IEEE Control Systems Letters paper presented at
2023 American Control Conference (ACC)
San Diego, CA, USA. May 31 - June 2, 2023

An event-triggered distributed nonsmooth resource allocation algorithm
for second-order multi-agent systems
Xiasheng Shi1 , Lifu Ding2 and Zhiyun Lin3,∗
projection operator, a class of initialization-free distributed
algorithms is designed for the economic dispatch or resource
allocation problem [5]–[8]. As a branch of the constrained
optimization problem, the penalty function method (interior
point method) is introduced [9], where a suboptimal solution
is obtained within a predeﬁned time using a time-based
generator to establish an adjustable control parameter. On
the other hand, the outer-point method is considered in [10]–
[12], in which the penalty factor is adaptively increased. In
recent years, more and more systems involve both distributed
optimization algorithm design and individual agent’s dynamics. One typical example is the double-integrator MAS [13],
[14] that may be more proper to describe the mathematical
model of mobile robots, four-rotor UAVs, hovercraft, etc.
However, the above-mentioned algorithms are generally designed for ﬁrst-order MASs rather than second-order MASs.
Therefore, studying distributed optimization of second-order
MASs has essential theoretical and practical signiﬁcance.
Different from ﬁrst-order MASs, the state/position 𝑥𝑖 of
second-order MASs is driven by the velocity 𝑣 𝑖 instead of
the control input 𝑢 𝑖 . Therefore, the existing projection-based
methods for ﬁrst-order MASs cannot be directly extended
to handle distributed optimization over second-order MASs.
In particular, it is challenging to deal with the local constraints for distributed optimization over second-order MASs.
Based on the tracking deviation control idea, a differential
projection-based optimization algorithm is designed in [15].
In addition, a penalty function with a constant penalty factor
is used in [16]. Unfortunately, to guarantee optimality, the
initial state in the above two algorithms must satisfy the
local constraints. Moreover, continuous-time communication
among agents is ideal and impractical. Therefore, some
recent studies adopt the idea of event-triggered control due
to its on-demand communication property [17]–[19].
Motivated by recent progress, this brief focuses on the design and convergence analysis of event-triggered distributed
optimization over second-order MASs by using the adaptive
control approach. The contributions are summarized as follows: 1) Compared with the continuous-time communication
algorithm [15], [16], a novel initialization-free distributed
nonsmooth resource allocation algorithm with an eventtriggered communication scheme is designed to reduce the
communication burden. 2) Compared with the differential
projection operator in [15], the local constraints of each agent
can be easily solved by an adaptive control scheme.
The remainder of this brief is organized as follows. A
network description and problem formulation are provided
in Section II. Section III presents the algorithm design and

Abstract— This brief aims to solve the distributed resource
allocation problem for second-order multi-agent systems over
an undirected network, where the global objective function
is strongly convex but not necessarily Lipschitz continuous
for its subgradient. The resource state is subject to a global
equality constraint and several local inequality constraints with
a convex function. Unlike existing tracking deviation control
strategies, a dynamic event-triggered and initialization-free
distributed resource allocation algorithm is proposed to reduce
the communication burden among agents. The local constraints
are solved by an adaptive control approach based on the
Karush-Kuhn-Tucker condition. It is shown that the proposed
algorithm asymptotically converges to the optimal solution by
using the set-valued LaSalle’s invariance principle. Moreover,
it is guaranteed that Zeno behavior is ruled out for any agent.
Finally, a simulation example shows the effectiveness of the
proposed algorithm.
Index Terms— Resource allocation, event-triggered, adaptive,
initialization-free

I. INTRODUCTION
Recently, cooperative optimization (especially distributed
resource allocation) for multi-agent systems (MASs) has
received extensive attention owing to its wide applications
[1] in energy internet, wireless networks, robot clusters,
etc. Compared to classical centralized methods, distributed
schemes exhibit higher scalability and ﬂexibility. Distributed
resource allocation focuses on designing a distributed optimization algorithm via exchanging information with only
neighbors to minimize the global objective function by
allocating appropriate quantity of resources over multiple
agents [2]. By taking into consideration of continuous-time
dynamics such as in cyber-physical systems (CPSs), the design of continuous-time distributed optimization algorithms
has received increasing interest in the ﬁeld [3]. Thus, tasks
can be collaboratively executed by MASs with dynamics.
Inspired by the primal-dual control concept [4], several
excellent algorithms have been developed. Based on the
This work was supported in partly by the National Natural Science
Foundation of China(62173118), partly by the Foundation for Innovative
Research Groups of the National Science of China(61921004), and partly
by the Laboratory of Intelligent Control for Industrial Equipment, Ministry
of Education, Dalian University of Technology(LICO2022TB02).
1 School of Artiﬁcial Intelligence, Anhui University, Hefei, China and
Key Laboratory of Intelligent Control and Optimization for Industrial
Equipment, Ministry of Education, Dalian University of Technology,

shixiasheng@zju.edu.cn
2 College of Electrical Engineering, Zhejiang University, Hangzhou,
China, 3140104040@zju.edu.cn
3 Department of Electronic and Electrical Engineering, Southern University of Science and Technology, and Peng Cheng Laboratory, Shenzhen,
China, linzy@sustech.edu.cn (∗ Corresponding author: Zhiyun
Lin)

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detailed convergence analysis. Simulation and conclusion are
shown in Section IV and V, respectively.
Notation: 𝜕 𝑓 (·) denotes subdifferential of the function 𝑓 .
ℝ denotes the real number set. 1𝑛 , 0𝑛 , and 𝐼𝑛 are all one, all
zero vectors, and an identity matrix with 𝑛 dimensions. The
transpose symbol is represented by T . For a column vector
𝑥 ∈ R𝑛 , (𝑥) + = [max{𝑥1 , 0}, ..., max{𝑥 𝑛 , 0}] T . For a vector
𝑥 ∈ R𝑛 , 𝑥  0𝑛 means that each component of vector 𝑥 is less
than or equal to zero. The symbol ,  and ≺ have similar
deﬁnitions. For column vectors 𝑥 1 , ..., 𝑥 𝑛 , col(𝑥1 , ..., 𝑥 𝑛 ) =
[𝑥1T , ..., 𝑥 𝑛T ] T is the aggregate vector.

dynamics [15]. The dynamics of each agent considered in
this brief are described by
𝑥¤𝑖 (𝑡) = 𝑣 𝑖 (𝑡), 𝑣¤ 𝑖 (𝑡) = 𝑢 𝑖 (𝑡),

where 𝑥 𝑖 (𝑡) ∈ R𝑠 , 𝑣 𝑖 (𝑡) ∈ R𝑠 , and 𝑢 𝑖 (𝑡) ∈ R𝑠 are position,
velocity and control input of agent 𝑖, respectively. For convenience, the implicit time 𝑡 has been omitted in the following.
It is noted that 𝑥𝑖 is driven by the velocity 𝑣 𝑖 instead of
the control input 𝑢 𝑖 . Therefore, it is difﬁcult to handle the
local constraint 𝑔𝑖 (𝑥𝑖 )  0𝑚𝑖 in problem (1). To this end, the
following proper assumption is proposed.
Assumption 1: 𝑓 (𝑥) is 𝜇−strongly convex and 𝑔𝑖, 𝑗 , ∀𝑖 ∈
V, 𝑗 = 1, .., 𝑚 𝑖 is convex. Moreover, at least one point 𝑥˜
∑𝑛
∑𝑛
𝑑𝑖 and inequality
𝑥˜ = 𝑖=1
exists such that the equality 𝑖=1
∑𝑛 𝑖
𝑔( 𝑥)
˜ ≺ 0𝑚 holds, where 𝑚 = 𝑖=1 𝑚 𝑖 and 𝑥˜ = [𝑥˜1 , 𝑥˜2 , ..., 𝑥˜ 𝑛 ].
Remark 1: In this brief, 𝑓 (𝑥) is assumed to be strongly
convex but not necessary to have Lipschitz continuous gradients. This assumption is the same as the ones in [6], [8],
[10] but is weaker than the ones made in [9], [15], [16],
[18]. To solve the local constraints with a continuous-time
communication mechanism, differential projection operator
and penalty function method are used in [15], [16]. However,
the computation complexities of these two methods are high.
Besides, the algorithm with a ﬁxed penalty factor in [16] may
oscillate at the stable point, thus leading to low accuracy.
Lemma 2: [11] Under Assumption 1, 𝑥 ∗ is the optimal solution to problem (1) iff there exists a real vector
(𝑥 ∗ , 𝑦 ∗ , 𝜏 ∗ ) ∈ R𝑛𝑠 × R𝑠 × R𝑚 such that the following results
are satisﬁed:

II. P ROBLEM FORMULATION
For the distributed resource allocation problem, it aims
to ﬁnd the optimal solution through information exchanging
among neighboring agents. For convenience, the communication network is represented by a time-invariant graph
𝐺 = (V, 𝐸, A), with a node set V = {1, 2, ..., 𝑛}, and edge
set 𝐸 ⊂ V × V, and an adjacency matrix A = [𝑎 𝑖 𝑗 ] 𝑛×𝑛 ∈
R𝑛×𝑛 . The element of A is deﬁned as 𝑎 𝑖 𝑗 = 1 if there
exists a communication link (𝑖, 𝑗); Otherwise, 𝑎 𝑖 𝑗 = 0. The
communication link is undirected if the information of any
two agents is directly accessible from each other. Moreover,
it is assumed that 𝑎 𝑖𝑖 = 0, ∀𝑖 ∈ V and 𝑎 𝑖 𝑗 = 𝑎 𝑗𝑖 , ∀𝑖, 𝑗 ∈
V for an undirected graph. For the undirected graph 𝐺,
Laplacian matrix 𝐿 = [𝐿 𝑖 𝑗 ] 𝑛×𝑛 ∈ R𝑛×𝑛 is deﬁned with
∑
𝐿 𝑖 𝑗 = −𝑎 𝑖 𝑗 , 𝐿 𝑖𝑖 = 𝑛𝑗=1 𝑎 𝑖 𝑗 . In addition, the eigenvalues
of 𝐿 for a connected undirected graph 𝐺 are ordered as
0 = 𝜌1 < 𝜌2 <, . . . , < 𝜌 𝑛 .
Lemma 1: [20] For an undirected network 𝐺, there exists
an orthogonal matrix 𝑄 = [ √1𝑛 1𝑛 , 𝑅] ∈ R𝑛×𝑛 to support the
following equalities: 𝑅Λ1−1 𝑅 T 𝐿 = 𝐿𝑅Λ1−1 𝑅 T = 𝐾𝑛 , where
𝐾𝑛 = 𝐼𝑛 − 𝑛1 1𝑛 1T𝑛 and Λ1 = diag{𝜌2 , ..., 𝜌 𝑛 }.
In this brief, the resource allocation problem can be
summarized as follows:
min 𝑓 (𝑥) =
𝑠.𝑡.

𝑛
∑
𝑖=1

𝑛
∑

𝑥𝑖 =

𝜁 ∗ +(𝜂∗ ) T 𝜏 ∗ − 1𝑛 ⊗ 𝑦 ∗ = 0𝑛𝑠 ,
𝑛
𝑛
∑
∑
𝑑𝑖 ,
𝑥 𝑖∗ =
𝑖=1

𝑛
∑

(3)

𝑖=1

𝑔(𝑥 ∗ ) 0𝑚 , 𝜏 ∗  0𝑚 , (𝜏 ∗ ) T 𝑔(𝑥 ∗ ) = 0,
where 𝜁 ∗ ∈ 𝜕 𝑓 (𝑥 ∗ ), 𝜂∗ ∈ 𝜕𝑔(𝑥 ∗ ) and 𝑥 ∗ = [𝑥 1∗ , 𝑥 2∗ , ...., 𝑥 𝑛∗ ].
Remark 2: Under Assumption 1, the optimal solution of
problem (1) is unique. Besides, if at least one local objective
function 𝑓𝑖 (𝑥𝑖 ) is smooth at the optimal point 𝑥𝑖∗ , then the
optimal Lagrangian multiplier 𝑦 ∗ is unique. Otherwise, if all
the local objective functions are nonsmooth at the optimal
point, then 𝑦 ∗ may not be unique (that is, the element of the
subgradient intersection is not unique).
Now, we introduce some concepts and propositions about
differential inclusion to facilitate the following convergence
analysis. Considering the following differential inclusion
system:
𝜚¤ ∈ F ( 𝜚(𝑡)),
(4)

𝑓𝑖 (𝑥 𝑖 ),

𝑖=1

(2)

(1)
𝑑𝑖 , 𝑔𝑖 (𝑥𝑖 )  0𝑚𝑖 , ∀𝑖 ∈ V,

𝑖=1

where 𝑔𝑖 (𝑥𝑖 ) = [𝑔𝑖,1 (𝑥𝑖 ), ..., 𝑔𝑖,𝑚𝑖 (𝑥 𝑖 )] T : R𝑠 → R𝑚𝑖 and
𝑚 𝑖 denotes the number of inequality constraints for agent
𝑖. 𝑓𝑖 (𝑥𝑖 ) : R𝑠 → R is the local objective function and
the global objective function 𝑓 (𝑥) is strongly convex but not
necessary to be smooth. 𝑑𝑖 ∈ R𝑠 denotes the local resource
demand. For example, the economic dispatch problem in
a smart grid aims at distributing the total power demand
∑𝑛
𝑖=1 𝑑 𝑖 among generators to minimize the global operating
cost under certain security constraints, where 𝑑𝑖 , as the power
demand of the user side, can be predicted in advance through
neural networks
and∑ other methods. The global equality
∑𝑛
𝑛
constraint 𝑖=1
𝑥 𝑖 = 𝑖=1
𝑑𝑖 indicates that the electric energy
production capacity on the generation side should be exactly
equal to the consumption on the user side. Besides, the
turbine-generator system can be described as a second-order

where F : R𝑛 → R𝑛 denotes a set-valued map. If the map
𝜚 : [0, 𝑇] → R𝑛 satisﬁes (4) for almost all 𝑡 ∈ [0, 𝑇], then
𝜚 is the Caratheodory solution of the system (4). Moreover,
the set-valued Lie derivative of (4) is denoted by L F 𝑉 =
{𝜍 T ∇𝑉 |𝜍 ∈ F ( 𝜚)} for a continuously differentiable function
𝑉 : R𝑛 → R. The following set-valued LaSalle invariance
principle is used to support the convergence analysis of our
algorithm.
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∑
𝑎
where 𝜎𝑖 ∈ [0, 1), 𝜃 𝑖 > 0, 𝑒 𝑖 = 𝜆ˆ 𝑖 − 𝜆𝑖 , 𝑞ˆ 𝑖 = 𝑛𝑗=1 2𝑖 𝑗 (𝜆ˆ 𝑖 −
2
𝜆ˆ 𝑗 ) , 𝜒𝑖 ∈ R, and the dynamic of variable 𝜒𝑖 (𝑡) is given by

Lemma 3: [22] For a continuously differentiable function
𝑉 : R𝑛 → R and a strongly positively invariant set 𝑆 for system (4), if the Lie derivative satisﬁes max L F 𝑉 ( 𝜚) ≤ 0, ∀𝜚 ∈
𝑆, then every solution 𝑥(𝑡) approaches the largest weakly
positively invariant set 𝑆 ∩ { 𝜚 ∈ R𝑛 |0 ∈ max L F 𝑉 ( 𝜚)}.

𝜒¤ 𝑖 = −𝛾𝑖 𝜒𝑖 + 𝛿𝑖 (

III. A LGORITHM DESIGN AND CONVERGENCE ANALYSIS

(8)

in which the control parameters satisfy 𝛾𝑖 > 1−𝜃𝑖𝛿𝑖 , 𝛿𝑖 ∈
[0, 1] and 𝜒𝑖 (0) > 0. Substituting (7) into (8), we obtain 𝜒¤ 𝑖 ≥ −𝛾𝑖 𝜒𝑖 − 𝛿𝜃𝑖𝑖 𝜒𝑖 . Therefore, we have 𝜒𝑖 (𝑡) ≥
𝜒𝑖 (0)exp(− 𝛾𝑖 𝜃𝜃𝑖𝑖+ 𝛿𝑖 𝑡) ≥ 0, ∀𝑡 ≥ 0.
Remark 3: In contrast to the static state-based eventtriggered algorithm [18], a dynamic event-triggered optimization algorithm is designed for second-order MASs in
this brief, in which the threshold function includes both
information 𝜆 𝑗 and auxiliary variable 𝜒𝑖 . The equivalent form
𝑖
of (7) is given as k𝑒 𝑖 k 2 ≤ 𝐿2𝜎
𝑞ˆ 𝑖 + 𝜃𝑖 (𝐿4𝑖𝑖 +4) 𝜒𝑖 . Additionally,
𝑖𝑖 +4
if 𝜒𝑖 ≡ 0 or 𝛿𝑖 = 0, the mechanism (7) degenerates into static
state-based and time-based event-triggered mechanism in
[17] and [7], respectively. Thus, our proposed event-triggered
algorithm can be seen as a combination of the methods and
obtains a larger inter-event time.

A. Algorithm design
From Lemma 2, it can be seen that the global Lagrangian
multiplier 𝑦 ∗ plays an important role in ﬁnding the optimal
solution 𝑥 ∗ . Therefore, we design one local Lagrangian multiplier 𝜆𝑖 ∈ R𝑠 for each agent 𝑖. Based on the consensus idea
of MASs, all local Lagrangian multipliers 𝜆𝑖 cooperatively
achieve the global optimal Lagrangian multiplier 𝑦 ∗ through
information communication among agents. Based on the
intrinsic dynamics of each agent, the control input 𝑢 𝑖 , ∀𝑖 ∈ V
is designed as follows:
𝑥¤𝑖 = 𝑣 𝑖 , 𝑣¤ 𝑖 = 𝑢 𝑖 ,
𝑢 𝑖 = −𝑣 𝑖 − (𝛼𝜁𝑖 + 𝜂𝑖T (𝜏𝑖 + 𝑔𝑖 (𝑥 𝑖 + 𝑣 𝑖 )) + − 𝜆𝑖 ),
𝑛
∑
𝑎 𝑖 𝑗 (𝜆ˆ 𝑖 − 𝜆ˆ 𝑗 ) − 𝑧 𝑖 − 𝛽(𝑥𝑖 + 𝑣 𝑖 − 𝑑𝑖 ),
𝜆¤ 𝑖 = −

𝜎𝑖
𝐿 𝑖𝑖 +4
𝑞ˆ 𝑖 −
k𝑒 𝑖 k 2 ),
2
4

(5)

𝑗=1

𝑧¤𝑖 =

𝑛
∑

𝑎 𝑖 𝑗 (𝜆ˆ 𝑖 − 𝜆ˆ 𝑗 ),

B. Convergence analysis

𝑗=1

Now, we aim to claim that the state variable 𝑥, generated by algorithm (6), converges to the optimal solution of
problem (1). Besides, the Zeno behavior means an unlimited
accumulation of executions within ﬁnite time, and thus it
is not allowed in event-triggered control [21]. Therefore,
we will also prove that the Zeno behavior is excluded for
triggering condition (7) in this subsection.
In the following, the convergence proof consists of three
steps. Step I: We ﬁrst show that the variable 𝜏𝑖 𝑗 , 𝑗 =
1, 2, ..., 𝑚 𝑖 , ∀𝑖 ∈ V, is nonnegative. Step II: We show that the
state variable 𝑥 ∗ converges to the optimal solution of problem
(1) by using the set-valued LaSalle invariance principle. Step
III: We show that the Zeno behavior of (7) is excluded by
using the contradiction method.
Theorem 1: Suppose that the communication graph is
undirected and connected, Assumption 1 holds, and the
1
control parameters satisfy 𝛼 > 4𝜇
, 𝜇 > 𝛽 > 0. Then the
optimization algorithm (5) over second-order MASs with the
event-triggered control scheme (7) asymptotically converges
to the optimal solution of problem (1). Moreover, the Zeno
behavior never occurs.
Proof: Step I. We prove the nonnegative property of
the variable 𝜏𝑖 𝑗 , 𝑗 = 1, 2, ..., 𝑚 𝑖 , ∀𝑖 ∈ V in algorithm (5).
Case 1: 𝑔𝑖 𝑗 (𝑥𝑖 + 𝑣 𝑖 ) > 0. From the deﬁnition of (·) + , we have
−𝜏𝑖 𝑗 + (𝜏𝑖 𝑗 + 𝑔𝑖 𝑗 (𝑥 𝑖 + 𝑣 𝑖 )) + > 0. Then, 𝜏𝑖 grows until 𝑥𝑖 + 𝑣 𝑖
is pulled back to 𝑔𝑖 𝑗 (𝑥𝑖 + 𝑣 𝑖 ) ≤ 0. Case 2: 𝑔𝑖 𝑗 (𝑥𝑖 + 𝑣 𝑖 ) = 0.
𝜏𝑖 remains unchanged. Case 3: 𝑔𝑖 𝑗 (𝑥𝑖 + 𝑣 𝑖 ) < 0. Similarly,
we have −𝜏𝑖 𝑗 + (𝜏𝑖 𝑗 + 𝑔𝑖 𝑗 (𝑥𝑖 + 𝑣 𝑖 )) + < 0. 𝜏𝑖 𝑗 decreases until
𝜏𝑖 𝑗 = 0. In conclusion, it is obtained that 𝜏𝑖 𝑗 (𝑡) ≥ 0, ∀𝑡 ≥ 0
under the initial value 𝜏𝑖 𝑗 (0) ≥ 0.
Step II. Let (𝑥 ∗ , 𝑣 ∗ , 𝜆∗ , 𝑧 ∗ , 𝜏 ∗ ) be the equilibrium point of

𝜏¤𝑖 = −𝜏𝑖 + (𝜏𝑖 + 𝑔𝑖 (𝑥𝑖 + 𝑣 𝑖 )) + ,
where 𝜁𝑖 ∈ 𝜕 𝑓𝑖 (𝑥𝑖 + 𝑣 𝑖 ), 𝜂𝑖 ∈ 𝜕𝑔𝑖 (𝑥𝑖 + 𝑣 𝑖 ) ⊂ R𝑚𝑖 ×𝑠 , 𝑧 𝑖 ∈
R𝑠 , 𝜏𝑖 = [𝜏𝑖1 , 𝜏𝑖2 , ..., 𝜏𝑖𝑚𝑖 ] ∈ R𝑚𝑖 , and 𝑧𝑖 (0) = 0𝑠 , 𝜏𝑖 (0) 
0𝑠 . 𝛼 and 𝛽 are two positive ﬁxed control parameters. 𝜁𝑖
drives agent 𝑖 to ﬁnd the optimum of problem (1) and 𝜏𝑖
ensures that the optimal solution 𝑥 ∗ satisﬁes the inequality
constraints of problem (1). To reduce communication burden,
the last triggering state is used. Namely, 𝜆ˆ 𝑖 = 𝜆𝑖 (𝑡 𝑖𝑘 ), ∀𝑡 ∈
[𝑡 𝑖𝑘 , 𝑡 𝑖𝑘+1 ), 𝜆ˆ 𝑗 = 𝜆 𝑗 (𝑡 𝑙𝑗 ), 𝑡 𝑙𝑗 = max{𝑡 𝑙𝑗 : 𝑡 ≥ 𝑡 𝑙𝑗 }. The auxiliary
variable 𝑧 𝑖 balances the difference between the real resource
𝑥𝑖 and the demand resource 𝑑𝑖 , which ensures that the control
parameter 𝛽 can be set as a constant. Besides, 𝑧𝑖 can be
considered as an integral term in the dynamic of 𝜆¤ 𝑖 and
thus the proportional integral control strategy is constructed
for 𝜆¤ 𝑖 . In addition, the initial states 𝑥 𝑖 (0) and 𝜆𝑖 (0) are
ˆ 𝑧, 𝜏, 𝜁, 𝜂 and 𝑔 be the
arbitrarily designed. Let 𝑥, 𝑣, 𝜆, 𝜆,
aggregate vectors of 𝑥 𝑖 , 𝑣 𝑖 , 𝜆 𝑖 , 𝜆ˆ 𝑖 , 𝑧 𝑖 , 𝜏𝑖 , 𝜁𝑖 , 𝜂𝑖 and 𝑔𝑖 , ∀𝑖 ∈ V.
Then the system (5) can be rewritten as
𝑥¤ = 𝑣,
𝑣¤ = −𝑣 − (𝛼𝜁 + 𝜂T (𝜏 + 𝑔(𝑥 + 𝑣)) + − 𝜆),
𝜆¤ = −𝐿 𝜆ˆ − 𝑧 − 𝛽(𝑥 + 𝑣 − 𝑑),

(6)

ˆ
𝑧¤ = 𝐿 𝜆,
𝜏¤ = −𝜏 + (𝜏 + 𝑔(𝑥 + 𝑣)) + .
It can be concluded that system (6) is an autonomous system.
The triggered time sequence {𝑡𝑖𝑘+1 } is determined by the
following event.
𝑡 𝑖𝑘+1 = max{𝑟 : 𝜃 𝑖 (
𝑟 ≥𝑡𝑖𝑘

𝐿 𝑖𝑖 +4
𝜎𝑖
k𝑒 𝑖 k 2 − 𝑞ˆ 𝑖 ) ≤ 𝜒𝑖 , ∀𝑡 ∈ [𝑡 𝑖𝑘 , 𝑟]}, (7)
4
2
224

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inequality. Besides, the result −𝜆T (𝐿 ⊗ 𝐼𝑠 ) 𝜆ˆ ≤ −
∑𝑛 𝐿𝑖𝑖
2
𝑖=1 2 k𝑒 𝑖 k is deduced as follows.

(6). Then, we have
∗

0𝑛𝑠 = 𝑣 ,
0𝑛𝑠 = −𝑣 ∗ − (𝛼𝜁 ∗ + (𝜂∗ ) T (𝜏 ∗ +𝑔(𝑥 ∗ +𝑣 ∗ )) + −𝜆∗ ),

0𝑛𝑠 = −𝐿𝜆∗ − 𝑧∗ − 𝛽(𝑥 ∗ + 𝑣 ∗ − 𝑑),

(9)

0𝑛𝑠 = −𝜏 ∗ + (𝜏 ∗ + 𝑔(𝑥 ∗ + 𝑣 ∗ )) + ,

where 𝜁 ∗ ∈ ∇ 𝑓 (𝑥 ∗ ). Based on the deﬁnition of (·) + , 𝜏 ∗ =
(𝜏 ∗ + 𝑔(𝑥 ∗ )) + is equivalent to (𝜏 ∗ ) T 𝑔(𝑥 ∗ ) = 0. Besides, we
have 𝜏 ∗  0𝑚 , and 𝑔(𝑥 ∗ )  0𝑚 . ∑
Because 𝐺 is
∑𝑛undirected
𝑛
𝑧𝑖 (𝑡) = 𝑖=1
𝑧𝑖 (0) =
and connected, one can see that 𝑖=1
0𝑠 , ∀𝑡 ≥ 0 under the zero initial state 𝑧𝑖 (0) = 0𝑠 and the
fact 1T𝑛 𝐿 = 0T𝑛 by the deﬁnition
𝐿. Furthermore,
∑𝑛 ∗ of∑matrix
𝑛
the equality constraint 𝑖=1
𝑥𝑖 = 𝑖=1
𝑑𝑖 holds. Similarly,
𝜆∗ = 1𝑛 ⊗ 𝑦 0 , 𝑦 0 ∈ R𝑠 is obtained. In addition, the result
𝜁 ∗ + (𝜂∗ ) T (𝜏 ∗ +𝑔(𝑥 ∗ )) + −1𝑛 ⊗ 𝑦 0 = 0𝑛𝑠 follows from the term
−𝑣 ∗ −(𝛼𝜁 ∗ +(𝜂∗ ) T (𝜏 ∗ +𝑔(𝑥 ∗ +𝑣 ∗ )) + −𝜆∗ ) = 0𝑛𝑠 . Therefore, there
exists a vector 𝑦 ∗ = 𝑦 0 such that the ﬁrst equality of (3) in
Lemma 2 holds. Finally, it is concluded that 𝑥 ∗ is the optimal
solution of problem (5) with the aid of Lemma 2.
Next, we prove that 𝑥𝑖 (𝑡) converges to 𝑥𝑖∗ . Let the considered function 𝑊1 be 𝑊1 = 𝛽2 k𝑥 − 𝑥 ∗ k 2 + 𝛽2 k𝑥 + 𝑣 − 𝑥 ∗ − 𝑣 ∗ k 2 . It
is claimed that 𝑊1 is positive semideﬁnite. Besides, the time
derivative of 𝑊1 over system (6) yields

≤ −2

𝑖=1
𝑛
∑

𝑞ˆ 𝑖 +

𝑖=1

≤−

𝑛
∑
𝑖=1

𝑞ˆ 𝑖 +

(12)

𝑖=1 𝑗=1
𝑛
𝑛 ∑
∑

𝑎𝑖 𝑗
(k𝑒 𝑖 k 2 + k 𝜆ˆ 𝑖 − 𝜆ˆ 𝑗 k 2 )
2
𝑖=1 𝑗=1

𝑛
∑
𝐿 𝑖𝑖
𝑖=1

2

k𝑒 𝑖 k 2 ,

where the fact 𝑒 𝑖 (𝜆ˆ 𝑖 − 𝜆ˆ 𝑗 ) ≤ 21 k𝑒 𝑖 k 2 + 12 k 𝜆ˆ 𝑖 − 𝜆ˆ 𝑗 k 2 obtained
from the Young inequality is used in the ﬁrst inequality
above. Now, let the candidate
Lyapunov function be 𝑉 =
∑𝑛
𝑊1 + 𝑊2 + 𝛽2 k𝜏 − 𝜏 ∗ k 2 + 𝑖=1
𝜒𝑖 (𝑡). Based on the nonnegative
of variable 𝜒𝑖 , it is concluded that 𝑉 is positive semideﬁnite
and radially unbounded. The set-valued Lie derivative of 𝑉
for the system (6) is denoted as L(6)𝑉. For any 𝜉 ∈ L(6)𝑉,
together with (10) and (11), we obtain
𝑛
𝑛
∑
∑
1 − 𝜎𝑖
1 − 𝛿𝑖
) 𝜒𝑖 −
𝑞ˆ 𝑖
(𝛾𝑖 −
𝜉 ≤ −k𝑣 − 𝑣 ∗ k 2 −
𝜃
2
𝑖
𝑖=1
𝑖=1
1
− (𝛼𝜇 − ) 𝛽k𝑥 − 𝑥 ∗ + 𝑣 − 𝑣 ∗ k 2
4
𝛽
1
)k𝑧 − 𝑧∗ k 2
−( −
4 4𝜇
− 𝛽(𝑥 + 𝑣 − 𝑥 ∗ − 𝑣 ∗ ) T (𝜂T 𝜏¯ − (𝜂∗ ) T 𝜏¯ ∗ )

𝑊¤ 1 = −𝛽k𝑣 − 𝑣 ∗ k 2 + 𝛽(𝑥 + 𝑣 − 𝑥 ∗ − 𝑣 ∗ ) T (𝜆 − 𝜆∗ )
− 𝛼𝛽(𝑥 + 𝑣 − 𝑥 ∗ − 𝑣 ∗ ) T (𝜁 − 𝜁 ∗ )
≤ −𝛽k𝑣 − 𝑣 ∗ k 2 − 𝛼𝛽𝜇k𝑥 − 𝑥 ∗ + 𝑣 − 𝑣 ∗ k 2

𝑖=1 𝑞ˆ 𝑖 +

− 𝜆T (𝐿 ⊗ 𝐼𝑠 ) 𝜆ˆ
= −𝜆ˆ T (𝐿 ⊗ 𝐼𝑠 ) 𝜆ˆ − 𝑒 T (𝐿 ⊗ 𝐼𝑠 ) 𝜆ˆ
𝑛
𝑛 ∑
𝑛
∑
∑
𝑎 𝑖 𝑗 𝑒 𝑖 (𝜆ˆ 𝑖 − 𝜆ˆ 𝑗 )
𝑞ˆ 𝑖 +
= −2

0𝑛𝑠 = 𝐿𝜆∗ ,

− 𝛽(𝑥 + 𝑣 − 𝑥 ∗ − 𝑣 ∗ ) T (𝜂T 𝜏¯ − (𝜂∗ ) T 𝜏¯ ∗ )

∑𝑛

(10)

(13)

+ 𝛽(𝜏 − 𝜏 ∗ ) T (−𝜏 + 𝜏),
¯

− 𝛽(𝑥 + 𝑣 − 𝑥 ∗ − 𝑣 ∗ ) T (𝜂T 𝜏¯ − (𝜂∗ ) T 𝜏¯ ∗ )+
+ 𝛽(𝑥 + 𝑣 − 𝑥 ∗ − 𝑣 ∗ ) T (𝜆 − 𝜆∗ )

where the result 𝐿𝑖𝑖4+4 k𝑒 𝑖 k 2 − 𝜎2𝑖 𝑞ˆ 𝑖 ≤ 𝜃1𝑖 𝜒𝑖 is used in the
above inequality. In the following, we simplify the term 𝜔 =
−(𝑥 + 𝑣 − 𝑥 ∗ − 𝑣 ∗ ) T (𝜂T 𝜏¯ − (𝜂∗ ) T 𝜏¯ ∗ ) + (𝜏 − 𝜏 ∗ ) T (−𝜏 + 𝜏)
¯ in
(13).

where 𝜏¯ = (𝜏 + 𝑔(𝑥 + 𝑣)) + and 𝜏¯ ∗ = (𝜏 ∗ + 𝑔(𝑥 ∗ + 𝑣 ∗ )) + . The
strongly convex property (𝑥 +𝑣 −𝑥 ∗ −𝑣 ∗ ) T (𝜕 𝑓 (𝑥 +𝑣) −𝜕 𝑓 (𝑥 ∗ +
𝑣 ∗ )) ≥ 𝜇k𝑥 + 𝑣 − 𝑥 ∗ − 𝑣 ∗ k 2 in Assumption 1 is used in the
above inequality. Next, Let 𝑊2 = 14 k𝜆 − 𝜆∗ k 2 + 41 k𝜆 − 𝜆∗ +
𝑧 − 𝑧∗ k 2 + 12 (𝑧 − 𝑧 ∗ ) T 𝑅Λ1−1 𝑅 T (𝑧 − 𝑧∗ ). Similarly, based on the
positive deﬁniteness of matrix 𝑅Λ1−1 𝑅 T , we also can know
that 𝑊2 is positive semideﬁnite. Then, similarly, we have

𝜔 = −(𝑥 + 𝑣 − 𝑥 ∗ − 𝑣 ∗ ) T (𝜂T 𝜏¯ − (𝜂∗ ) T 𝜏¯ ∗ )+
(𝜏 − 𝜏 ∗ )(−𝜏 + 𝜏)
¯
= −(𝑥 + 𝑣 − 𝑥 ∗ − 𝑣 ∗ ) T (𝜂T 𝜏¯ − (𝜂∗ ) T 𝜏¯ ∗ )+
( 𝜏¯ − 𝜏¯ ∗ )(−𝜏 + 𝜏)
¯ − k𝜏 − 𝜏k
¯ 2

1
𝑊¤ 2 = (𝜆 − 𝜆∗ ) T (−(𝐿 ⊗ 𝐼𝑠 )(𝜆ˆ − 𝜆∗ ) + (𝑧 − 𝑧∗ ) T (𝜆ˆ − 𝜆∗ )
2
− (𝑧 − 𝑧∗ ) − 𝛽(𝑥 − 𝑥 ∗ + 𝑣 − 𝑣 ∗ ))
1
+ (𝜆−𝜆∗ +𝑧−𝑧∗ ) T (−𝛽(𝑥 −𝑥 ∗ +𝑣 −𝑣 ∗ ) − 𝑧 + 𝑧 ∗ )
2
𝑛
1∑
≤−
𝑞ˆ 𝑖 − 𝛽(𝜆 − 𝜆∗ ) T (𝑥 + 𝑥 ∗ − 𝑣 − 𝑣 ∗ )
(11)
2 𝑖=1

(14)

= −k𝜏− 𝜏k
¯ 2 + 𝜏¯ T (−𝜏+ 𝜏−𝜂(𝑥
¯
+𝑣 −𝑥 ∗ −𝑣 ∗ ))−
( 𝜏¯ ∗ ) T (−𝜏 + 𝜏¯ − 𝜂∗ (𝑥 + 𝑣 − 𝑥 ∗ − 𝑣 ∗ )),
where the fact 𝜏 − 𝜏 ∗ = 𝜏 − 𝜏¯ + 𝜏¯ − 𝜏 ∗ is used in the second
equality above. Based on the deﬁnition of (·) + , which yields
−𝜏 + 𝜏¯ = 𝑔(𝑥 + 𝑣) + (−𝜏 − 𝑔(𝑥 + 𝑣)) + . By substituting the
above result into (14), one gets the following conclusion.

𝛽
1
−
)k𝑧 − 𝑧∗ k 2
4 4𝜇
𝑛
∑
𝐿 𝑖𝑖
𝛽𝜇
k𝑥 − 𝑥 ∗ + 𝑣 − 𝑣 ∗ k 2 +
+ 1)k𝑒 𝑖 k 2 ,
(
+
4
4
𝑖=1

𝜔 = −k𝜏 − 𝜏k
¯ 2 + 𝜏¯ T (𝑔(𝑥 ∗ + 𝑣 ∗ ) + (−𝜏 − 𝑔(𝑥 + 𝑣)) + )

−(

+ 𝜏¯ T (𝑔(𝑥 +𝑣) −𝑔(𝑥 ∗ +𝑣 ∗ ) −𝜂(𝑥 +𝑣 −𝑥 ∗ −𝑣 ∗ ))
− ( 𝜏¯ ∗ ) T 𝑔(𝑥 +𝑣) −𝑔(𝑥 ∗ +𝑣 ∗ ) −𝜂∗ (𝑥 +𝑣 −𝑥 ∗ −𝑣 ∗ ))

(15)

− ( 𝜏¯ ∗ ) T (𝑔(𝑥 ∗ + 𝑣 ∗ ) + (−𝜏 − 𝑔(𝑥 + 𝑣)) + ).

𝛽𝜇
4 k𝑥 +

where we used − 12 𝛽(𝑧 − 𝑧∗ ) T (𝑥 + 𝑣 − 𝑥 ∗ − 𝑣 ∗ ) ≤
𝛽
𝑣 − 𝑥 ∗ − 𝑣 ∗ k 2 + 4𝜇
k𝑧 − 𝑧∗ k 2 based on the young’s inequality
∑𝑛
∑𝑛 𝐿𝑖𝑖
2
T
and −𝜆 (𝐿 ⊗ 𝐼𝑠 ) 𝜆ˆ ≤ − 𝑖=1
𝑞ˆ 𝑖 + 𝑖=1
2 k𝑒 𝑖 k in the above

Besides, with the help of the convexity of the function 𝑔(𝑥 +
𝑣) and the nonnegativity of 𝜏,
¯ 𝜏¯ ∗ , one can see that 𝜏¯ T (𝑔(𝑥 +
∗
∗
∗
𝑣) − 𝑔(𝑥 + 𝑣 ) − 𝜂(𝑥 + 𝑣 − 𝑥 − 𝑣 ∗ )) ≤ 0 and −( 𝜏¯ ∗ ) T 𝑔(𝑥 + 𝑣) −
225

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𝑔(𝑥 ∗ + 𝑣 ∗ ) − 𝜂∗ (𝑥 + 𝑣 − 𝑥 ∗ − 𝑣 ∗ )) ≤ 0. Then, from Lemma 2,
the results 𝑔(𝑥 ∗ + 𝑣 ∗ ) ≤ 0, 𝜏¯ T 𝑔(𝑥 ∗ + 𝑣 ∗ ) ≤ 0 and ( 𝜏¯ ∗ ) T 𝑔(𝑥 ∗ +
𝑣 ∗ ) = 0 are obtained. Finally, based on the deﬁnition of (·) + ,
it can be obtained that 𝜏¯ T (−𝜏 − 𝑔(𝑥 + 𝑣)) + = ((𝜏 + 𝑔(𝑥 +
𝑣)) + ) T (−𝜏 − 𝑔(𝑥 + 𝑣)) + = 0 and ( 𝜏¯ ∗ ) T (−𝜏 − 𝑔(𝑥 + 𝑣)) + ≥ 0.
Following, the results 𝜏¯ T (𝑔(𝑥 ∗ + 𝑣 ∗ ) + (−𝜏 − 𝑔(𝑥 + 𝑣)) + ) ≤ 0
and −( 𝜏¯ ∗ ) T (𝑔(𝑥 ∗ + 𝑣 ∗ ) + (−𝜏 − 𝑔(𝑥 + 𝑣)) + ) ≤ 0 are hold in
term 𝜔. As a result, we have 𝜔 ≤ −k𝜏 − 𝜏k
¯ 2 . Inserting the
above result into (13), which yields
𝑛
∑
1 − 𝛿𝑖
𝜉 ≤ −𝛽k𝑣 − 𝑣 ∗ k 2 −
(𝛾𝑖 −
) 𝜒𝑖
𝜃𝑖
𝑖=1
−

𝑛
∑
1 − 𝜎𝑖
𝑖=1

2

𝑞ˆ 𝑖

sidered. For each agent 𝑖, the local objective 𝑓𝑖 (𝑥 𝑖 ) =
𝑎 𝑖1 𝑥𝑖2 + 𝑎 𝑖2 |𝑥 𝑖 − 𝑎 𝑖3 | with 𝜇𝑖 = 2𝑎 𝑖1 is known by themselves. In addition, the local constrained convex function is
deﬁned by 𝑔𝑖 (𝑥 𝑖 ) = (𝑥𝑖 − 𝑥𝑖max )(𝑥𝑖 − 𝑥 𝑖max ), 𝑖 = 1, 2, ..., 6
with lower and upper bounds 𝑥 𝑖min and 𝑥 𝑖max . Overall, the
parameters are set by 𝑎 1 = [2, 1, 0.5, 1.5, 1, 1.5], 𝑎 2 =
[3, 4, 5, 2, 3.5, 4.5], 𝑎 3 = [30, 28, 45, 35, 35, 40, 35], 𝑥 min =
[20, 25, 35, 25, 20, 28], 𝑥 max = [40, 35, 50, 45, 47, 42], where
𝑥 max = [𝑥 1max , 𝑥 2max , ..., 𝑥 6max ] and 𝑥 min = [𝑥1min , 𝑥 2min , ..., 𝑥 6min ].
Let 𝛼 = 0.5, 𝛽 = 0.5, 𝛿𝑖 = 0.8, 𝜃 𝑖 = 1, 𝛾𝑖 = 0.22, 𝜎𝑖 =
0.8, ∀𝑖 ∈ V and let the step size be 0.01s. The trajectories
of 𝑥(𝑡) are shown in Fig. 1 under zero initialization. It
can be seen that 𝑥(𝑡) converges to the optimal solution
𝑥 ∗ = [23.4853, 35, 50, 30.9804, 43.7206, 31.8137]. Besides,
the triggering time sequences {𝑡 𝑖𝑘 }, ∀𝑖 ∈ V are depicted
in Fig. 2. From Fig. 2, it can be concluded that the total
communication times are 627.

(16)

1
− (𝛼𝜇 − ) 𝛽k𝑥 − 𝑥 ∗ + 𝑣 − 𝑣 ∗ k 2
4
1
𝛽
−( −
)k𝑧 − 𝑧∗ k 2 − 𝛽k𝜏 − 𝜏k
¯ 2.
4 4𝜇

50

0 𝑖

resource variable x i

Based on the arbitrariness of 𝜉, we have max L(6)𝑉 ≤ 0.
From the deﬁnition of 𝑉, we have 𝑉 ≥ 𝛽2 k𝑥 − 𝑥 ∗ k 2 +
𝛽
∗
𝑣 ∗ k 2 + 14 k𝜆 − 𝜆∗ k 2 + 41 k𝜆 + 𝑧 − 𝜆∗ − 𝑧∗ k 2 +
2 k𝑥 + 𝑣 − 𝑥 − ∑
𝑛
1
∗
2
𝑖=1 𝜒𝑖 ≥ 0 and 𝑉 (𝑥, 𝑣, 𝜆, 𝑧, 𝜏, 𝜒) = 0 iff
2𝜌2 k𝑧 − 𝑧 k +
(𝑥, 𝑣, 𝜆, 𝑧, 𝜏, 𝜒) = (𝑥 ∗ , 𝑣 ∗ , 𝜆∗ , 𝑧 ∗ , 𝜏 ∗ , 0𝑛 ). By recalling (16), it
follows that the set M ≜ {(𝑥, 𝑣, 𝜆, 𝑧, 𝜏, 𝜒) ∈ R𝑛𝑠 × R𝑛𝑠 ×
R𝑛𝑠 × R𝑛𝑠 × R𝑚 × R𝑛 |𝑉 (𝑥, 𝑣, 𝜆, 𝑧, 𝜏, 𝜒)) ≤ 𝑐} with 𝑐 > 0 is
strongly positive invariant and bounded. Therefore, from any
initial point (𝑥(0), 𝑣(0), 𝜆(0), 𝑧(0), 𝜏(0), 𝜒(0)), it is ensured
that the trajectory (𝑥(𝑡), 𝑣(𝑡), 𝜆(𝑡), 𝑧(𝑡), 𝜏(𝑡), 𝜒(𝑡)) generated
by (6) is bounded for all 𝑡 ≥ 0. Eventually, according to
LaSalle’s invariance principle, one can claim that 𝑥 converges
to the optimal solution 𝑥 ∗ under algorithm (6).
Step III. The following proves that the Zeno behavior is
excluded in system (7) using the contradiction method. If
the Zeno behaviour occurs at time 𝑇0 for agent 𝑖, which
yields that lim 𝑘→∞ 𝑡𝑖𝑘 = 𝑇0 . For any positive number 𝜖, 𝑡𝑖𝑘 ∈
[𝑇0 − 𝜖, 𝑇0 ], ∀𝑘 ≥ 𝑁 (𝜖) with positive integer 𝑁 (𝜖). That is,
𝑡 𝑖𝑘+1 − 𝑡𝑖𝑘 ≤ 𝜖.
From Step II, we can see that the trajectory
(𝑥(𝑡), 𝑣(𝑡), 𝜆(𝑡), 𝑧(𝑡), 𝜏(𝑡), 𝜒(𝑡)) generated by (5) is
bounded for all 𝑡 ≥ 0. Then, it can be assumed that there
exists a constant 𝑀0 such that k 𝜆¤ 𝑖 k ≤ 𝑀0 , ∀𝑖 ∈ V. From
the deﬁnition of 𝑒 𝑖 (𝑡), k𝑒 𝑖 ((𝑡𝑖𝑘+1 )k ≤ 𝑀0 (𝑡𝑖𝑘+1 − 𝑡𝑖𝑘 ). Besides,
𝑖 (𝑡 )
from scheme (7), we have k𝑒 𝑖 (𝑡 𝑖𝑘+1 )k 2 ≥ 𝜃𝑖4𝜒
(𝐿𝑖𝑖 +4) ≥
𝛾𝑖 𝜃𝑖 + 𝛿𝑖
4𝜒𝑖 (0)
𝑡). Therefore, we obtain that
𝜃𝑖 (𝐿𝑖𝑖 +4) exp(−
𝜃𝑖
√
𝛾𝑖 𝜃 𝑖 + 𝛿 𝑖
4𝜒𝑖 (0)
𝑡) > 0. (17)
𝑡 𝑖𝑘+1 − 𝑡𝑖𝑘 ≥
exp(−
2
2𝜃 𝑖
𝑀0 𝜃 𝑖 (𝐿 𝑖𝑖 + 4)
√
𝜒𝑖 (0)
𝜃𝑖 + 𝛿𝑖
Let 𝜖 =
𝑡). Then we have 𝑡 𝑖𝑘+1 −
exp(− 𝛾𝑖 2𝜃
𝑖
𝑀 2 𝜃 (𝐿 +4)

45
agent 1
agent 3
agent 5

40
35

agent 2
agent 4
agent 6

30
25
0

20

40

60

80

t(s)

Fig. 1.

The trajectories of resource variable 𝑥𝑖 for each agent in case 1.

7
6

Agent number

5
4
3
2
1
0
0

Fig. 2.

10

20

30

40
tki (s)

50

60

70

80

The trajectories of triggering time {𝑡𝑖𝑘 } for each agent in case 1.

Case 2. The simulation results in a four-agent system with
a two dimensional (𝑥 𝑖 = [𝑥𝑖1 , 𝑥 𝑖2 ] T ∈ R2 ) resource allocation problem (ring communication topology) are provided
[7]. The local objective function and inequality constrains
are deﬁned as 𝑓1 (𝑥1 ) = k𝑥 1 k 2 + k𝑥1 − [2, 2] T k, 𝑓2 (𝑥2 ) =
𝑥2
𝑥2
k𝑥 2 k 2 + √ 212 + √ 222 , 𝑓3 (𝑥 3 ) = k𝑥3 − [2, 3] T k 2 , 𝑓4 (𝑥 4 ) =

𝑖𝑖

𝑡 𝑖𝑘 ≥ 2𝜖, which contradicts to the fact that 𝑡 𝑖𝑘+1 − 𝑡𝑖𝑘 ≤ 𝜖.
Thus, it is known that the Zeno behavior is avoided.

20

𝑥21 +1

20

𝑥22 +1

ln(𝑒 −0.05𝑥41 + 𝑒 0.05𝑥41 ) + ln(𝑒 −0.05𝑥42 + 𝑒 0.05𝑥42 ) + k𝑥4 k 2 and
𝑔1 (𝑥 1 ) = k𝑥1 −[2, 2] T k 2 −4 ≤ 0, 𝑔21 (𝑥2 ) = (𝑥21 −1)(𝑥22 −2) ≤
0, 𝑔22 (𝑥2 ) = 𝑥 22 (𝑥22 − 1) ≤ 0, 𝑔31 (𝑥3 ) = 0.5 − 𝑥31 ≤

IV. S IMULATIONS
Case 1. The economic dispatch problem for a six-agent
grid network with ring communication topology is con226

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0, 𝑔32 = 1 − 𝑥32 ≤, 𝑔33 (𝑥3 ) = 6 − 𝑥31 − 𝑥 32 ≤ 0, 𝑔4 (𝑥4 ) =
k𝑥 4 − [3, 5] T k 2 − 4 ≤ 0. Besides, the local expected resource
is set as 𝑑 = [[2, 1], [2, 3], [2, 4], [1, 5]] T ∈ R8 . The control
parameters are the same as case 1, and the trajectories of
𝑥(𝑡) are shown in Fig. 3 under zero initialization. It can
also be seen that 𝑥(𝑡) converges to the optimal solution
𝑥 ∗ = [[1.88, 3.47], [1.82, 1], [1.44, 4.56], [1.85, 3.97]] T ∈
R8∑, which is depicted by red ∗ in Fig. 3. Let 𝑒(𝑡) =
𝑛
1
∗ 2
2 𝑖=1 (𝑥 𝑖 (𝑡) − 𝑥 𝑖 ) . The convergence trajectories between
our algorithm and the existing differential-projection-based
and penalty-function-based algorithms [15], [16] are shown
in Fig. 3. As depicted in Fig. 3, the convergence speed
of the continuous time communication algorithm is almost
the same as the event-triggered communication algorithm.
As discussed in Remark 1, the convergence speed of the
algorithm with the deviation control structure in [15] is lower
than ours, and the ﬁxed penalty factor in [16] may lead to a
slight oscillation at the equilibrium point.

x22

x12

2

1
0.5
0

0
0

2
x11

4

1

1.5
x21

2

6

4

x42

x32

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1.5

4

4

2
2

4

2

x31

Fig. 3.

to guarantee convergence. In the future, distributed resource
allocation algorithms over an unbalanced directed network
will be considered.

4
x41

The trajectories of resource variable 𝑥𝑖 for each agent in case 2.

Discrete time (5) of case 1
Continuous time (5) of case 1
Algorithm in [15] of case 1
Algorithm in [16] of case 1
Discrete time (5) of case 2
Algorithm in [15] of case 2

Error e

10 0

10 -5

10 -10

0

0.5

1
1.5
communication round

2
10 5

Fig. 4. Comparison between our algorithm and several existing algorithms.

V. C ONCLUSION
This brief focuses on the algorithm design and convergence analysis of distributed optimization over secondorder MASs. To this end, unlike existing projection-operatorbased and penalty-function-based optimization algorithms,
an adaptive distributed optimization algorithm with eventtriggered communication is provided. In this paper, the bidirectional communication property of the network is assumed
227

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