4 research outputs found
On extension results for n-cyclically monotone operators in reflexive Banach spaces
In this paper we provide some extension results for n-cyclically monotone
operators in reflexive Banach spaces by making use of the Fenchel duality. In
this way we give a positive answer to a question posed by Bauschke and Wang in
[4]
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
On the convergence rate improvement of a primal-dual splitting algorithm for solving monotone inclusion problems
We present two modified versions of the primal-dual splitting algorithm
relying on forward-backward splitting proposed in \cite{vu} for solving
monotone inclusion problems. Under strong monotonicity assumptions for some of
the operators involved we obtain for the sequences of iterates that approach
the solution orders of convergence of O(1/n) and O(\omega^n), for , respectively. The investigated primal-dual algorithms are fully
decomposable, in the sense that the operators are processed individually at
each iteration. We also discuss the modified algorithms in the context of
convex optimization problems and present numerical experiments in image
processing and support vector machines classification.Comment: 24 page