Distributed optimization algorithms for heterogeneous linear

multi-agent systems with inequality constraints

ZHENGQUAN YANG1, WENJIE YU2, ZHIYUN GAO1

1College of transportation science and Engineering, Civil Aviation University of China,

Tianjin 300300, CΗΙΝΑ

2College of Science, Civil Aviation University of China, Tianjin 300300, CΗΙΝΑ

Abstract: - In this paper, for heterogeneous linear multi-agent systems, a distributed constrained optimization

problem about digraphs is studied. Every agent only utilizes local interaction and information such that all agents

can achieve the global objective function. The state of each agent is limited to a local inequality constraint set.

First, this paper proposes a distributed continuous-time optimization algorithm by designing a left eigenvector

corresponding to the zero eigenvalue of the Laplacian matrix, which removes the imbalance of the communication

graph. Next, the asymptotical convergence about the algorithm is demonstrated using Lyapunov stability. Finally,

two numerical examples are given to illustrate the effectiveness of the algorithm.

Key-Words: - Asymptotical convergence, Weight-unbalanced digraphs, Multi-agent systems, Distributed

optimization,

Received: June 2, 2023. Revised: September 1, 2023. Accepted: October 1, 2023. Published: October 20, 2023.

1 Introduction

Distributed optimization has broad application

prospects in the fields of economic dispatch in

smart grid, [1], wireless sensor networks, [2], UAV

formation, [3], etc., so it has become a research

hotspot at present. Several types of distributed

algorithms based on the framework of multi-agent

systems have been proposed successively and are

used to solve various optimization problems.

Up to now, in most of the literature (see, [4], [5]),

many works have been derived based on the setting

of discrete-time. Owing to the fact that continuous-

time dynamic systems are more convenient to

study convergence and their applications in

physical systems are more flexible, distributed

continuous-time algorithms have also received

widespread attention from scholars. In [6], the

authors put forward the continuous-time distributed

adaptive coordination protocols to address the given

optimization problems. Aiming at the distributed

optimization problem with general constraints,

a proportional-integral (PI) consensus protocol

was proposed by [7]. However, in light of these

distributed strategies depend on the differentiability

of local objective functions, they may not be able

to solve the problems that local objective functions

are non-smooth. The dual decomposition technique

was given to decouple the equality constraints and

a new initialization-free distributed continuous-

time algorithm was developed in [8]. In [9], the

authors adopted the Laplacian-gradient method

and designed an adaptive control to resolve the

resource allocation problem, and constructed a

distance-based exact penalty function to handle local

convex sets. Different from the above distributed

optimization problems based on single integrator

multi-agent systems, the Euler-Lagrange systems

were considered in [10], and the system dynamics

were second-order dynamical systems in [11].

These above researches are all based on undirected

networks, which require the information transmission

between agents to be symmetrical and bidirectional.

However, in the actual systems, the information

exchange between agents is often one-way and easy

to be interfered with external factors, such as data

packet loss and signal attenuation. Therefore, in

recent years, distributed optimization on digraphs

has attracted more and more attention. In [12],

the authors proposed a subgradient optimization

algorithm of distributed average consistency and in

[13], the authors established an improved subgradient

push algorithm by introducing push-sum protocol

and distributed subgradient algorithm for the

deterministic directed switching network. In [14],

the authors solved the constrained optimization

problems without any initialization conditions and in

[15], the authors researched the resource allocation

problems with the help of a projection operator over

weighted balanced digraphs. In [16], [17], [18], [19],

the directed network is generalized to the situation

of weight-unbalanced. To avoid knowing certain

information of both in-neighbors and out-neighbors,

the effort was made in [16], to design a continuous-

time coordination algorithm. In [17], the authors

designed an adaptive algorithm by virtue of dynamic

coupling gains and the requirements of the Lipschitz

constants, meanwhile, the network connectivity

was removed. Resorting to the gradient-tracking

WSEAS TRANSACTIONS on MATHEMATICS

DOI: 10.37394/23206.2023.22.83

Zhengquan Yang, Wenjie Yu, Zhiyun Gao

E-ISSN: 2224-2880

756

Volume 22, 2023

and PI strategies, in [18], the authors established a

distributed optimization algorithm with a fixed step-

size, which ensured that the convergence rate was

not affected by the decreasing step-size. In [19], the

authors designed an adaptive distributed continuous-

time algorithm, in which the out-Laplacian matrix

was adopted to overcome the difficulties brought

about by a weight-unbalanced communication graph.

As an important carrier of distributed information

processing, linear multi-agent systems have achieved

many research results. For homogeneous linear

multi-agent systems on general digraphs, some

relevant studies have been conducted in [20].

Actually, different multi-agents have different system

states and even diverse dimensions of state space.

Many works in [21], [22], [23], have been derived

for heterogeneous linear multi-agent systems. By

means of the KKT condition and primal-dual control

scheme, in [22], the authors proposed the state-based

and output-based adaptive control algorithms without

initialization. Based on output feedback, the PI

control technique was used to handle distributed

optimal output consensus problem in [23]. This did

not require the convexity of the local cost functions,

but the global control parameters were adopted.

For more related works on distributed optimization

algorithms, readers can refer to some literature

reviews, [24], [25], [26], [27], [28], [29].

Most of theses above-mentioned researches

are limited to distributed optimization problems

under undirected graphs or unconstrained digraphs.

The design of constrained distributed algorithms

under weight-unbalanced digraphs still has some

limitations. When the balance of communication

topology is damaged, the original distributed

algorithms suitable for undirected and balanced

directed graphs may become invalid. The Laplacian

matrix for the unbalanced communication graph is

asymmetric, so the equilibrium point of the general

distributed algorithm is not equal to the optimal

solution. Hence, it is quite necessary to extend

the constrained distributed optimization algorithms

to the weight-unbalanced digraphs. Besides, most

systems are generally heterogeneous in practice, in

the meantime, notice that more and more scholars

focus on the heterogeneity of multi-agent systems.

Generally speaking, on account of the more complex

dynamics involved in systems, dealing with the

consistency of heterogeneous systems is undoubtedly

a challenge. thus, the study of distributed constrained

optimization problems over weight-unbalanced

digraphs considering the heterogeneous structure of

multi-agent systems can be further discussed.

To handle the distributed optimization problem

with inequality constraints, in which the local

objective functions are strongly convex, this

paper aims to design a novel continuous-time

optimization algorithm for heterogeneous linear

multi-agent systems. Simultaneously, each agent

interacts information with other agents through a

communication network modeled by the weight-

unbalanced digraph. The main contributions of this

paper are given as follows.

(1) Compared with [6], [7], [8], [22], which require

the graphs to be undirected, our algorithm is

designed over weight-unbalanced digraphs. In

addition, some of the methods used to prove

convergence cannot be applied to unbalanced

graphs such as the symmetry of undirected

graphs in [6]. The distributed optimization

problems on unbalanced digraphs have also

been studied in [16], [17], [23]. However, in

contrast to [16], [17], [23], which can only deal

with unconstrained optimization problems, these

approaches they used do not enable agents to

reach consensus on the intersection of feasible

sets subject to set constraints.

(2) We devote ourselves to researching the

heterogeneous linear multi-agent systems in

this study while the works of [15], [20] are

restricted to homogeneous linear multi-agent

systems. The subsystems of this paper have

different dynamics, so the method is also

suitable for agents with identical dynamics.

We arrange the remaining parts of this paper in the

following order. Section 2 introduces some useful

preliminaries. Section 3 formulates the constrained

optimization problem and gives some indispensable

assumptions and lemmas. The main result of

this article presented in Section 4 is to seek the

optimal output of a given problem by designing a

distributed continuous time optimization algorithm,

and to analyze its asymptotic convergence.Then in

Section 5, the effectiveness of the proposed algorithm

is illustrated via two numerical examples. Finally,

Section 6 discusses our conclusions and future work.

2 Preliminaries

2.1 Notations

Let R,Rn,Rn×mstand for the sets of real

numbers, n-dimensional real vectors, and n×m-

dimensional real matrices, respectively. Let In∈

Rn×n,1n∈Rn,0n∈Rnrepresent the identity

matrix, the vector with entries equal to one and

zero, respectively. A>and k · k are respectively

the transpose of a matrix Aand the Euclidean norm.

col(x1, . . . , xn) = (x>

1, . . . , x>

n)>is a column vector

sequentially stacked by vectors x1, . . . , xn.(p)+=

p, if p > 0, and (p)+= 0 otherwise. ⊗represents the

Kronecker product.

WSEAS TRANSACTIONS on MATHEMATICS

DOI: 10.37394/23206.2023.22.83

Zhengquan Yang, Wenjie Yu, Zhiyun Gao

E-ISSN: 2224-2880

757

Volume 22, 2023

2.2 Convex Analysis and Projection

Operation

For a set Ω⊆Rd, if τ x1+ (1 + τ)x2∈Ωfor

any x1, x2∈Rd,x16=x2and τ∈(0,1), then Ω

is a convex set. f:Rd→Ris a continuously-

differentiable function, if there exists m > 0such

that (x1−x2)>(∇f(x1)−∇f(x2)) ≥mkx1−x2k2,

for any x1, x2∈Rdholds, the function fis said to

be strongly convex over a convex set Ω⊆Rd. And

moreover, if there exists x, M > 0such that for

any x1, x2∈ B(x, x), there is |f(x1)−f(x2)| ≤

Mkx1−x2k, then we say fis locally Lipschitz at

x∈Rd.

We signify PΩ(x) = argminy∈Ωkx−ykas the

projection of a vector x, where Ωdenotes a closed

convex set. A useful lemma on the based on the

definition of PΩ(x)is following:

Lemma 1 ([7]) Define Ω⊆Rdto be a nonempty

closed convex set. Then x=PΩ(y)if and only if

x∈Ωand (x−y)>(%−x)≥0holds for any %∈Ω.

2.3 Graph Theory

We describe G= (V, ε, A)as a communication

graph consisting of Nagents with node set V=

{1, . . . , N}and edge set ε⊆ V × V, and A=

[aij ]∈RN×Nis a weighted adjacency matrix.

Particularly, aij >0if (i, j)∈ε, and aij = 0

otherwise. A directed path from agent ito agent

jis a sequence of ordered edges in the form of

(i, i1),(i1, i2), . . . , (il, j). If there exists a directed

path connecting every pair of nodes, then Gis called

a strongly connected directed graph. Let din

j=1 aij and dout

i=PN

j=1 aji respectively be the

in-degree and out-degree of agent i. The Laplacian

matrix L= [lij ]∈RN×Nassociated with G

is denoted by L=Din −A, where Din =

diag(din

1, din

2, . . . , din

N). For all i∈ V, if din

i=dout

then we say the directed graph is weight-balanced,

otherwise weight-unbalanced.

Lemma 2 ([16]) For a strongly connected graph G,

and L∈RN×Nis its Laplacian matrix. Then, the

following properties hold.

(1) L1N=0N;

(2) there is a positive left eigenvector ξ=

(ξ1, ξ2, . . . , ξN)>concerning eigenvalue zero

such that ξ>L=0>

Nand PN

i=1 ξi= 1;

(3) ˜

L=ΞL+L>Ξ

2is a valid Laplacian matrix

for a strongly connected and balanced graph,

where Ξ = diag(ξ1, . . . , ξN), and ˜

Lis positive

semidefinite. Zero is a simple eigenvalue of

Land λ2(˜

L) = min1>

N=0N

x>˜

x>xis its second

smallest eigenvalue.

3 Problem formulation

We consider that a multi-agent system is

composed of Nagents, and each agent iis given the

following heterogeneous linear dynamics:

˙xi=Aixi+Biui,

yi=Cixi,(1)

where xi∈Rniis the state variable of agent i,ui∈

Rpiis the control input of agent i, and yi∈Rdis the

control output of agent i.Ai∈Rni×ni,Bi∈Rni×pi,

Ci∈Rd×niare the state, input, and output matrices,

respectively, which are constant matrices.

Our goal is to design a proper controller ui(t)

for each agent i, where each agent only knows its

own information and local interaction, such that all

agents can reach the optimal output y∗under the local

inequality constraints. The optimization problem

with inequality constraints is formulated as follows:

min f(y) =

i=1

fi(y),

s.t. gi(y)≤0, i = 1, . . . , N,

(2)

where y∈Rd,fi:Rd→R,gi:Rd→Rri.

Here, fi(y)is known as the local objective function

of ith agent, and gi(y)=(gi1(y), . . . giri(y))>is

the local inequality constraints of ith agent, where ri

denotes the number of local constraints. Clearly, the

optimization problem can be resolved in a distributed

way due to both information of the local objective

functions fi(y)and local constraints gi(y)are only

obtained by agent i.

Before the introduction of our main results, we

make a few assumptions about the communication

graph, the local objective functions and the local

constraint functions.

Assumption 1 The communication network Gis

strongly connected.

Assumption 2 The function fi(y), i = 1, . . . , N

is mi-strongly convex. Local constraint function

gi(y), i = 1, . . . , N is convex.

Assumption 3 The gradient function of the local

objective function ∇fi, i = 1, . . . , N as well as the

local constraint function ∇gi, i = 1, . . . , N is Mi-

Lipschitz with Mi>0.

Under the whole output variable y=

col(y1, y2, . . . , yN)∈RN d, the global

WSEAS TRANSACTIONS on MATHEMATICS

DOI: 10.37394/23206.2023.22.83

Zhengquan Yang, Wenjie Yu, Zhiyun Gao

E-ISSN: 2224-2880

758

Volume 22, 2023

objective function incurred by all agents is

f(y) = PN

i=1 fi(yi). Under Assumption 1, we

can reformulate the problem (2) as

min f(y) =

i=1

fi(yi),

s.t. (L⊗Id)y=0, gi(yi)≤0, i = 1, . . . , N.

(3)

The equality constraint (L⊗Id)y=0ensures that

y1=y2=. . . =yN, the problem (2) and problem

(3) are equivalent in terms of the optimal solution

set. Then to solve the problem (2), we can solve the

problem (3) instead.

Remark 1 Define Y=TN

i=1 Yiwith Yi={yi∈

Rd|gi(yi)≤0, i = 1, . . . , N}as the global constraint

set on the output variable. And the optimal set of the

considered optimization problem is denoted by Y∗.

Assumption 1 permits the graph to be unbalanced.

Assumption 2 ensures that the optimal solution to (3)

is unique and the optimal set Y∗is convex.

Assumption 4 (Ai, Bi)is controllable, and

rank(CiBi) = d, i = 1, . . . , N. (4)

Lemma 3 ([23]) Under Assumption 4, the following

matrix equations

CiBiΥi=CiAi,

CiBiΨi=Id.(5)

exist solutions Υi∈Rpi×ni,Ψi∈Rpi×d.

4 Distributed General

Continuous-Time Convex

Optimization Algorithm

In this part, we give a novel distributed

optimization algorithm to solve the problem (3)

and prove its convergence property in detail. The

distributed optimization algorithm is given as follows

ui=−Υixi+ Ψi(−∇fi(yi)

−(∇gi(yi))>(µi+ ˙µi)

−α

j=1

aij (yi−yj)−

j=1

aij (ηi−ηj)),

˙ηi=α

j=1

aij (yi−yj),

˙zi=−

j=1

aij (zi−zj),

˙µi= (µi+gi(yi))+−µi,

(6)

where ηi∈Rdand µi∈Rr

+with r=PN

i=1 riare the

auxiliary states of agent i,zi∈RNand zi

iis the ith

component of zi,xi, ui, yiare same defined as (1). α

is a positive parameter. Υi,Ψiare feedback matrices

based on Lemma 3. ∇fiis the gradient of fi,∇giis

the gradient of gi. The simulation structure diagram

of the control algorithm (6) is shown in Figure 1

(Appendix)

Remark 2 In comparison, the related work in [7],

[8], [22] only give the results on undirected graphs,

while (6) is designed over unbalanced digraphs. We

evaluate the left eigenvector associated with the

zero eigenvalue of the Laplacian matrix ˜

Lthrough

(6) by designing the variable zi, which removes

the imbalance of the communication graph, and

ensures that the constrained optimization algorithm

can converge to the optimal output y∗without

knowing the information of the left eigenvector.

However, in [20], the information of ξneeds to

be obtained in advance. And moreover, the µiis

applied to handle local inequality constraints. It

should be noted that the algorithms in [16], [17],

[23], do not take the local state constraints into

account. Furthermore, for some other unconstrained

optimization problems over weight-balanced directed

graphs or undirected graphs, the algorithm (6) we

designed is still applicable.

Let x=col(x1, . . . , xN),y=col(y1, . . . , yN),

z=col(z1, . . . , zN),η=col(η1, . . . , ηN),

µ=col(µ1, . . . , µN),Ψ = diag(Ψ1, . . . , ΨN),A=

diag(A1,Υ = diag(Υ1, . . . , ΥN), . . . , AN),

B=diag(B1, . . . , BN),C=

diag(C1, . . . , CN),ZN=diag(z1

1, . . . , zN

N),

g(y) = col(g1(y1), . . . , gN(yN),∇f(y) =

col(∇f1(y1), . . . , ∇fN(yN)),∇g(y) =

col(∇g1(y1), . . . , ∇gN(yN)), and ¯µ= (µ+g(y))+.

On the basis of the above definition, the compact

form of the algorithm (6) can be written as

˙x= (A−BΥ)x+BΨ(−(Z−1

N⊗Id)∇f(y)

−(∇g(y))>¯µ−α(L⊗Id)y−(L⊗Id)η),

(7a)

˙η=α(L⊗Id)y, (7b)

y=Cx, (7c)

˙z=−(L⊗IN)z, (7d)

˙µ= ¯µ−µ. (7e)

Pre-multiplying (7a) by C, substituting Lemma 3

into the above system, then we can get the following

WSEAS TRANSACTIONS on MATHEMATICS

DOI: 10.37394/23206.2023.22.83

Zhengquan Yang, Wenjie Yu, Zhiyun Gao

E-ISSN: 2224-2880

759

Volume 22, 2023

result.

˙y=−(Z−1

N⊗Id)∇f(y)−(∇g(y))>¯µ

−α(L⊗Id)y−(L⊗Id)η,

˙η=α(L⊗Id)y,

˙z=−(L⊗IN)z,

˙µ= ¯µ−µ.

(8)

Remark 3 In order to guarantee the existence of

Z−1

N, it is essential that the initial value z(0) satisfies

i= 1 for i= 1, . . . , N and zj

i= 0 for all i6=j.

With Assumption 1, it can be readily obtained that

e−Lt(t > 0) is a nonnegative matrix with positive

diagonal entries. This implies that zi

i>0for all

t≥0. Therefore, there exists (zi

i)−1, that is, Z−1

is well-defined.

Next, the optimality condition is given in Lemma 4

which plays a crucial part in the following analysis.

Lemma 4 Let the Lagrangian function of problem

(3) be L(y, η, µ) = f(y)+ 1

2k(Ξ1

2⊗Id)¯µk2+y>(˜

L⊗

Id)η, where η∈RNd and µ∈RNr

+are Lagrange

multipliers. Then, if Assumptions 1-4 hold, the point

y∗is an optimal solution of the optimization problem

(3) iff there exist Lagrangian multipliers (η∗, µ∗)∈

RNd ×RNr

+such that the following KKT condition

holds:

0=∇f(y∗)+(µ∗)>(Ξ ⊗Id)∇g(y∗)+(˜

L⊗Id)η∗,

µ∗≥0, g(y∗)≤0,(µ∗)>g(y∗) = 0.(9)

Proof 1 The second formula of (9) can be discussed

in the following two cases.

(1) It follows from µ∗≥0, g(y∗)=0that

(µ∗)>g(y∗)=0, then we can get that (µ∗+

g(y∗))+=max(µ∗+g(y∗),0) = µ∗+g(y∗) =

µ∗.

(2) It follows from µ∗= 0, g(y∗)≤0that

(µ∗)>g(y∗)=0, by similar discussion, we can

also have that (µ∗+g(y∗))+= 0 = µ∗.

And vice versa.

Combined with the definition of (·)+and pre-

multiplying (9) by (Ξ−1⊗Id), (9) is equivalent to

0= (Ξ−1⊗Id)∇f(y∗)+(∇g(y∗))>µ∗

+ (L⊗Id)η∗,

µ∗= (µ∗+g(y∗))+.

(10)

The lemma below gives the relation between an

optimal solution to the optimization problem (3) and

an equilibrium point of the algorithm (8).

Lemma 5 Assume that Assumptions 1-4 hold. Given

z(0) in Remark 3 and the parameter α(α > 0),

if (y∗, η∗, µ∗, z∗)is the equilibrium point of the

algorithm (8), then y∗is an optimal solution of the

distributed constrained optimization problem (3).

Proof 2 Let (y∗, η∗, µ∗, z∗)is the equilibrium point

of the algorithm (8), so

0=−((Z−1

N)∗⊗Id)∇f(y∗)−(∇g(y∗))>¯µ∗

−α(L⊗Id)y∗−(L⊗Id)η∗,

0=α(L⊗Id)y∗,

0=−(L⊗IN)z∗,

0= ¯µ∗−µ∗.

(11)

It follows from [16], that limt→∞ z=

limt→∞ e−(L⊗IN)tz(0) = (1Nξ>⊗IN)z(0) =

1N⊗ξ, which implies that limt→∞ Z−1

N= Ξ−1.

Thus, z∗= 1N⊗ξand (Z−1

N)∗= Ξ−1. Based on the

above discussion and combined with the definition

of ¯µ= (µ+g(y))+, then we have

0=−(Ξ−1⊗Id)∇f(y∗)−(∇g(y∗))>µ∗−(L⊗Id)η∗,

(12a)

0= (L⊗Id)y∗,(12b)

¯µ∗=µ∗= (µ∗+g(y∗))+.(12c)

It is obvious that (12b) generates y∗=1N⊗˜y

with ˜y∈Rd. Furthermore, by comparing (12a), (12c)

with (10), we can obtain that the equilibrium point

of (8) satisfies the condition (10). By Lemma 4, we

know that y∗is an optimal solution to the problem (3).

Then, Theorem 1 provides the proof of convergence.

Theorem 1 Under Assumptions 1-4, given z(0) in

Remark 3 and the parameter α(α > 0), then

the output variable y(t)asymptotically converge to

the optimal solution y∗of (3) for any initial value

(y(0), η(0), µ(0)) ∈RN d ×RN d ×RN r

Proof 3 Let θ=col(y, η, µ),it follows from (8) that

θsatisfies

θ=f(θ) + g(t, θ) + v(t),(13)

where

f(θ) = f1(θ)

f2(θ)

f3(θ)!.

f1(θ) = −(Ξ−1⊗Id)∇f(y)−(∇g(y))>¯µ−α(L⊗

Id)y−(L⊗Id)η, f2(θ) = α(L⊗Id)y, f3(θ) = ¯µ−µ,

WSEAS TRANSACTIONS on MATHEMATICS

DOI: 10.37394/23206.2023.22.83

Zhengquan Yang, Wenjie Yu, Zhiyun Gao

E-ISSN: 2224-2880

760

Volume 22, 2023

g(t, θ) = 



((Ξ−1−Z−1

N)⊗Id)(∇f(y)− ∇f(y∗))

0

,

v(t) = 



((Ξ−1−Z−1

N)⊗Id)∇f(y∗)

0

.

First, consider the stability of the system:

θ=f(θ).(14)

Let θ∗=col(y∗, η∗, µ∗)be an equilibrium point

of system (14) and the Lyapunov candidate is defined

V1=1

2(y−y∗)>(Ξ ⊗Id)(y−y∗),(15)

Then we have

V1= (y−y∗)>(Ξ ⊗Id)(−(Ξ−1⊗Id)∇f(y)

−(∇g(y))>¯µ−α(L⊗Id)y−(L⊗Id)η)

=−(y−y∗)>(∇f(y)− ∇f(y∗))

−(y−y∗)>(Ξ ⊗Id)((∇g(y))>¯µ−(∇g(y∗))>µ∗)

−α

2(y−y∗)>((ΞL+L>Ξ) ⊗Id)(y−y∗)

−1

2(y−y∗)>((ΞL+L>Ξ) ⊗Id)(η−η∗)

=−(y−y∗)>(∇f(y)− ∇f(y∗))

−(y−y∗)>(Ξ ⊗Id)((∇g(y))>¯µ−(∇g(y∗))>µ∗)

−α(y−y∗)>(˜

L⊗Id)(y−y∗)

−(y−y∗)>(˜

L⊗Id)(η−η∗),

(16)

where ˜

L=ΞL+L>Ξ

2we have used in Lemma 2.

Next, define V2=1

2(µ−µ∗)>(Ξ ⊗Id)(µ−µ∗).

According to the definition of (·)+and Lemma 1, we

can conclude that

( ˙µ+µ−(µ+g(y)))>(µ∗−˙µ−µ)≥0,(17)

expand the equation (17), that is,

˙µ>(µ∗−µ)− k ˙µk2−g(y)>(µ∗−¯µ)≥0,(18)

pre-multiplying (7a) by (Ξ ⊗Id), it can be obtained

that

˙µ>(Ξ ⊗Id)(µ∗−µ)− k(Ξ ⊗Id)1

2˙µk2

−g(y)>(Ξ ⊗Id)(µ∗−¯µ)≥0,(19)

by combining (19) and the definition of V2, it is

simplified as

V2≤ −k(Ξ ⊗Id)1

2˙µk2−g(y)>(Ξ ⊗Id)(µ∗−¯µ).

(20)

Take V3=1

2α(η−η∗)>(Ξ ⊗Id)(η−η∗), where

α > 0. Then the derivative of V3along with (14) is

V3=1

2α·α(y−y∗)>((ΞL+L>Ξ) ⊗Id)(η−η∗)

= (y−y∗)>(˜

L⊗Id)(η−η∗).

(21)

Let

V=V1+V2+V3(22)

be a Lyapunov function candidate, obviously, which

is positive semi-definitive. By utilizing (16),(20) and

(21), and from (22), we can obtain that

V=−(y−y∗)>(∇f(y)− ∇f(y∗))

−α(y−y∗)>(˜

L⊗Id)(y−y∗)

− k(Ξ ⊗Id)1

2˙µk2−g(y)>(Ξ ⊗Id)(µ∗−¯µ)

−(y−y∗)>(Ξ ⊗Id)((∇g(y))>¯µ

−(∇g(y∗))>µ∗).

(23)

Since the function fis strongly convex, we have

(y−y∗)>(∇f(y)− ∇f(y∗)) ≥mky−y∗k2.(24)

Utilized the fact that the communication graph is

a connected digraph, one can obtain that

α(y−y∗)>(˜

L⊗Id)(y−y∗)≥αλ2(˜

L)ky−y∗k2,

(25)

where λ2(˜

L)is the second smallest eigenvalue of ˜

Next analyzing ω=−g(y)>(Ξ ⊗Id)(µ∗−¯µ)−

(y−y∗)>(Ξ ⊗Id)((∇g(y))>¯µ−(∇g(y∗))>µ∗).

By rearranging these terms, it follows that

ω= +(µ∗)>(Ξ ⊗Id)((∇g(y∗))>(y−y∗)−g(y))

−¯µ>(Ξ ⊗Id)((∇g(y))>(y−y∗)−g(y)).

(26)

By adding and subtracting g(y∗)in (26), one has

ω= (¯µ∗)>(Ξ ⊗Id)((∇g(y∗))>(y−y∗)

+g(y∗)−g(y)) −(¯µ∗)>(Ξ ⊗Id)g(y∗)

−¯µ>(Ξ ⊗Id)((∇g(y))>(y−y∗)

+g(y∗)−g(y)) + ¯µ>(Ξ ⊗Id)g(y∗)

(27)

where the convex property of g(y)and the

nonnegative property of ¯µand µ∗are used in the first

and third term. It follows from Lemma 4 that y∗is

an optimal solution to the optimization problem (3).

This makes it easy to verify that g(y∗)≤0,

¯µ>(Ξ ⊗Id)g(y∗)≤0, and (¯µ∗)>(Ξ ⊗Id)g(y∗) = 0.

Hence, what we can gain from the above analysis is

that

V≤ −k(Ξ ⊗Id)1

2˙µk2−(m+αλ2)ky−y∗k2

≤0.(28)

WSEAS TRANSACTIONS on MATHEMATICS

DOI: 10.37394/23206.2023.22.83

Zhengquan Yang, Wenjie Yu, Zhiyun Gao

E-ISSN: 2224-2880

761

Volume 22, 2023

Due to ˙

Vis continuous and negative for any

α > 0,V(t)is bounded, then we can conclude

that y(t),µ(t)and η(t)are bounded. This

together with the Lipschitz condition of fiand

gi, which imply that ∇fiand ∇giare bounded,

then −(Z−1

N⊗Id)∇f(y)−(∇g(y))>¯µ−α(L⊗

Id)y−(L⊗Id)ηis also bounded, and denotes

σ, σ > 0as its upper bound. Furthermore, we have

that kx(t)k ≤ kx0ke−t+ (1 −e−t)σ≤ kx0k+σ

according to the expression of ˙x(t), so x(t)is also

bounded. For the reason that Vis radially unbounded,

then by using LaSalle Invariance Principle, it can be

found that limt→∞ yi(t) = y∗, for i= 1, . . . , N.

Next, we discuss the asymptotical stability of the

terms g(t, θ)and v(t).

From [16], there exist two positive constants %1

and ι1such that max|ξ−1

i−(zi

i)−1| ≤ %1e−ι1tholds.

Thus, g(t, θ)satisfies the inequality

kg(t, θ)k=k((Ξ−1−Z−1

N)⊗Id)(∇f(y)− ∇f(y∗))k

≤max|ξ−1

i−(zi

i)−1| · k∇f(y)− ∇f(y∗)k

≤M%1e−ι1tky−y∗k.

(29)

v(t)satisfies

kv(t)k=k((Ξ−1−Z−1

N)⊗Id)∇f(y∗)k

≤max|ξ−1

i−(zi

i)−1| · k∇f(y∗)k

≤%1k∇f(y∗)ke−ι1t.

(30)

In light of (29), (30), it is evident that

limt→∞ g(t, θ) = 0and limt→∞ v(t) = 0. As

already indicated above, one has θ(t)asymptotically

converges to θ∗as t→ ∞, which means that the

proposed algorithm is effective for solving the

problem (3). This completes the proof.

5 Numerical Examples

Two numerical examples are provided to

demonstrate the effectiveness of the presented

algorithm in this section.

Example 1 We consider a heterogeneous multi-agent

system consisting of six agents in R2inspired by [23],

where A1=A2=1 0

1 1,A3=A4=0 1

−2 1,

A5=A6="110

011

102#,B1=B2=−1

2,B3=

B4=1

−1,B5=B6="1

2#,C1=C2= [1 1],

C3=C4= [3 1],C5=C6=1

2−1 0. The

local objective functions referred to [11] and the local

convex inequality functions of six agents are given by

f1(y1) = 1

2y2

11 +1

2y2

12,

f2(y2) = 1

2(y21 + 1)2+1

2y2

22,

f3(y3) = 1

2y2

31 +1

2(y32 + 1)2,

f4(y4) = 1

2(y41 + 1)2+1

2(y42 + 1)2,

f5(y5) = 1

4y4

51 +1

4y4

52,

f6(y6) = 1

4(y61 + 1)4+1

4y4

62,

gi(yi) = (y1−2)2+ (y2−2)2−5, i = 1, . . . , 6.

(31)

Obviously, all of the local objective functions

and inequality functions are continuously

differentiable. Due to the strong convexity of

function f(y) = PN

i=1 fi(yi), the global minimizer

y∗is unique. And the minimum value f(y) = 5.2347

and the optimal solution y∗= [0.2945,0.5539]>

can be acquired through some simple and easy

calculations.

In addition, the feedback gain matrixes of each

agent can be selected by γ1=γ2= [−2−1],

γ3=γ4= [−1 2],γ5=γ6= [1−1 2].

Ψ1= Ψ2=[−1],Ψ3= Ψ4=1

2,Ψ5= Ψ6=[2].

The initial states of xi(0), ηi(0) ∈R2are chosen

randomly and the initial value µi(0) = 0 is set.

The communication network among these six

agents is unbalanced and depicted in Figure 2

(Appendix) setting the weight of all edges as 1.

Let α1= 7 and α2= 10, the state trajectories

of these agents by executing algorithm (6) are given

in Figure 3(Appendix), it can be seen that the

outputs of all agents converge uniformly to the

global optimal solution y∗of optimization problem

2. Meanwhile, compared with the two subplots in

Figure 3 (Appendix), we can see that, the curve

corresponding to gain parameter α= 7 converges

around t= 20 s and the curve corresponding to α=

10 converges around at the time t= 15 s, namely,

the convergence rate is improved when αis increased

from 7 to 10. This indicates that the convergence

rate can be faster by choosing larger constant control

parameters.

Example 2 In this example, we also consider a

system consisting of six agents in R2. Assume that the

coefficient matrixes and the feedback gain matrixes

are the same as that chosen in Example 1, so is

the communication graph. Each agent ihas the

WSEAS TRANSACTIONS on MATHEMATICS

DOI: 10.37394/23206.2023.22.83

Zhengquan Yang, Wenjie Yu, Zhiyun Gao

E-ISSN: 2224-2880

762

Volume 22, 2023

following objective function fi, i = 1, . . . , 6and

convex inequality function gi, i = 1, . . . , 6.

f1(y1) = y2

ln(y2

11 + 6),

f2(y2) = 1

2(y21 + 1)2ln(y2

11 + 3) + y2

22,

f3(y3) = y2

py2

31 + 1 +1

5y2

32,

f4(y4) = y4

41 + 2y2

42 + 5,

f5(y5) = 1

10y2

51 +e1

10 y52 ,

f6(y6) = 1

2e−1

2y61 +2

5e3

10 y62 ,

gi(yi) = (y1−3)2+ (y2−5)2−4, i = 1, . . . , 6.

(32)

Besides, configuring the initial states of

xi(0), ηi(0) ∈R2randomly and setting the initial

value to µi(0) = 0, αi(0) = 10. Figure 4 (Appendix)

depicts the evolution trajectories of the six agents

by executing algorithm (6), and it is clear to

observe that the output decision of each agent also

effectively converges to the exact optimal value

y∗= [1.606,3.566]>.

6 Conclusion

To handle the distributed optimization problem

with inequality constraints of heterogeneous linear

multi-agent systems over weight-unbalanced

digraphs, a distributed continuous-time optimization

algorithm is presented. Additionally, a variable is

designed to evaluate the left eigenvector associated

with the zero eigenvalue of the Laplacian matrix,

which removes the imbalance of the communication

graph. When the local cost functions are assumed to

be strongly convex with global Lipschitz gradients

and the local constraint functions are assumed to be

convex, by resorting to the Lyapunov stability, it is

proved that the state of agents will asymptotically

converge consensus at an optimal solution of the

given optimization problem.

In the future, we will focus on the

nonsmooth constrained optimization problems

with communication delays and practical safety

constraints of each agent against a collision.

References:

[1] P. Yi, Y. Hong, and F. Liu, Initialization-free

distributed algorithms for optimal resource

allocation with feasibility constraints and

application to economic dispatch of power

systems, Automatica, Vol. 74, pp. 259-269,

2016.

[2] Y. Zhang, Y. Lou, Y. Hong, and L. Xie,

Distributed projection-based algorithms for

source localization in wireless sensor networks,

IEEE Transactions on Wireless Communication,

Vol. 14, No. 6, pp. 3131-3142, 2015.

[3] Y. Huang, H. Wang, and P. Yao, Energy-optimal

path planning for solarpowered uav with tracking

moving ground target, Aerospace Science &

Technology, Vol. 53, pp. 241-251, 2016.

[4] A.Nedic, and A. Ozdaglar, Distributed

subgradient methods for multi-agent

optimization, IEEE Transactions on Automatic

Control, Vol. 54, No. 1, pp. 48-61, 2009.

[5] A.Nedic, A. Ozdaglar, and P. Parrilo, Constrained

consensus and optimization in multi-agent

networks, IEEE Transactions on Automatic

Control, Vol. 55, No. 4, pp. 922-938, 2010.

[6] A. Huang, F. Chen, and W. Lan, An Adaptive

dynamic protocol for distributed convex

optimization, in Proceedings of the 34th

Chinese Control Conference (CCC), China, pp.

1318-1322, 2015.

[7] W. Xu, and F. Yang, Projection-based dynamics

for distributed optimization subject to general

constraints, Proceedings of the 37th Chinese

Control Conference, China, pp. 2474-2478, 2018.

[8] G. Chen, Q. Yang, Y. Song, and F. Lewis,

A distributed continuous-time algorithm for

nonsmooth constrained optimization, IEEE

Transactions on Automatic Control, 2020, doi:

10.1109/TAC.2020.2965905.

[9] M. Lian, Z. Guo, X. Wang, S. Wen, and T.

Huang, Adaptive exact penalty design for optimal

resource allocation, IEEE Transactions on Neural

Networks and Learning Systems, vol. 34, no. 3,

pp. 1430 – 1438, 2023.

[10] Y. Zou, B. Huang, and Z. Meng, Distributed

continuous-time algorithm for constrained

optimization of networked Euler-Lagrange

systems, IEEE Transactions on Control of

Network Systems, Vol. 8, No. 2, pp. 1034-1041,

2021.

[11] Z. Yang, Q. Zhang, and Z. Chen, Distributed

constraint optimization with flocking

behavior, Hindawi, Complexity, 2018,

https://doi.org/10.1155/2018/1579865.

[12] S. Ram, V. Veeravalli, and A. Nedic,

Distributed non-autonomous power control

through distributed convex optimization, in

WSEAS TRANSACTIONS on MATHEMATICS

DOI: 10.37394/23206.2023.22.83

Zhengquan Yang, Wenjie Yu, Zhiyun Gao

E-ISSN: 2224-2880

763

Volume 22, 2023

Proceedings of the 28th conference on computer

communications, Brazil, pp. 3001-3005, 2009.

[13] A. Nedic, and A. Olshevsky, Distributed

optimization over time-varying directed graphs,

in 52nd IEEE Conference on Decision and

Control, Italy, pp. 6855-6860, 2013.

[14] Q. Yang, G. Chen, and J. Ren, Continuous-

time algorithm for distributed constrained

optimization over directed graphs, in 2019 IEEE

15th International Conference on Control and

Automation (ICCA), Uk, pp. 1020-1025, 2019.

[15] Y. Zhu, W. Ren, W. Yu, and G. Wen, Distributed

resource allocation over directed graphs via

continuous-time algorithms, IEEE Transactions

on Systems, Man, and Cybernetics: Systems, Vol.

51, No. 2, pp. 1090-1106, 2021.

[16] Y. Zhu, W. Yu, G. Wen, and W. Ren, Continuous-

time coordination algorithm for distributed

convex optimization over weight-unbalanced

directed networks, IEEE Transactions on Circuits

and Systems II: Express Briefs, Vol. 66, No. 7,

pp. 1202-1206, 2019.

[17] Z. Li, Z. Ding, J. Sun, and Z. Li, Distributed

adaptive convex optimization on directed

graphs via continuous-time algorithms, IEEE

Transactions on Automatic Control, Vol. 63, No.

5, pp. 1434-1441, 2018.

[18] J. Xia, and W. Ni, Distributed optimization

algorithms over weighted unbalanced

directed networks, Journal of Nanchang

University(Natural Science), Vol. 46, No. 3, pp.

303-308, 2022.

[19] M. Lian, Z. Guo, S. Wen, and T. Huang,

Distributed adaptive algorithm for resource

allocation problem over weight-unbalanced

graphs, IEEE Transactions on Network

Science and Engineering, pp. 1-10, 2023,

doi:10.1109/TNSE.2023.3300736

[20] J. Zhang, L. Liu, and H. Ji, Exponential

convergence of distributed optimal coordination

for linear multi-agent systems over general

digraphs, in Proceedings of the 39th Chinese

Control Conference (CCC), China, pp. 5047-

5051, 2020.

[21] Z. Li, Z. Wu, Z. Li, and Z. Ding, Distributed

optimal coordination for heterogeneous linear

multiagent systems with event-triggered

mechanisms, IEEE Transactions on Automatic

Control, Vol. 65, No. 4, pp. 1763-1770, 2020.

[22] B. Shao, M. Li, and X. Shi, Distributed

resource allocation algorithm for general

linear multiagent systems, IEEE Access,

Vol. 10, pp. 74691-74701, 2022, https://doi:

10.1109/ACCESS.2022.3191909.

[23] L.Li, Y. Yu, X. Li, and L. Xie, Exponential

convergence of distributed optimization for

heterogeneous linear multi-agent systems

over unbalanced digraphs, Automatica, 2022,

https://doi.org/10.1016/j.automatica.2022.110259.

[24] Y. Hong, and Y. Zhang, Distributed

optimization: algorithm design and convergence

analysis, Control Theory & Applications, Vol.

31, No. 7, pp. 850-857, 2014.

[25] P. Yi, and Y. Hong, Distributed cooperative

optimization and its application, Chinese

Science: mathematics, Vol. 46, No. 10, pp. 1547-

1564, 2016.

[26] P. Xie, K. You, Y. Hong, and L. Xie, Research

progress of networked distributed convex

optimization algorithms, Control Theory

&Applications, Vol. 35, No. 7, pp. 918-927,

2018.

[27] L. Wang, K. Lu, and Y. Guan, Distributed

optimization via multi-agent systems, Control

Theory & Applications, Vol. 36, No. 11, pp. 1820-

1833, 2019.

[28] X. Jiang, X. Zeng, J. Sun, and J. Chen, Research

status and prospect of distributed optimization

for multiple aircraft, Acta Aeronautica et

Astronautica Sinica, Vol. 42, No. 4, pp. 1-16,

2021.

[29] Q. Yang, J. Yu, Y. Gao, An adaptive optimization

algorithm for heterogeneous linear multi-

agent systems with inequality constraints,

in Proceedings of the 42th Chinese Control

Conference (CCC), China, pp. 6123-6128, 2023.

WSEAS TRANSACTIONS on MATHEMATICS

DOI: 10.37394/23206.2023.22.83

Zhengquan Yang, Wenjie Yu, Zhiyun Gao

E-ISSN: 2224-2880

764

Volume 22, 2023

Project Grant No. 62173332).

Conflict of Interest

The authors have no conflicts of interest to declare

that are relevant to the content of this article.

Creative Commons Attribution

License 4.0 (Attribution 4.0

International , CC BY 4.0)

This article is published under the terms of the

Creative Commons Attribution License 4.0

https://creativecommons.org/licenses/by/4.0/deed.en_US

WSEAS TRANSACTIONS on MATHEMATICS

DOI: 10.37394/23206.2023.22.83

Zhengquan Yang, Wenjie Yu, Zhiyun Gao

E-ISSN: 2224-2880

765

Volume 22, 2023

Acknowledgments

Zhengquan Yang proposed the idea of the method

and checked the correctness of the manuscript.

Wenjie Yu gave the method and wrote the article.

Zhiyu Gao carried out the simulation and the

optimization.

Sources of funding for research

presented in a scientific article or

scientific article itself

The authors acknowledge the financial support

from Natural Science Foundation of China (Key

Figure 1: The controller to solve the optimization problem (3).

Figure 2: Communication graph among six agents.

0 5 10 15 20 25 30

t/s

0.5

1.5

2.5

3.5

states of agnets yi(i=1,...,6)

Agent xi1

xi1

Agent xi2

xi2

0 5 10 15 20 25 30

t/s

0.5

1.5

2.5

3.5

states of agnets yi(i=1,...,6)

=10

Agent xi1

xi1

Agent xi2

xi2

Figure 3: The state trajectories of output variable yiwith respect to time tfor different values of αby executing

algorithm (6).

WSEAS TRANSACTIONS on MATHEMATICS

DOI: 10.37394/23206.2023.22.83

Zhengquan Yang, Wenjie Yu, Zhiyun Gao

E-ISSN: 2224-2880

766

Volume 22, 2023

APPENDIX

0 5 10 15 20 25 30

t/s

0.5

1.5

2.5

3.5

states of agnets yi(i=1,...,6)

=10

Agent xi1

xi1

Agent xi2

xi2

Figure 4: The state trajectories of output variable yirespect to time tby executing algorithm (6).

WSEAS TRANSACTIONS on MATHEMATICS

DOI: 10.37394/23206.2023.22.83

Zhengquan Yang, Wenjie Yu, Zhiyun Gao

E-ISSN: 2224-2880

767

Volume 22, 2023