure weak convergence in real separable Hilbert
spaces. Applications to composite monotone inclusion and minimization problems
are demonstrated.Comment: arXiv admin note: text overlap with arXiv:1210.298
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