A distributed algorithm for computing a common fixed point of a family of strongly quasi-nonexpansive maps

Abstract

This paper addresses the problem of seeking a common fixed point for a collection of nonexpansive operators over time-varying multi-agent networks in real Hilbert spaces, where each operator is only privately and approximately known to each individual agent, and all agents need to cooperate to solve this problem by propagating their own information to their neighbors through local communications over time-varying networks. To handle this problem, inspired by the centralized inexact Krasnosel'skiȋ-Mann (IKM) iteration, we propose a distributed algorithm, called distributed inexact Krasnosel'skiȋ-Mann (D-IKM) iteration. It is shown that the D-IKM iteration can converge weakly to a common fixed point of the family of nonexpansive operators. Moreover, under the assumption that all operators and their own fixed point sets are (boundedly) linearly regular, it is proved that the D-IKM iteration converges with a rate O(1/k ln(1/ξ) ) for some constant ξ ∈ (0, 1), where k is the iteration number. To reduce computational complexity and burden of storage and transmission, a scenario, where only a random part of coordinates for each agent is updated at each iteration, is further considered, and a corresponding algorithm, named distributed inexact block-coordinate Krasnosel'skiȋ-Mann (D-IBKM) iteration, is developed. The algorithm is proved to be weakly convergent to a common fixed point of the group of considered operators, and, with the extra assumption of (bounded) linear regularity, it is convergent with a rate O(1/k ln(1/ξ) ). Furthermore, it is shown that the convergence rate O(1/k ln(1/ξ) ) can still be guaranteed under a more relaxed (bounded) power regularity condition.Index Terms-Distributed algorithms, multi-agent networks, Krasnosel'skiȋ-Mann iteration, nonexpansive operators, fixed point, optimization.weakly to a fixed point of T under mild conditions [21], [25], [36], for example, when ∞ j=1 α j (1 − α j ) = ∞ for the KM iteration [36]. Now, let us introduce the graph theory for describing the communication pattern among all agents [11]. Specifically, the communication mode among N agents can be modeled by a digraph G = (V, E), where V = [N ] is the node or vertex set, and E ⊂ V × V is the edge set. An edge (i, j) ∈ E means that agent i is capable of transmitting its information to agent j, in which case agent i is called a neighbor of agent j. A directed path from i 1 to i l is a sequence of edges of the form (i 1 , i 2 ), (i 3 , i 4 ), . . . , (i l−1 , i l ). A graph is called strongly connected if there exists at least a directed path from any node to any other node in this graph. In this paper, the communication graph for all agents is assumed to be time-varying, that is, any two agents can have different communication status at different time steps. In this case, the graph is denoted asAt each time k ∈ N, there exists an adjacency matrix A k = (a ij,k ) ∈ R N ×N such that a ij,k > 0 if (j, i) ∈ E k , and a ij,k = 0 otherwise. Assume that a ii,k > 0 for all i ∈ [N ] and all k ∈ N. For communication graphs, we...

Asynchronous and Distributed Tracking of Time-Varying Fixed Points

“…The problem (40) seeks a trade off between tracking of a reference r (t) associated with y (t) (x) and costs associated with some control actions x. Similar to, e.g., (5), assuming that h (t) (x) is strongly convex and η = 0, an online gradient descent method involves a sequential execution of the following step:…”

DOT and DOP: Linearly Convergent Algorithms for Finding Fixed Points of Multi-Agent Operators

“…with Remark 3. It should be noticed that the problem considered in this paper is more general than the common fixed point finding problem in [16]- [18], [20], where all local operators are assumed to have at least one common fixed point, while this is dropped in this paper. Moreover, no convergence speeds are provided in [16]- [18].…”

“…For instance, distributed algorithms were developed in [16], [17] to find a common fixed point for a finite group of paracontraction operators. Meanwhile, a finite faimily of strongly quasi-nonexpansive operators were addressed in [18] for the common fixed point seeking problem. It should be noted that many interesting problems can boil down to the common fixed point finding problem, such as convex feasibility problems [22], [23] and the problem of solving linear algebraic equations in a distributed fashion [24]- [26], and so forth.…”

“…Compared with the closely related work [21], where no convergence speed is provided, a different algorithm from that in [21] is developed here and shown to be convergent at a linear rate. It should be noted that the problem here is more general than the common fixed point seeking problem [16]- [18], [20], where all local operators are assumed to have at least one common fixed point, while this assumption is dropped here. As a special case, linear convergence can also be ensured for the common fixed point seeking problem.…”

Distributed Algorithms for Computing a Fixed Point of Multi-Agent Nonexpansive Operators

“…To be specific, the problem of seeking a common fixed point for a finite number of paracontraction operators was addressed in [10,11], and a distributed algorithm was designed along with convergence analysis. Also, the common fixed point finding problem for a family of strongly quasi-nonexpansive operators was addressed in [22]. The aforementioned common fixed point seeking problem can find numerous applications, such as, in convex feasibility problems [15,31] and solving linear algebraic equations in a distributed approach [1,30,[40][41][42], and so on.…”