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IEEE TRANSACTIONS ON CONTROL OF NETWORK SYSTEMS, VOL. 8, NO. 3, SEPTEMBER 2021

Nonsmooth Resource Allocation of Multiagent
Systems With Disturbances: A Proximal
Approach
Yanan Zhu , Student Member, IEEE, Guanghui Wen , Senior Member, IEEE,
Wenwu Yu , Senior Member, IEEE, and Xinghuo Yu , Fellow, IEEE

Abstract—This article aims to solve the nonsmooth resource allocation problem in the presence of a global network resource constraint and local set constraints in the
framework of multiagent system optimization. It is assumed
that multiagent systems are subject to some external disturbances, and the control inputs of the agents satisfy
Lispchitz continuity. These two distinguished features render the existing distributed optimization algorithms, especially the subgradient-based algorithms inapplicable due to
the employment of discontinuity of subgradients. To solve
such a challenging resource allocation problem, a new kind
of continuous-time proximal algorithm is designed with the
aid of convex optimization theory and the internal-model
technique. The proximal algorithm is further augmented by
introducing an event-based communication scheme such
that the continuous-time communication among the agents
is avoided successfully. The theoretical analysis shows that
the multiagent systems under the proposed algorithms can
converge to the optimal solution of the considered problem,
while the external disturbances are rejected. Besides, the
Zeno behavior can be excluded for the proximal algorithm
with event-based communication. Finally, the numerical
simulations are given to verify the established theoretical
results.
Index Terms—Disturbance rejection, event-based communication, multiagent systems, proximal algorithm, resource allocation.
Manuscript received November 17, 2020; accepted February 13,
2021. Date of publication March 23, 2021; date of current version
September 17, 2021. This work was supported in part by the Startup
Foundation for Introducing Talent of the Nanjing University of Information
Science and Technology under Grant 2020r012, in part by the Natural Science Foundation of the Jiangsu Higher Education Institutions
of China under Grant 20KJB120004, in part by the Natural Science
Foundation of Jiangsu Province under Grant BK20200809, in part by the
National Natural Science Foundation of China under Grants 62073076
and 62073079, in part by the Six Talent Peaks of Jiangsu Province under
Grant 2019-DZXX-006, in part by the Australian Research Council under
Grant DP200101199, and in part by the Jiangsu Provincial Key Laboratory of Networked Collective Intelligence under Grant BM2017002. Recommended by Associate Editor W. X. Zheng. (Corresponding authors:
Guanghui Wen; Wenwu Yu.)
Yanan Zhu is with the School of Automation, Nanjing University of
Information Science and Technology, Nanjing 210044, China (e-mail:
zhuxiaoyazhuyanan@163.com).
Guanghui Wen and Wenwu Yu are with the Jiangsu Provincial Key Laboratory of Networked Collective Intelligence, School of
Mathematics, Southeast University, Nanjing 210096, China (e-mail:
wenguanghui@gmail.com; wwyu@seu.edu.cn).
Xinghuo Yu is with the School of Engineering, RMIT University, Melbourne, VIC 3001, Australia (e-mail: x.yu@rmit.edu.au).
Digital Object Identifier 10.1109/TCNS.2021.3068349

I. INTRODUCTION
ISTRIBUTED optimization has been extensively investigated via plenty of distributed algorithms in recent years
(see [1] and the references therein). In fact, most results have
been previously concerned with designing distributed optimization, while the dynamics of agents have received less attention. Recently, distributed optimization with the dynamics of
agents has been studied by designing continuous-time control algorithms for many types of multiagent systems, for instance, single-integrator dynamics [2]–[5]; Euler–Lagrange systems [6], [7]; double-integrator dynamics [8]; linear multiagent
systems [9]; and nonlinear multiagent systems [10]–[12].
As one of the most important research topics in distributed
optimization, resource allocation has a wide range of applications in network systems such as communication networks [13],
[14] and power systems [15]–[18]. Recently, a lot of results
on resource allocation have been established by considering
both smooth cost functions and nonsmooth cost functions, for
example, gradient-based distributed algorithms for smooth resource allocation [19]–[22] and subgradient-based distributed
algorithms for nonsmooth resource allocation [23]–[27]. Moreover, when the dynamics of agents are described by second-order
systems, the resource allocation problem with network resource
constraints was studied in [28]. The result provided in [28]
was further extended in [29], where both the network resource
constraints and the local constraints were considered.
It is well-known that disturbances are ubiquitous in reality.
In general, it is challenging to make the states of the multiagent
systems converge to the optimal solutions of distributed optimization problems in the presence of external disturbances. How
to develop effective algorithms to obtain the optimal solutions
of distributed optimization problems subject to disturbances
has received increasing attention in recent years. In [2], [3],
and [10]–[12], the internal model techniques were integrated
into distributed optimization algorithms for solving the unconstrained distributed optimization problem of multiagent systems
with external disturbances, where the mechanism of internal
model techniques can be found in [30]. By utilizing Nash
equilibrium seeking algorithms with disturbance rejection, distributed game problems of multiagent systems were addressed
in [31]–[34]. Also, the resource allocation problem of multiagent systems with a global network resource constraint and

D

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ZHU et al.: NONSMOOTH RESOURCE ALLOCATION OF MULTIAGENT SYSTEMS

exogenous disturbances was studied in [35]. Still, the existing
results on resource allocation of multiagent systems in the
presence of local set constraints rarely considered the effect of
disturbances.
On the other hand, for the case when the objective functions
of resource allocation problems are nonsmooth, distributed subgradient algorithms were utilized in [23], [26], and [27]. However, due to the discontinuity of subgradients, these subgradient
algorithms are inapplicable to the situation, in which the dynamics of agents are required to satisfy the Lipschitz continuity.
Actually, the Lipschitz continuity ensures that there is no violent vibration in multiagent dynamical systems. This facilitates
applying Lyapunov stability theory to analyze the convergence
property of multiagent systems. Note that the proximal mapping
of a nonsmooth convex function satisfies the Lipschitz continuity [36], which has become an effective tool in designing smooth
algorithms for nonsmooth optimization problems [37]–[40]. In
particular, a continuous-time smooth algorithm was proposed
in [41] to deal with the distributed nonsmooth optimization
problem in the presence of convex intersection constraints. To
the best of our knowledge, how to construct smooth algorithms
in a continuous-time setting to solve the nonsmooth resource
allocation problem of multiagent systems is still challenging.
Based on the above analysis, we aim to address the nonsmooth
resource allocation problem with the global network resource
constraint and local set constraints for a class of multiagent
systems subject to external disturbances. To this end, we first propose a novel continuous-time proximal algorithm by skillfully
combining some convex optimization methods and an internal
model technique. Then, to avoid continuous communication
among neighboring agents, we extend the proposed proximal algorithm by introducing an event-based communication scheme,
where the event-based communication scheme is partly motivated by the authors of [9] and [42]–[44]. We show that the
multiagent systems can succeed in finding the optimal solution
under the proposed algorithms while rejecting disturbances. To
summarize, the contributions of this article are given as follows.
In the problem setup, the present work systematically considers
the case with nonsmooth local objective functions, local set
constraints, and disturbances in comparison with most of the existing results on resource allocation of multiagent systems [23]–
[29], [35]. In algorithm design, different from subgradient-based
nonsmooth algorithms [23]–[27], the dynamics of the present
algorithms satisfy the Lipschitz continuity, even if the agents’
subgradients are discontinuous. Moreover, the employment of
the event-based communication scheme significantly reduces
the communication frequencies among neighboring agents. By
using tools from internal model control theory and Lyapunov
stability analysis, it is proved that the optimal solution of the
considered problem can be achieved, and meanwhile, the effect
of external disturbances is effectively eliminated. Besides, the
Zeno behavior can be avoided for the proposed algorithm with
the event-based communication mechanism.
The rest of this article is organized as follows. In Section II,
some notations and basic terminologies are introduced. In Section III, the nonsmooth resource allocation problem of multiagent systems with external disturbances is described. To solve

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the problem, two proximal algorithms with continuous-time
communication and event-based communication are proposed
in Sections IV and V, respectively. These two algorithms are
illustrated by performing numerical simulations in Section VI.
Finally, Section VII concludes this article.
II. PRELIMINARIES
This section introduces some notations throughout this article
and some basic terminologies from proximal mapping and graph
theory.
Notations: Let R, R>0 , Rn , and Rm×n denote the set of real
numbers, positive real numbers, real n-dimensional vectors, and
real m × n matrixes, respectively. We denote  · 1 and  ·  as
the 1-norm and 2-norm on Rn , respectively. Let N represent
the set of natural numbers. For a symmetric matrix C ∈ Rn×n ,
λmin (C) and λmax (C) denote its smallest and largest eigenvalues, respectively. For any matrixes A and B, A ⊗ B represents their Kronecker product. 1n ∈ Rn (respectively, 0n ∈ Rn )
represents the vector, whose entries are all one (respectively,
zero), and In ∈ Rn×n represents the identity matrix. For column
vectors u1 , u2 , . . . , uN , col(u1 , u2 , . . . , uN ) is the aggregated
vector. Let Γ0 (Rn ) : Rn → (−∞, +∞) be the set of proper
lower semicontinuous convex functions. For f ∈ Γ0 (Rn ), ∂f
represents the subdifferential of f . A function f : Rn → is
differentiable if the derivative of f exists at any point x ∈
Rn . A differentiable function f : Rn → R is δ-strongly convex (δ ∈ R>0 ) if and only if (u − u )T (∇f (u ) − ∇f (u )) ≥
δu − u 2 , which is equivalent to f (u ) − f (u ) ≥ (u −
u )T ∇f (u ) + 2δ u − u 2 for all u, u ∈ Rn . A vectorvalued function f : Rn → Rn is -Lipschitz ( ∈ R>0 ) if and
only if f (u ) − f (u ) ≤ u − u  for all u, u ∈ Rn . For
a nonempty closed convex set C, let PC (u) = arg minv∈C u −
v and NC (u) = {w ∈ Rn : wT (v − u) ≤ 0 ∀v ∈ C} denote
the projection of the vector v ∈ Rn into C and the normal
cone of C at u ∈ C, respectively. The following are some useful
properties with respect to PC : i) PC is nonexpansive, that is,
PC (u) − PC (v) ≤ u − v ∀ u, v ∈ Rn ; ii) PC (u − v) = u
if and only if −v ∈ NC (u); and iii) (v − PC (v))T (PC (v) −
u) ≥ 0 for all u ∈ C.
Proximal mapping: We review some preliminaries on proximal mapping from [36]. For f ∈ Γ0 (Rn ) and v ∈ Rn ,
let Proxf (v) ∈ Rn be the proximal mapping of f at
v. Then, Proxf (v) satisfies f (v) = inf u∈Rn f (u) + 12 v −
u2 = f (Proxf (v)) + 12 v − Proxf (v)2 . The definition of
Proxf (v) implies that v − Proxf (v) ∈ ∂f (Proxf (v)). Also,
Proxf is nonexpansive. This means that Proxf (u) −
Proxf (v) ≤ u − v for any u, v ∈ Rn .
Graph theory: We introduce some graph-theoretic notions
from [45] and [46]. The communication network among agents
can be modeled by an undirected graph G = (V, E), where
V = {1, 2, . . . , N } and E ⊂ V × V are the set of agents and
edges, respectively. An undirected edge (i, j) ∈ E means that
the information between agent i and agent j can be shared.
A path connecting agent i and agent j is composed of a sequence of some edges in the form of (i, i1 ), (i1 , i2 ), . . ., (ik , j).
G is connected if and only if there is a path between any two

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agents. Let A = [aij ] ∈ RN ×N denote the adjacency matrix of
G, where aij = aji > 0 if (i, j) ∈ E, and aij = 0, otherwise.
Also, we assume that G has no self-loop, which implies that
aii = 0. Correspondingly, the Laplacian
 matrix of G is defined
by L = [lij ] ∈ RN ×N , where lii = N
j=1 aij for i ∈ V, and
lij = −aij if i = j. From the definition of L, it follows that
L1N = 0N . Note that L is a real symmetric matrix, and it has
a zero eigenvalue. Therefore, its eigenvalues λi , i ∈ V, can be
ordered by 0 = λ1 ≤ λ2 ≤ λ3 ≤ · · · ≤ λN . If G is connected,
we have λ2 > 0 and
uT Lu ≥ λ2 u2

(1)

for any u satisfying 1TN u = 0.

Assumption 2:

III. PROBLEM FORMULATION
We consider a class of multiagent systems with N agents
communicating over an undirected graph. The dynamics of the
agents are described by
ẋi = ui + di , i ∈ V

(2)

where xi ∈ Rn and ui ∈ Rn are agent i’s decision variable and
control input, respectively, and di ∈ Rn is its locally bounded
disturbance. To characterize the dynamic behavior of di , we
assume that di can be generated by the following system:
ẇi = Si wi , di = Ci wi , i ∈ V

(3)

where wi ∈ Rmi , Si ∈ Rmi ×mi , and Ci ∈ Rn×mi . In general,
system (3) is assumed to be neutrally stable and observable, as
pointed out in [47]. Also, the initial condition wi (0) belongs to
a compact set Ki .
The objective of the multiagent system (2) is to find the
optimal solution of the following resource allocation problem:
min

xi ,i∈V

N


fi (xi ) = min

xi ,i∈V

i=1

s.t.

N

i=1

xi =

N


the agents’ dynamics with external disturbances. The injection
of the disturbances described in problem (4) cannot be effectively dealt with by the proposed continuous-time algorithms
in [15]–[24], [26], and [27]. Moreover, the multiagent system (2)
requires that the control protocol ui satisfies Lipschitz continuity. This renders the subgradient-based control protocols [23]–
[27] unavailable because of the discontinuity of subgradients.
This motivates us to seek a novel algorithm design to solve
problem (4).
To solve problem (4), we made the following proper
assumptions.
Assumption
1: Fori ∈ V, there exists an interior x̃i ∈ Ωi

N
such that N
i=1 x̃i =
i=1 bi .

N


0n ∈ ∇fi1 (x∗i ) + ∂fi2 (x∗i ) + NΩi (x∗i ) + z∗0

(fi1 (xi ) + fi2 (xi ))

N


i=1

bi , xi ∈ Ωi , i ∈ V

1) For i ∈ V, fi1 (xi ) is μi -strongly convex with μi ∈ R>0 .
2) For i ∈ V, ∇fi1 (xi ) is νi -Lipschitz with νi ∈ R>0
Assumption 3: For i ∈ V, fi2 (xi ) ∈ Γ0 (Rn ) and Proxfi2 is
solvable.
Remark 3: Assumption 1 guarantees the existence of the
solutions of problem (4). Assumption 3 facilitates the calculation
of the proximal mapping of fi2 (xi ). In fact, there are many
functions satisfying Assumption 3, for example, fi2 (xi ) is in
the form of  · 1 or  ·  of the decision variable xi . With
Assumptions 2 and 3, fi (xi ) is strongly convex. Thus, problem
(4) has a unique optimal solution.
Assumption 4: The communication graph among agents is
undirected and connected.
By Assumptions 1–3, we get the optimal conditions of problem (4) based [48, Th. 3.25].
Lemma 1: With Assumptions 1–3, x∗i ∈ Ωi is the optimal
solution to problem (4) if and only if there exists a Lagrangian
multiplier z∗0 ∈ Rn satisfying

(4)

i=1

where fi (xi ) : Rn → R is agent i’s local nonsmooth cost function that can be viewed as the sum of a differentiable function
fi1 (xi ) and a nondifferentiable function fi2 (xi ); bi ∈ Rn and
Ωi ⊂ Rn are its local resource and local constraint, of which Ωi
is a closed convex set. Also, agent i’s individual information,
including fi (xi ), bi , and Ωi , is not known to other agents.
Remark 1: The form of the objective function in problem
(4) arises widely in practical applications such as the structured
optimal control and inverse problems involved in identifying
and controlling dynamical representations of systems evolving
in real time. These problems can be viewed as the nonsmooth
composite optimization problems that have been studied in many
references such as [37]–[41].
Our aim is to design the control protocol ui such that the
multiagent system (2) with (3) can find the optimal solution of
problem (4) and the disturbances are rejected simultaneously.
Remark 2: Unlike most of the nonsmooth resource allocation
problems such as in [23], [26], and [27], we also consider

x∗i =

i=1

N


bi , i ∈ V.

(5)
(6)

i=1

IV. CONTINUOUS-TIME PROXIMAL ALGORITHM DESIGN
The purpose of this section is to design ui such that the multiagent system (2) can solve problem (4) by rejecting disturbances.
Inspired by [31] and [32], we consider the following internal
model mechanism to reject disturbance:
v̇i = −Si (Ki xi − vi ) + Ki (ui − Ci (Ki xi − vi )), i ∈ V
(7)
where vi ∈ Rmi , and Ki ∈ Rmi ×n is selected such that Si +
Ki Ci is Hurwitz, i.e., the real parts of all eigenvalues of Si +
Ki Ci are negative.
For agent i ∈ V, we design the following control protocol ui :


ui = PΩi xi − α∇fi1 (xi ) − zi + yi − xi + Ci (Ki xi − vi )
ẏi = Proxαfi2 (xi − yi ) − xi + Ci (Ki xi − vi ) + Ci wi
żi = xi − bi −

N


βij aij (zi − zj ) − pi

j=1

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ZHU et al.: NONSMOOTH RESOURCE ALLOCATION OF MULTIAGENT SYSTEMS



+ PΩi xi − α∇fi1 (xi ) − zi + yi − xi
ṗi =

N


βij aij (zi − zj ), pi (0) = 0n

j=1

β̇ij = γij aij zi − zj 2
v̇i = −Si (Ki xi − vi )
+ Ki (PΩi (xi − α∇fi1 (xi ) − zi + yi ) − xi )

(8)

where yi ∈ Rn , zi ∈ Rn , pi ∈ Rn , α > μ1 with μ =
min{μ1 , μ2 , . . . , μN }, βij is the time-varying coupling coefficient for the edge (i, j) ∈ E with βij (0) = βji (0), and γij =
γji ∈ R>0 .
Remark 4: The design of algorithm (8) can be roughly interpreted as follows.
1) The role of yi is to estimate a subgradient of −αfi2 . With
the aid of yi , −α∇fi1 + yi acts a decent direction of agent
i. This drives agent i’s decision variable xi to the optimal
solution of problem (4).
2) zi is a consensus variable for estimating a global Lagrangian multiplier associated with the network resource
consensus
constraint in (4). pi assists zi in reaching
N ex
actly by driving the error between N
i=1 xi and
i=1 bi
to zero, which guarantees that (6) holds.
3) The terms Ci (Ki xi − vi ) + Ci wi and PΩi (xi −
α∇fi1 (xi ) − zi + yi ) − xi , involved, respectively, in
ẏi in żi , play a key role in ensuring the convergence
of algorithm (8). To be precise, the introduction of the
above two terms guarantees that some unfavorable terms
in the derivative of the Lyapunov function with regard to
ẋi , ẏi , and żi can be mutually neutralized.
4) The time-varying gain βij is to ensure that the convergence of (8) does not rely on the global information of the
network, and its introduction is motivated by the authors
of [42] and [43]. vi is in charge of rejecting disturbances
caused by system (3).
By combining algorithm (8) with system (2), we have the
following closed-loop system:


ẋi = PΩi xi − α∇fi1 (xi ) − zi + yi − xi
ẏi = Proxαfi2 (xi − yi ) − xi + Ci (Ki xi − vi + wi )
N


βij aij (zi − zj ) − pi

j=1



+ PΩi xi − α∇fi1 (xi ) − zi + yi − xi
ṗi =

N


βij aij (zi − zj ), pi (0) = 0n

ẏ = Proxαf 2 (x − y) − x + Cr
ż = x − b − (Lβ ⊗In )z − p


+ PΩ x − α∇f 1 (x) − z + y − x
ṗ = (Lβ ⊗ In )z, p(0) = 0N n
β̇ij = γij aij zi − zj 2
ṙ = (S + KC)r.

(10)

Remark 6: Let col(x∗ , y∗ , z∗ , p∗ , r∗ ) be an equilibrium point
of algorithm (10). Then, we have r∗ = 0N mi by the inverti=1
ibility of S + KC. Meanwhile, col(x∗ , y∗ , z∗ , p∗ ) satisfies


PΩ x∗ − α∇f 1 (x∗ ) − z∗ + y∗ = x∗
(11)

Proxαf 2 (x∗ − y∗ ) = x∗
∗

x −b=p

∗

(Lβ ⊗ In )z∗ = 0N n .

(12)
(13)
(14)

−y∗ ∈ α∂f 2 (x∗ ).

j=1

Using ii) for (11) results in

β̇ij = γij aij zi − zj 2

−α∇f 1 (x∗ ) − 1N ⊗ z0 + y∗ ∈ NΩ (x∗ )

v̇i = − Si (Ki xi − vi )
+ Ki (PΩi (xi − α∇fi1 (xi ) − zi + yi ) − xi ).

Remark 5: Notice that the projection mapping PΩi and the
proximal mapping Proxαfi2 are nonexpansive. Therefore, the
right-hand sides of ẋi and ẏi satisfy the Lipschitz continuity. In
other words, algorithm (9) is smooth even if fi (xi ) is nonsmooth.
For subsequent analysis, define x = col(x1 , x2 , . . . ,
xN ), y = col(y1 , y2 , . . . , yN ), z = col(z1 , z2 , . . . , zN ),
p = col(p1 , p2 , . . . , pN ), v = col(v1 , v2 , . . . , vN ), w =
r = w + Kx − v
with
K=
col(w1 , w2 , . . . , wN ),
diag(K1 , K2 , . . . , KN ), b = col(b1 , b2 , . . . , bN ), C = diag
(C1 , C2 , . . . , CN ), S = diag(S1 , S2 , . . . , SN ), ν = max
j
(xN ))
{ν1 , ν2 , . . . , νN }, f j (x) = col(f1j (x1 ), f2j (x2 ), . . . , fN
1
(xN )),
for j = 1, 2, ∇f 1 = col(∇f11 (x1 ), ∇f21 (x2 ), . . . , ∇fN
2
(xN )), Proxf 2 =
∂f 2 (x) = col(∂f12 (x1 ), ∂f22 (x2 ), . . . , ∂fN
col(Proxf12 , Proxf22 , . . . , ProxfN2 ), PΩ = col(PΩ1 , PΩ2 , . . . ,
PΩN ) with Ω = Ω1 × Ω2 × . . . × ΩN , and Lβ = [lβij ] ∈

RN ×N is denoted as lβii = N
j=1 βij aij for i ∈ V and
ij
lβ = −βij aij for i = j.
With x, y, z, p, r, and Lβ , we can obtain from (9) the following
form:


ẋ = PΩ x − α∇f 1 (x) − z + y − x + Cr

It is evident that Lβ is also a symmetric Laplacian matrix
according to its definition and Assumption 4. Therefore, we
have z∗ = 1N ⊗ z0 with z0 ∈ Rn by (14). Notice that (1TN ⊗
In )ṗ = (1TN ⊗ In )(Lβ ⊗ In )z = 0n by Assumption 4. Thus,


N
p = N
p (0) = 0n with pi (0) = 0n . This implies
i=1
i=1
Ni
Ni
N
∗
∗
that
x
−
i=1 i
i=1 bi =
i=1 pi = 0n by (13). Equivalently, (6) holds.
According to (12) and the definition of the proximal operator,
we get

+ Ci (Ki xi − vi + wi )

żi = xi − bi −

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(9)

where z∗ = 1N ⊗ z0 is used. Thus, we have −α∇fi1 (x∗i ) −
α∂fi2 (x∗i ) − z0 ∈ NΩi (x∗i ) for i ∈ V. Setting z∗0 = α1 z0 , then

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0n ∈ ∇fi1 (x∗i ) + ∂fi2 (x∗i ) + z∗0 + NΩi (x∗i ). That is, (5) holds.
By Lemma 1, it follows that x∗ is the optimal solution of problem
(4).
The following theorem shows that the closed-loop system
(9) can guarantee that system (2) converges to the optimal
solution of problem (4). Meanwhile, the disturbances generated
by system (3) are rejected.
Theorem 1: Suppose that Assumptions 1–4 hold. Under
algorithm (9), system (2) asymptotically converges to the optimal solution of problem (4) while rejecting the disturbances
generated by system (3).
Proof: To obtain the conclusion in Theorem 1, we resort to
analyzing system (10). To do this, we starting by transforming
system (10) into


x̄˙ = PΩ x̄ + x∗ − α∇f 1 (x̄ + x∗ ) − z̄ − z∗ + ȳ + y∗
− x̄ − x∗ + Cr

+

ρ+1 1
where P1 = ( αμ−1+ 1 −1
1 ) and P2 = ( 1 1 ) are positivedefinite matrixes. This implies that V1 is positive definite and
radially unbounded with respect to col(x̄, ȳ, z̄, p̄).
Let Δ = x̄ + x∗ − α∇f 1 (x̄ + x∗ ) − z̄ − z∗ + ȳ + y∗ and
∗
Δ = x∗ − α∇f 1 (x∗ ) − z∗ + y∗ . By iii), we have

(PΩ (Δ) − x̄∗ )T (PΩ (Δ) − Δ) ≤ 0

(20)

(−α∇f 1 (x̄∗ ) − z∗ + y∗ )T (PΩ (Δ) − x̄∗ ) ≤ 0.

(21)

Define υ1 = (x̄ + α∇f 1 (x̄ + x∗ ) − α∇f 1 (x∗ ) − ȳ)T
(PΩ (Δ) − x̄∗ ),
υ2 = (x̄ + α∇f 1 (x̄ + x∗ ) − α∇f 1 (x∗ ) −
T
ȳ) x̄, and υ3 = (x̄ + α∇f 1 (x̄ + x∗ ) − α∇f 1 (x∗ ) − ȳ)T Cr.
Then, we can derive
υ1 = (x̄ + x∗ − PΩ (Δ) − z̄ + x̄)T (PΩ (Δ) − x̄∗ )

ȳ˙ = Proxαf 2 (x̄ + x∗ − ȳ − y∗ ) − x̄ − x∗ + Cr

z̄˙ = x̄ − (Lβ ⊗ In )z̄ − p̄ + PΩ x̄ + x∗ − α∇f 1 (x̄ + x∗ )

+ (PΩ (Δ) − Δ)T (PΩ (Δ) − x̄∗ )

− z̄ − z∗ + ȳ + y∗ ) − x̄ − x∗

+ (−α∇f 1 (x̄∗ ) − z∗ + y∗ )T (PΩ (Δ) − x̄∗ )

p̄˙ = (Lβ ⊗ In )z̄

(20), (21)
≤(x̄ + x∗ − PΩ (Δ) − z̄ + x̄)T (PΩ (Δ) − x̄∗ )

2

β̇ij = γij aij z̄i − z̄j 
ṙ = (S + KC)r

= − PΩ (Δ) − x̄ − x∗ 2 + x̄2 − z̄T (PΩ (Δ) − x∗ )

(15)

where x̄ = x − x∗ , ȳ = y − y∗ , z̄ = z − z∗ , and p̄ = p − p∗ .
The proof in detail consists of the following three steps.
Step I: Consider
V1 =

ρ+1
(x̄2 + ȳ2 − 2x̄T ȳ)
2


+ α(ρ+1) 1TN f 1 (x̄+x∗ )−1TN f 1 (x∗i )− x̄T ∇f 1 (x∗ )

ρ
1
+ z̄2 + z̄ + p̄2
2
2


1
where ρ > max λmin1(P ) − 1, 0 with P = ( αμ
1 1 ).

(16)

(18)

υ2 ≥ x̄2 + αμx̄2 − ȳT x̄
(19)

υ3 ≤ (x̄ − ȳ)T Cr + ανCx̄r.
1 T ˙
Substituting υ1 , υ2 , and υ3 into ( ∂V
∂ x̄ ) x̄ yields



∂V1
∂ x̄

T

x̄˙ = (ρ + 1)(υ1 − υ2 + υ3 )
≤ −(ρ+1)PΩ (Δ) − x̄ − x∗ 2 − (ρ+1)αμx̄2
+ (ρ + 1)ȳT x̄ − (ρ + 1)z̄T (PΩ (Δ) − x∗ )

From Assumption 2, we have

+ (ρ + 1)(x̄ − ȳ)T Cr + (ρ + 1)ανCx̄r.
(22)

μ
1TN f 1 (x̄ + x∗ ) − 1TN f 1 (x∗ ) − x̄T ∇f 1 (x∗ ) ≥ x̄2
2

(17)
x̄T (∇f 1 (x̄ + x∗ ) − ∇f (x̄∗ )) ≥ μx̄2
(18)
∇f 1 (x̄ + x∗ ) − ∇f (x̄∗ ) ≤ νx̄.
(19)
Using (17) in (16), we obtain
ρ
ρ+1
(αμx̄2 + ȳ2 − 2x̄T ȳ) + z̄2
2
2
1
+ z̄ + p̄2
2
(ρ + 1)λmin (P1 )
(x̄2 + ȳ2 )
≥
2

V1 ≥

λmin (P2 )
(z̄2 + p̄2 )
2

Notice that
ȳ + x̄ + x∗ − Cr = Proxαf 2 (x̄ + x∗ − ȳ − y∗ )
x∗ = Proxαf 2 (x∗ − y∗ ).
According to the definition of proximal mapping, we have
−ȳ − y∗ − ȳ˙ + Cr ∈ α∂f 2 (ȳ˙ + x̄ + x∗ − Cr) and −y∗ ∈
α∂f 2 (x∗ ). As a result, it holds that (ȳ˙ + x̄ − Cr)T (ȳ˙ + ȳ −
Cr) ≤ 0 by using the convexity of f 2 . Simplifying the above
inequality leads to
ȳT ȳ˙ ≤ −Proxαf 2 (Υ) − x̄ − x∗ 2 − x̄T ȳ + ȳT Cr
− x̄T (Proxαf 2 (Υ) − x̄ − x∗ )

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(23)

ZHU et al.: NONSMOOTH RESOURCE ALLOCATION OF MULTIAGENT SYSTEMS

1 T ˙
where Υ = x̄ + x∗ − ȳ − y∗ . Using (23) in ( ∂V
∂ ȳ ) ȳ = (ρ +
˙ we can arrive at
1)(ȳ − x̄)T ȳ,

T
∂V1
ȳ˙ ≤ −(ρ + 1)Proxαf 2 (Υ) − x̄ − x∗ 2
∂ ȳ

− (ρ + 1)x̄T ȳ
− 2(ρ + 1)x̄T (Proxαf 2 (Υ) − x̄ − x∗ )
− (ρ + 1)(x̄ − ȳ)T Cr.


Besides, we have

T
∂V1
∂V1
z̄˙ +
∂z̄
∂ p̄

T

(24)

p̄˙ = −(ρ + 1)z̄T p̄ − ρz̄T (Lβ ⊗In )z̄
+ (ρ+1)z̄T (PΩ (Δ)−x∗ )+ p̄T x̄
+ p̄T (PΩ (Δ) − x̄ − x∗ ) − p̄2 .
(25)

∂V1 T ˙
1 T ˙
Applying (22), (24), and (25) to V̇1 = ( ∂V
∂ x̄ ) x̄ + ( ∂ ȳ ) ȳ +
∂V1 T ˙
1 T ˙
( ∂V
∂z̄ ) z̄ + ( ∂ p̄ ) p̄ leads to

V̇1 ≤ − (ρ + 1)PΩ (Δ) − x̄ − x∗ 2 − (ρ + 1)λmin (P )x̄2
− (ρ+1)λmin (P )Proxαf 2 (Υ)− x̄−x∗ 2 −(ρ+1)z̄T p̄
− ρz̄T (Lβ ⊗In )z̄+ p̄T x̄+ p̄T (PΩ (Δ)− x̄−x∗ )−p̄2
+ (ρ + 1)ανCx̄r.
Let z̄ = z̄⊥ + z̄ , where z̄ = 1N ⊗ z̄0 and (z̄⊥ )T z̄ = 0.
Then, it follows that −(ρ + 1)z̄T p̄ = −(ρ + 1)(z̄⊥ )T p̄ ≤ (ρ +
1)2 z̄⊥ 2 + 14 p̄2 by using (1) and (1TN ⊗ In )p̄ = 0n . Also,
we can verify that p̄T x̄ ≤ 14 p̄2 + x̄2 , p̄T (PΩ (Δ) − x̄ −
x∗ ) ≤ 14 p̄2 + PΩ (Δ) − x̄ − x∗ 2 , and
(ρ + 1)ανCx̄r ≤

(ρ + 1)λmin (P ) − 1
x̄2 +
2

2(ρ + 1)2 α2 ν 2 C2
r2 .
(ρ + 1)λmin (P ) − 1
Thus, V̇1 is bounded by
V̇1 ≤ − ρPΩ (Δ) − x̄ − x∗ 2 +

2((ρ + 1)αν)2 C2
r2
(ρ + 1)λmin (P ) − 1

1
− p̄2 − (ρ + 1)λmin (P )Proxαf 2 (Υ) − x̄ − x∗ 2
4
+ (ρ + 1)2 z̄⊥ 2 − ρz̄T (Lβ ⊗In )z̄
(ρ + 1)λmin (P ) − 1
x̄2 .
(26)
2
√
2σ 2
N

( ρβij − √
ρ)
Step II: Consider V2 = N
, where
i=1
j=1,j =i
4γij
−

2

σ > (ρ+1)
2λ2 . Then, the derivative V̇2 along the trajectory βij is
V̇2 =

N 
N


N 
N


ρ
βij aij z̄i − z̄j 2 − σ
aij z̄i − z̄j 2
2 i=1 j=1
i=1 j=1

= ρz̄T (Lβ ⊗ In )z̄ − 2σz̄T (L ⊗ In )z̄

1459

≤ ρz̄T (Lβ ⊗ In )z̄ − 2σλ2 z̄⊥ 2

(27)

where the last inequality is based on (1) and z̄T (L ⊗ In )z̄ =
(z̄⊥ )T (L ⊗ In )z̄⊥ ≥ λ2 z̄⊥ 2 .
Step III: Since S + KC is Herwitz, there is a positive matrix
Q such that

2(ρ + 1)2 α2 ν 2 C2
T
+ 1 Im
Q(S + KC) + (S + KC) Q = −
(ρ + 1)λmin (P ) − 1

T
with m = N
i=1 mi . Taking V3 = r Qr and computing its
derivative with respect to ṙ yields

2(ρ + 1)2 α2 ν 2 C2
+ 1 r2 .
V̇3 = −
(28)
(ρ + 1)λmin (P ) − 1
From (26)–(28), the derivative of V = V1 + V2 + V3 with
respect to system (15) is given by
(ρ + 1)λmin (P ) − 1
x̄2
2
1
− (ρ + 1)λmin (P )Proxαf 2 (Υ) − x̄ − x∗ 2 − p̄2
4

V̇ ≤ −ρPΩ (Δ) − x̄ − x∗ 2 −

− (2σλ2 − (ρ + 1)2 )z̄⊥ 2 − r2
≤ 0.

(29)

It is not difficult to find V is radially unbounded with regard to col(x̄, ȳ, z̄, p̄, r) and βij . Together with (29), it follows that col(x̄, ȳ, z̄, p̄, r) and βij are bounded. Note that βij
is monotonously nondecreasing. Thus, limt→∞ βij exists. In
the set {V̇ = 0}, we have x̄ = 0N n . By LaSalle’s invariance
principle [49], we have limt→∞ x̄ = 0N n . This implies that x
approaches x∗ as t → ∞. In other words, the conclusion in
Theorem 1 follows.

V. PROXIMAL ALGORITHM WITH EVENT-TRIGGERED
COMMUNICATION
In Section IV, the proximal algorithm (9) depends on
continuous-time communication among agents. However, communication is only allowed at discrete-time instants in some
practical systems such as digital networks with limited channel
capacity. It is well known that the event-based communication
mechanism provides a useful means in avoiding continuous-time
communication [42], [44]. In [9] and [43], distributed optimal
coordination based on event-based communication schemes was
studied, where the agents minimize the sum of the local objective
functions.
Motivated by [9] and [43], for agent i ∈ V, we revisit (9) under
the following event-based communication strategy:


ẋi = PΩi xi − α∇fi1 (xi ) − zi + yi − xi
+ Ci (Ki xi − vi + wi )
ẏi = Proxαfi2 (xi − yi ) − xi + Ci (Ki xi − vi + wi )
żi = xi − bi −

N


βij aij (ẑi − ẑj ) − pi

j=1



+ PΩi xi − α∇fi1 (xi ) − zi + yi − xi

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ṗi =

N


ȳ˙ = Proxαf 2 (x̄ + x∗ − ȳ − y∗ ) − x̄ − x∗ + Cr

z̄˙ = x̄ − (Lβ ⊗ In )z̃ − p̄ + PΩ x̄ + x∗ − α∇f 1 (x̄ + x∗ )

βij aij (ẑi − ẑj ), pi (0) = 0n

j=1

− z̄ − z∗ + ȳ + y∗ ) − x̄ − x∗

β̇ij = γij aij ẑi − ẑj 2

p̄˙ = (Lβ ⊗ In )z̃

v̇i = − Si (Ki xi − vi )
+ Ki (PΩi (xi − α∇fi1 (xi ) − zi + yi ) − xi )

(30)

where ẑi (t) = zi (tik ) is the latest known state of agent i for all
t ∈ [tik , tik+1 ); tik and {tik }k∈N denote the kth triggering time
instant and the triggering time sequence of agent i, respectively.
For agent i ∈ V, the measurement error between ẑi and zi (t)
is denoted as ei = ẑi − zi . We consider the following eventtriggered condition:
N


N

1
βij aij ei  ≥
aij ẑi − ẑj 2 + θe−ιt
ςi ei  + ςi
4
j=1
j=1
2

2

(31)
where ςi ∈ R>0 , θ ∈ R>0 and ι ∈ R>0 . The triggering time
sequence {tik }k∈N is defined as tik+1 = inf{t > tik |(31) holds}.
Once the event-triggered condition (31) holds, agent i broadcasts
its current state zi to its neighbors and updates its controller
using its current state zi and the latest received states from its
neighbors. Meanwhile, ei is reset to zero.
Remark 7: In (31), the exponential decay term θe−ιt plays
a key role in avoiding Zeno behavior. Such a mechanism has
been widely adopted in event-based control laws such as [9] and
[42]–[44].
With x, y, z, p, r, and ẑ = col(ẑ1 , ẑ2 , . . . , ẑN ), algorithm
(30) is rewritten as the following equivalent form:


ẋ = PΩ x − α∇f 1 (x) − z + y − x + Cr

β̇ij = γij aij z̃i − z̃j 2
ṙ = (S + KC)r.

To obtain the conclusion in Theorem 2, we consider the
Lyapunov function V̄ = V1 + 12 V2 + V3 . Following the details
similar to the proof of Theorem 1, the derivative of V̄˙ with respect
to system (33) can be bounded by
(ρ + 1)λmin (P ) − 1
V̄˙ ≤ − ρPΩ (Δ) − x̄ − x∗ 2 −
x̄2
2
1
− (ρ + 1)λmin (P )Proxαf 2 (Υ) − x̄ − x∗ 2 − p̄2
2
− ρz̄T (Lβ ⊗ In )z̃ + (ρ + 1)z̄T p̄ − r2
ρ
+ z̃T (Lβ ⊗ In )z̃
2
N

N

σ 
σ
−
aij z̃i − z̃j 2 − z̃T (L ⊗ In )z̃.
4 i=1 j=1
2
−ρz̄T (Lβ ⊗In )z̃ = ρz̃T (Lβ ⊗In )e − ρz̃T (Lβ ⊗In )z̃
ρ
ρ
≤ eT (Lβ ⊗In )e − z̃T (Lβ ⊗In )z̃
2
2
eT (Lβ ⊗In )e =

≤

β̇ij = γij aij ẑi − ẑj 

βij aij ei 2 +

N 
N


N 
N


βij aij ej 2

i=1 j=1

βij aij ei 2 .

(36)

i=1 j=1

(32)

The following result illustrates that algorithm (30) under
the event-triggered condition (31) can ensure that system (2)
converges to the optimal solution of problem (4). Meanwhile,
the disturbances are rejected and the Zeno behavior is ruled out.
Theorem 2: Suppose that Assumptions 1–4 hold. Under
algorithm (30) with the event-triggered condition (31), system
(2) asymptotically converges to the optimal solution of problem
(4) while rejecting the disturbances generated by system (3).
Furthermore, the Zeno behavior is excluded.
Proof: Setting z̃ = ẑ − z∗ , e = col(e1 , e2 , . . . , eN ), and using the variables x̄, ȳ, z̄, z̃, p̄, and r can convert system (32)
into


x̄˙ = PΩ x̄ + x∗ − α∇f 1 (x̄ + x∗ ) − z̄ − z∗ + ȳ + y∗
− x̄ − x∗ + Cr

N

N 
N


=2

2

ṙ = (S + KC)r.

N

(35)

1 
βij aij ei − ej 2
2 i=1 j=1

i=1 j=1

ṗ = (Lβ ⊗ In )ẑ, p(0) = 0N n

(34)

It is readily verified that

ẏ = Proxαf 2 (x − y) − x + Cr
ż = x − b − (Lβ ⊗ In )ẑ − p


+ PΩ x − α∇f 1 (x) − z + y − x

(33)

Let z̃ = z̃⊥ + z̃ , where z̃ = 1N ⊗ z̃0 and (z̃⊥ )T z̃ = 0.
According to e = z̃ − z̄, (1TN ⊗ In )p̄ = 0n , and (1), we arrive
at
(ρ + 1)z̄T p̄ = (ρ + 1)z̃T p̄ − (ρ + 1)eT p̄
= (ρ + 1)(z̃⊥ )T p̄ − (ρ + 1)eT p̄
1
≤ 2(ρ + 1)2 z̃⊥ 2 + 2(ρ + 1)2 e2 + p̄2
4
(37)
z̃T (L ⊗ In )z̃ = (z̃⊥ )T (L ⊗ In )z̃⊥ ≥ λ2 z̃⊥ 2 .

(38)

By substituting (35)–(38) into (34), using the event-triggered
2
2
, maxi∈V 2(ρ+1)
},
condition (31) and setting σ ≥ max{ 4(ρ+1)
λ2
ςi
we have
V̇ ≤ − ρPΩ (Δ) − x̄ − x∗ 2 −

(ρ + 1)λmin (P ) − 1
x̄2
2

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ZHU et al.: NONSMOOTH RESOURCE ALLOCATION OF MULTIAGENT SYSTEMS

1461

1
− (ρ + 1)λmin (P )Proxαf 2 (Υ) − x̄ − x∗ 2 − p̄2
4

σλ2
− 2(ρ + 1)2 z̃⊥ 2
− r2 −
2
+ 2(ρ + 1)2

N


ei 2

Fig. 1.

Undirected communication topology among agents.

i=1

+ρ

N 
N


βij aij ei 2 −

i=1 j=1

N

N

σ 
aij z̃i − z̃j 2
4 i=1 j=1

Accordingly, the event-triggered condition (31) holds when

ϑi + ϑβ 

(ρ + 1)λmin (P ) − 1
x̄2
≤ − ρPΩ (Δ) − x̄ − x  −
2
∗ 2

t

From (39), we can deduce V (t) ≤ V (0) + N σθ 0 e−ιτ dτ .
Thus, we get V (∞) ≤ V (0) + Nισθ . In other words,
V (∞) is bounded. Together with (39), we have
(ρ+1)λmin (P )−1 ∞
N σθ
2
0 x̄ dt ≤ V (0) − V (∞)+ ι . This implies
2
∞

that 0 x̄2 dt is bounded by combining with (ρ +
1)λmin (P ) − 1 > 0. Hence, x̄ tends to zero as t → ∞.
Equivalently, x converges to the optimal solution x∗ .
Now, we show that the Zeno behavior does not happen in the
closed-loop system (30). In other words, we just have to prove
that the time interval tik+1 − tik between two adjacent triggering
instants tik and tik+1 is strictly positive.
Similar to the analysis given in [43], we demonstrate that
tik+1 − tik is strictly positive by discussing the following two
cases.
Case I: Agent i’s neighbors are all not be triggered between
tik and tik+1 . For t ∈ [tik , tik+1 ), we have ei (tik ) = 0n and
ėi = xi − bi −

N


βij aij (ẑi − ẑj ) − pi

j=1



+ PΩi xi − α∇fi1 (xi ) − zi + yi − xi .
Thus, we have
t

ei (t) =

tik

−



(PΩi xi − α∇fi1 (xi ) − zi + yi − bi − pi )dτ
N
t
tik j=1

βij aij (ẑi − ẑj )dτ.

(40)

By combining the boundedness of V with its radial unboundedness with respect to col(x, y, z, p) and βij , we can get that x,
y, z, p, and βij are also bounded. Thus, there are ϑi ∈ R>0 and
ϑβ ∈ R>0 such that PΩi (xi − α∇fi1 (xi ) − zi + yi ) − bi −
pi  ≤ ϑi and βij ≤ ϑβ . From (40), it follows that

ei  ≤

ϑi + ϑβ 

N

j=1

aij (ẑi − ẑj ) (t − tik ).

2

(t − tik )2

j=1

= 4
(39)

aij (ẑi − ẑj )

N
1

− (ρ + 1)λmin (P )Proxαf 2 (Υ) − x̄ − x∗ 2
1
− p̄2 + N σθe−ιt .
4

N


2
−ιt
i=1 aij ẑi − ẑj  + θe
.
N
ςi (1 + j=1 βij aij )

(41)

It can be found that the right-hand side of (41) is positive
when agent i does not agree with its neighbors. This implies
that tik+1 − tik is strictly positive.
Case II: There is at least a neighbor of agent i that is triggered. Without loss of generality, agent j ∈ Ni is assumed to be
triggered at tjk after tik for agent i. Then, we have tjk > tik . In
this regard, tik+1 − tik is also strictly positive.
To sum up, the Zeno behavior can be excluded with (31). This
completes the proof.

Remark 8: From the convergence analysis provided above, it
follows that the asymptotical convergence of algorithms (9) and
(30) can be ensured for any γij > 0. Generally speaking, larger
γij implies a faster convergence rate but will lead to an increased
communication frequency simultaneously. Such an observation
can be verified in the numerical simulations given in the next
section.
VI. SIMULATIONS
This section illustrates the results in Theorems 1 and 2 by a
numerical example.
We consider the nonsmooth resource allocation problem (4)
with N = 4 agents. Each agent’s individual information is listed
in Table I [27]. The communication among agents is depicted in
Fig. 1, and the corresponding Laplacian matrix is given by
⎛
⎞
2 −1 0 −1
⎜ −1 2 −1 0 ⎟
⎟
L=⎜
⎝ 0 −1 2 −1 ⎠ .
−1 0 −1 2
π

sin(t + 4 )
The disturbance di = ( ϕϕii cos(t
+ π ) ) is generated by system
4

√

2ϕ

i
0 1
1 0
√2
(3) with Si = ( −1
0 ), Ci = ( 0 1 ) and wi (0) = ( 2ϕi ), where
2 √
√
√
√
ϕ1 = ( 44√22 ), ϕ2 = ( 22√22 ), ϕ3 = ( 33√22 ), and ϕ4 = ( 99√22 ).
Let vi ∈ R2 for i = 1, 2, 3, 4. In light of the definition of

proximal mapping, we can figure out

T
1 −(2,2) )
, v1 − (2, 2)T  > α
Prox1αf 2 (v1 ) = v1 − α(v
v1 −(2,2)T 
1
1
Proxαf 2 (v1 ) = (2, 2)T ,
v1 − (2, 2)T  ≤ α
1

and Proxβfi2 (vi ) = vi for i = 2, 3, 4.

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1462

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TABLE I
AGENTS’ INDIVIDUAL INFORMATION

VII. CONCLUSION

Fig. 2. Profile of x under algorithm (9) for different γij . (a) Case with
γij = 0.001. (b) Case with γij = 0.005.

We have studied the non-smooth resource allocation problem
for a class of multiagent systems with external disturbances.
Specifically, we have proposed some new classes of proximal
algorithms under continuous-time communication and eventbased communication, respectively. The theoretical results have
shown that the multiagent systems equipped with the designed
algorithms can find the optimal solution of the considered problem in the presence of external disturbances. Moreover, we have
demonstrated that the proximal algorithm based on the eventbased communication scheme can avoid the Zeno behavior. The
numerical simulations have verified the established theoretical
results. Future work will focus on extending the present algorithms to solve the resource allocation problems under directed
communication topologies and cyber-physical attacks.
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Fig. 3. Profile of x under algorithm (30) for different γij . (a) Case with
γij = 0.001. (b) Case with γij = 0.005.

Fig. 4. Triggering time sequence of each agent for different γij . (a)
Case with γij = 0.001. (b) Case with γij = 0.005.

In simulations, α = 1, θ = 3, ι = 0.1, and ςi = 5 are set.
Also, the step size is chosen as 0.01. To illustrate the effect
of γij on convergence rate and triggering frequency, γij =
0.001 and γij = 0.005 are considered separately. In these
two cases, the trajectory of the agents’ decision variable x =
col(x1 , x2 , x3 , x4 ) and the triggering time instants for each
agent are depicted in Figs. 2– 4. It can be seen that in either case,
x converges to the optimal solution of the nonsmooth resource
allocation problem given in Table I. Moreover, it can be found
that larger γij implies a faster convergence rate but meanwhile
increases the communication frequency. This is accordance with
the observation given in Remark 8.

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Yanan Zhu (Student Member, IEEE) received
the Ph.D. degree in mathematics from the
School of Mathematics, Southeast University,
Nanjing, China, in 2019.
She is currently a Lecturer with the School
of Automation, Nanjing University of Information
Science and Technology, Nanjing. In 2017, she
was a Research Assistant with the City University of Hong Kong, Hong Kong. From 2017
to 2018, she was a Visiting Scholar with the
University of California, Riverside, CA, USA. In
2018 and 2019, she was a Visiting Student with the Royal Melbourne
Institute of Technology (RMIT) University, Melbourne, VIC, Australia. Her
research interests include multiagent systems, distributed optimization,
and game theory.

Guanghui Wen (Senior Member, IEEE) received the Ph.D. degree in mechanical systems and control from Peking University, Beijing,
China, in 2012.
He is currently a Young Endowed Chair Professor with the Department of Systems Science,
Southeast University, Nanjing, China. He has
been named a Highly Cited Researcher by Clarivate Analytics since 2018. His current research
interests include autonomous intelligent systems, complex networked systems, distributed
control and optimization, resilient control, and distributed reinforcement
learning.
Dr. Wen was awarded the National Natural Science Fund for Excellent Young Scholars, in 2017. Moreover, he was a recipient of the Australian Research Council Discovery Early Career Researcher Award in
2018, and a recipient of the Asia Pacific Neural Network Society Young
Researcher Award in 2019. He is a Reviewer for American Mathematical
Review and an active reviewer for many journals. He is an Associate
Editor for the IEEE JOURNAL OF EMERGING AND SELECTED TOPICS IN
INDUSTRIAL ELECTRONICS, the IEEE TRANSACTIONS ON SYSTEMS, MAN
AND CYBERNETICS: SYSTEMS, the IEEE OPEN JOURNAL OF THE INDUSTRIAL
ELECTRONICS SOCIETY, and the Asian Journal of Control.

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1464

IEEE TRANSACTIONS ON CONTROL OF NETWORK SYSTEMS, VOL. 8, NO. 3, SEPTEMBER 2021

Wenwu Yu (Senior Member, IEEE) received the
B.Sc. degree in information and computing science and the M.Sc. degree in applied mathematics from the Department of Mathematics,
Southeast University, Nanjing, China, in 2004
and 2007, respectively, and the Ph.D. degree in
electronic engineering from the Department of
Electronic Engineering, City University of Hong
Kong, Hong Kong, in 2010.
He is currently the Founding Director of the
Laboratory of Cooperative Control of Complex
Systems and the Deputy Associate Director of the Jiangsu Provincial
Key Laboratory of Networked Collective Intelligence, an Associate Dean
of the School of Mathematics, and a Full Professor with the Young
Endowed Chair Honor in Southeast University. He held several visiting
positions in Australia, China, Germany, Italy, The Netherlands, and USA.
He was listed by Clarivate Analytics/Thomson Reuters Highly Cited
Researchers in Engineering in 2014–2020. He authored or coauthored
about 100 IEEE TRANSACTIONS journal papers with more than 10 000
citations. His research interests include multiagent systems, complex
networks and systems, disturbance control, distributed optimization,
neural networks, game theory, cyberspace security, smart grids, intelligent transportation systems, and big-data analysis.
Dr. Yu is an Editorial Board Member of several flag journals, including the IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS II: EXPRESS
BRIEFS, the IEEE TRANSACTIONS ON INDUSTRIAL INFORMATICS, the IEEE
TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS: SYSTEMS, the Science China Information Sciences, and the Science China Technological
Sciences. He was the recipient of the Second Prize of State Natural
Science Award of China, in 2016.

Xinghuo Yu (Fellow, IEEE) received the B.Eng.
and M.Eng. degrees in electrical engineering
from the University of Science and Technology
of China, Hefei, China, in 1982 and 1984, respectively, and the Ph.D. degree in control science and engineering from Southeast University, Nanjing, China, in 1988.
He is currently an Associate Deputy ViceChancellor and Distinguished Professor with
Royal Melbourne Institute of Technology (RMIT)
University, Melbourne, VIC, Australia. His current research interests include variable structure and nonlinear control,
complex and intelligent systems, and smart energy systems.
Prof. Yu was a recipient of the 2018 M. A. Sargent Medal of Engineers
Australia, the 2013 Dr.-Ing. Eugene Mittelmann Achievement Award
of the IEEE Industrial Electronics Society, and a number of awards
and honors for his contributions. He was the President of the IEEE
Industrial Electronics Society from 2018 to 2019. He was an Associate
Editor for the IEEE TRANSACTIONS ON AUTOMATIC CONTROL, the IEEE
TRANSACTIONS ON INDUSTRIAL ELECTRONICS, the IEEE TRANSACTIONS ON
INDUSTRIAL INFORMATICS, and the IEEE TRANSACTIONS ON CIRCUITS AND
SYSTEMS I: REGULAR PAPERS.

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