16 research outputs found
Stochastic Quasi-Fej\'er Block-Coordinate Fixed Point Iterations with Random Sweeping
This work proposes block-coordinate fixed point algorithms with applications
to nonlinear analysis and optimization in Hilbert spaces. The asymptotic
analysis relies on a notion of stochastic quasi-Fej\'er monotonicity, which is
thoroughly investigated. The iterative methods under consideration feature
random sweeping rules to select arbitrarily the blocks of variables that are
activated over the course of the iterations and they allow for stochastic
errors in the evaluation of the operators. Algorithms using quasinonexpansive
operators or compositions of averaged nonexpansive operators are constructed,
and weak and strong convergence results are established for the sequences they
generate. As a by-product, novel block-coordinate operator splitting methods
are obtained for solving structured monotone inclusion and convex minimization
problems. In particular, the proposed framework leads to random
block-coordinate versions of the Douglas-Rachford and forward-backward
algorithms and of some of their variants. In the standard case of block,
our results remain new as they incorporate stochastic perturbations
Strong Convergence Theorems for Quasi-Bregman Nonexpansive Mappings in Reflexive Banach Spaces
We study a strong convergence for a common fixed point of a
finite family of quasi-Bregman nonexpansive mappings in the framework of
real reflexive Banach spaces. As a consequence, convergence for a common
fixed point of a finite family of Bergman relatively nonexpansive mappings is
discussed. Furthermore, we apply our method to prove strong convergence theorems
of iterative algorithms for finding a common solution of a finite family
equilibrium problem and a common zero of a finite family of maximal monotone
mappings. Our theorems improve and unify most of the results that have
been proved for this important class of nonlinear mappings
On Strong Convergence of Halpernâs Method for Quasi-Nonexpansive Mappings in Hilbert Spaces
In this paper, we introduce a Halpernâs type method to approximate common fixed points of a nonexpansive mapping T and a strongly quasi-nonexpansive mappings S, defined in a Hilbert space, such that I â S is demiclosed at 0. The result shows as the same algorithm converges to different points, depending on the assumptions of the coefficients. Moreover, a numerical example of our iterative scheme is given
Visual Analysis of Dynamics Behaviour of an Iterative Method Depending on Selected Parameters and Modifications
There is a huge group of algorithms described in the literature that iteratively find solutions of a given equation. Most of them require tuning. The article presents root-finding algorithms that are based on the Newton-Raphson method which iteratively finds the solutions, and require tuning. The modification of the algorithm implements the best position of particle similarly to the particle swarm optimisation algorithms. The proposed approach allows visualising the impact of the algorithm's elements on the complex behaviour of the algorithm. Moreover, instead of the standard Picard iteration, various feedback iteration processes are used in this research. Presented examples and the conducted discussion on the algorithm's operation allow to understand the influence of the proposed modifications on the algorithm's behaviour. Understanding the impact of the proposed modification on the algorithm's operation can be helpful in using it in other algorithms. The obtained images also have potential artistic applications