159 research outputs found
Distributed Algorithms for Computing a Fixed Point of Multi-Agent Nonexpansive Operators
This paper investigates the problem of finding a fixed point for a global
nonexpansive operator under time-varying communication graphs in real Hilbert
spaces, where the global operator is separable and composed of an aggregate sum
of local nonexpansive operators. Each local operator is only privately
accessible to each agent, and all agents constitute a network. To seek a fixed
point of the global operator, it is indispensable for agents to exchange local
information and update their solution cooperatively. To solve the problem, two
algorithms are developed, called distributed Krasnosel'ski\u{\i}-Mann (D-KM)
and distributed block-coordinate Krasnosel'ski\u{\i}-Mann (D-BKM) iterations,
for which the D-BKM iteration is a block-coordinate version of the D-KM
iteration in the sense of randomly choosing and computing only one
block-coordinate of local operators at each time for each agent. It is shown
that the proposed two algorithms can both converge weakly to a fixed point of
the global operator. Meanwhile, the designed algorithms are applied to recover
the classical distributed gradient descent (DGD) algorithm, devise a new
block-coordinate DGD algorithm, handle a distributed shortest distance problem
in the Hilbert space for the first time, and solve linear algebraic equations
in a novel distributed approach. Finally, the theoretical results are
corroborated by a few numerical examples
A new projection method for finding the closest point in the intersection of convex sets
In this paper we present a new iterative projection method for finding the
closest point in the intersection of convex sets to any arbitrary point in a
Hilbert space. This method, termed AAMR for averaged alternating modified
reflections, can be viewed as an adequate modification of the Douglas--Rachford
method that yields a solution to the best approximation problem. Under a
constraint qualification at the point of interest, we show strong convergence
of the method. In fact, the so-called strong CHIP fully characterizes the
convergence of the AAMR method for every point in the space. We report some
promising numerical experiments where we compare the performance of AAMR
against other projection methods for finding the closest point in the
intersection of pairs of finite dimensional subspaces
Implicit Iterative Method for Hierarchical Variational Inequalities
We introduce a new implicit iterative scheme with
perturbation for finding the approximate solutions of a hierarchical variational inequality, that is, a variational inequality over the
common fixed point set of a finite family of nonexpansive mappings. We establish some convergence theorems for the sequence
generated by the proposed implicit iterative scheme. In particular, necessary and sufficient conditions for the strong convergence
of the sequence are obtained
Strong Convergence Theorems for Mixed Equilibrium Problem and Asymptotically I
This paper aims to use a hybrid algorithm for finding a common element of a fixed point problem for a finite family of asymptotically nonexpansive mappings and the set solutions of mixed equilibrium problem in uniformly smooth and uniformly convex Banach space. Then, we prove some strong convergence theorems of the proposed hybrid algorithm to a common element of the above two sets under some suitable conditions
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