Linear and strong convergence of algorithms involving averaged nonexpansive operators

Bauschke, Heinz H.; Noll, Dominikus; Phan, Hung M.

doi:10.1016/j.jmaa.2014.06.075

Cited by 89 publications

(106 citation statements)

References 26 publications

(38 reference statements)

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“…where α c = max i∈[N ],k∈N α i,k . In view of (69), following the same arguments as that of Lemmas 3 and 4 in [38] for (39), one can conclude that…”

Section: A Proofs Of Theorems 1-3mentioning

confidence: 68%

“…denotes the set of interior points of set C. Please refer to [9], [39] for more details and [40] for another relevant notion, i.e., metric (sub-)regularity for set-valued mappings.…”

Section: Definition 1 ( [39]mentioning

confidence: 99%

“…Invoking (38) and ǫ i,k → 0 because of { ǫ i,k } k∈N ∈ ℓ 1 + , one readily obtains thatε k → 0. With reference to (39), by applying the same arguments as that of Lemmas 3 and 4 in [38] and using Lemma 3, one has that…”

Section: A Proofs Of Theorems 1-3mentioning

confidence: 99%

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

Liu

Fullmer

Nedić

et al. 2017

2017 American Control Conference (ACC)

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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...

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“…where α c = max i∈[N ],k∈N α i,k . In view of (69), following the same arguments as that of Lemmas 3 and 4 in [38] for (39), one can conclude that…”

Section: A Proofs Of Theorems 1-3mentioning

confidence: 68%

“…denotes the set of interior points of set C. Please refer to [9], [39] for more details and [40] for another relevant notion, i.e., metric (sub-)regularity for set-valued mappings.…”

Section: Definition 1 ( [39]mentioning

confidence: 99%

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

Liu

Fullmer

Nedić

et al. 2017

2017 American Control Conference (ACC)

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show abstract

“…For the algorithm (23): (i) The bounds (10), (11), and (18) hold with e T,t as in (28); (ii) Suppose further that f t is strongly convex with constant k t ; then, (20) hold with L t = min{|1 − νk t |, |1 − νK t |}.…”

Section: A Running Gradient Methods With Errorsmentioning

confidence: 99%

On the Convergence of the Inexact Running Krasnosel’skiĭ–Mann Method

Dall’Anese

Simonetto

Bernstein

2019

IEEE Control Syst. Lett.

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This paper leverages a framework based on averaged operators to tackle the problem of tracking fixed points associated with maps that evolve over time. In particular, the paper considers the Krasnosel'skiȋ-Mann method in a settings where: (i) the underlying map may change at each step of the algorithm, thus leading to a "running" implementation of the Krasnosel'skiȋ-Mann method; and, (ii) an imperfect information of the map may be available. An imperfect knowledge of the maps can capture cases where processors feature a finite precision or quantization errors, or the case where (part of) the map is obtained from measurements. The analytical results are applicable to inexact running algorithms for solving optimization problems, whenever the algorithmic steps can be written in the form of (a composition of) averaged operators; examples are provided for inexact running gradient methods and the forward-backward splitting method. Convergence of the average fixed-point residual is investigated for the nonexpansive case; linear convergence to a unique fixed-point trajectory is showed in the case of inexact running algorithms emerging from contractive operators.

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“…Selected references: [20,16,21]. (For other cyclic version of DR, see [7,9]. Also, if m = 2 and C 1 = C 2 = {0}, then (x k ) k∈N = ((λ /2) k x 0 ) k∈N is actually unbounded when x 0 = 0 and λ > 2.…”

Section: The Four Feasibility Algorithmsmentioning

confidence: 99%

Numerical Explorations of Feasibility Algorithms for Finding Points in the Intersection of Finite Sets

Bauschke

Gretchko

Moursi

2019

Splitting Algorithms, Modern Operator Theory, and Applications

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Projection methods are popular algorithms for iteratively solving feasibility problems in Euclidean or even Hilbert spaces. They employ (selections of) nearest point mappings to generate sequences that are designed to approximate a point in the intersection of a collection of constraint sets. Theoretical properties of projection methods are fairly well understood when the underlying constraint sets are convex; however, convergence results for the nonconvex case are more complicated and typically only local. In this paper, we explore the perhaps simplest instance of a feasibility algorithm, namely when each constraint set consists of only finitely many points. We numerically investigate four constellations: either few or many constraint sets, with either few or many points. Each constellation is tackled by four popular projection methods each of which features a tuning parameter. We examine the behaviour for a single and for a multitude of orbits, and we also consider local and global behaviour. Our findings demonstrate the importance of the choice of the algorithm and that of the tuning parameter.Goal of this paper The goal of this paper is to showcase the surprising numerical complexity of the most simple instance of (FP); namely, when each constraint set

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Linear and strong convergence of algorithms involving averaged nonexpansive operators

Cited by 89 publications

References 26 publications

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

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

On the Convergence of the Inexact Running Krasnosel’skiĭ–Mann Method

Numerical Explorations of Feasibility Algorithms for Finding Points in the Intersection of Finite Sets

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