IEEE TRANSACTIONS ON NETWORK SCIENCE AND ENGINEERING, VOL. 9, NO. 5, SEPTEMBER-OCTOBER 2022

3691

A Distributed Network System for Nonsmooth
Coupled-Constrained Optimization
Xiaoxuan Wang, Shaofu Yang , Member, IEEE, Zhenyuan Guo ,
Shiping Wen , and Tingwen Huang , Fellow, IEEE

Abstract—This paper addresses a class of distributed
nonsmooth optimization problems whose objective function is a
sum of convex local objective functions subjected to local set
constraints and heterogeneous coupled constraints, including
inequality and equality ones. To settle the problem, based on the
consensus protocol for the Lagrangian multipliers of coupled
constraints, we propose a distributed multi-agent network system
with projected output feedback, which is different from the
common projected primal-dual subgradient flow. It is proved
that the output vector of the system is convergent to the optimal
solution of the optimization problem from any initial state over
connected communication networks. Finally, the effectiveness of
the system is illustrated via two numerical examples.
Index Terms—Consensus protocol, coupled constraint, differential inclusions, distributed convex optimization, multi-agent
network.

I. INTRODUCTION

I

N RECENT decades, large-scale optimization problems
have attracted great attention owing to their appearance in
diverse application domains, such as sensor networks [1]–[4],
machine learning [5]–[7], resource allocation [8]–[10], and
intelligent transportation systems [11]–[13]. It is well known
that a large number of centralized systems have been developed [14]–[16]. Since there requires a central agent to collect/
store all data and make decisions, centralized systems are
unsuitable for solving large-scale optimization problems,
especially large-scale data analysis and network computation.

Manuscript received 26 May 2021; revised 27 March 2022; accepted 22
May 2022. Date of publication 26 May 2022; date of current version 9 September 2022. This work was supported in part by the National Natural Science
Foundation of China under Grants 61573003 and 62176056, in part by the
Natural Science Foundation of Hunan under Grant 2019JJ40022, in part by
Young Elite Scientists Sponsorship Program by CAST under Grant
2021QNRC001, and in part by Qatar National Research Fund under Grant
NPRP 8-274-2-107. Recommended for acceptance by Dr. Shiwen Mao. (Corresponding author: Zhenyuan Guo.)
Xiaoxuan Wang and Zhenyuan Guo are with the School of Mathematics,
Hunan Provincial Key Laboratory of Intelligent Information Processing and
Applied Mathematics, Hunan University, Changsha, Hunan 410082, China (email: xxwang@hnu.edu.cn; zyguo@hnu.edu.cn).
Shaofu Yang is with the School of Computer Science and Engineering,
Southeast University, Nanjing, Jiangsu 211189, China (e-mail: sfyang@seu.
edu.cn).
Shiping Wen is with the Centre for Artificial Intelligence, Faculty of Engineering Information Technology, University of Technology Sydney, Ultimo,
NSW 2007, Australia (e-mail: shiping.wen@uts.edu.au).
Tingwen Huang is with Science Program, Texas A&M University at Qatar,
Doha 23874, Qatar (e-mail: tingwen.huang@qatar.tamu.edu).
Digital Object Identifier 10.1109/TNSE.2022.3178107

Therefore, it becomes a significant work on developing distributed systems to deal with optimization problems [17]–
[23]. Among them, there is great interest in designing continuous-time systems, which are governed by ordinary differential
equations and process the qualification for real-time
computing.
Actually, many efforts have been devoted to designing distributed continuous-time systems for optimization problems.
Various gradient-based distributed continuous-time systems
are proposed for unconstrained optimization problems in
[24]–[26]. Due to the widespread presence of constraints in
optimization problems, numerous results on distributed constrained optimization are designed in [27]–[35]. To be specific, [27] and [28] propose a second-order projection system
and an adaptive system for distributed nonsmooth optimization problems subjected to set constraints, respectively. [29]
designs a distributed system including two decent directions
for optimization problems subjected to inequality constraints.
[30]–[32] apply the collective neurodynamic approach to nonsmooth optimization problems subjected to inequality and
affine equality constraints. Furthermore, [33]–[35] solve nonsmooth optimization problems subjected to general constraints, including equality and inequality as well as set
constraints.
It is worth noting that the aforementioned results on constrained optimization problems only address independent constraints, namely each agent has private constraints on its local
decision vector. In contrast, the coupled constraints, wherein
each agent has access to some constraints on the decision vectors of other agents, emerge in many practical applications,
such as wireless communication, neural computation, and networked robotics. Such a structure of constraints brings more
challenges in designing and analysis of distributed systems. It
is noticed that the supply-demand constraints in resource allocation problems are coupled equality constraints, which are
studied in [36]–[39]. [40]–[42] investigate extended monotropic optimization problems with weighted coupled equality
constraints. Moreover, some practical optimization problems
may have coupled inequality constraints. Along this line,
[43]–[45] solve smooth optimization problems subjected to
coupled inequality constraints by proposing distributed discrete-time algorithms. Distributed continuous-time systems
are designed in [46] and [47] for solving smooth optimization
problems subjected to coupled equality/inequality constraints
as well as set constraints. Since the smoothness of objective

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functions limits the scope of applications, [48] and [49] settle
nonsmooth optimization problems subjected to coupled
inequality constraints and set constraints. In [48], a distributed
differentiated projected continuous-time system is obtained by
the exact penalty method. In [49], a distributed projected primal-dual subgradient continuous-time system is proposed
with the proof of the existence of solutions.
Motivated by the above discussions, we further investigate
distributed coupled-constrained optimization problems with
local set constraints. Particularly, the coupled constraints considered here are more general, which consist of inequality and
equality ones. To tackle the coupled inequality and coupled
equality constraints, we introduce the consensus protocol for
the Lagrangian multipliers to decouple the constraints. Since
the convexity of the subgradient-based projection operator
cannot be guaranteed, it is difficult to ensure the existence of
the Caratheodory solution by the traditional projected primaldual subgradient flow. Therefore, we employ the projected
output feedback to handle the local set constraints, i.e. the projection of an auxiliary variable on the set constraints. In summary, the contributions of this paper are listed as follows:
1) We study distributed constrained optimization problems
subjected to local set constraints and heterogeneous
coupled constraints, including inequality and equality
ones. On one hand, the objective functions are nonsmooth and convex (not necessarily to be differentiable
[42]–[47] or strongly convex [36]–[38]). On the other
hand, such coupled constraints are more general than
the independent constraints considered in [29]–[35].
2) Based on the consensus protocol for the Lagrangian
multipliers of coupled constraints, we design a full distributed multi-agent network system with projected
output feedback. Instead of the subgradient-based projection operators adopted in [27], [34], [41], [49], the
projection operator in our system depends on the auxiliary variable, which ensures to be convexity all the
time, and then can handle nonsmooth objective
functions.
3) Compared with systems in [47]–[49] requiring the feasibility of initial states, our system is convergent to the optimal solution of the considered problem from any initial
state, which is more suitable for practical implementation.
The structure of this paper is described as follows.
Section II shows some preliminaries. Section III puts forward
the coupled-constrained optimization problems with local set
constraints. In Section IV, a full distributed multi-agent network system with projected output feedback is designed. The
main results on the convergence of the system are presented in
Section V. Next, two numerical examples are shown in
Section VI to substantiate the theoretical results. This paper is
finally concluded in Section VII.
Notations. Let RN be the N dimensional Euclidean space.
Let RN
þ be the collection of vectors with nonnegative entries
in RN . Denote 0N and 1N as the N dimensional vectors with
all elements being 0 and 1, respectively. Let IN be the N  N
identity matrix. Generally, B ¼ ½bij Nn is a N  n dimensional real matrix, where the i-th row and j-th column element

of B is bij . Denote colfx1 ; . . . ; xN g as the stacked column vector in the form of ½xT1 ; . . . ; xTN T , where xTi is the transpose of
xi . j  j, k  k and  represent the 1-norm, 2-norm and Kronecker product, respectively.
II. PRELIMINARIES
A. Graph Theory
Denote a multi-agent network as an undirected graph G ¼
ðV; E; AÞ, where V ¼ f1; 2; . . . ; Ng, E  V  V and A ¼
½aij NN are the node set, edge set and weighted adjacency
matrix, respectively. If ði; jÞ 2 E, i.e. there exist communication between nodes i and j, then aij > 0, otherwise, aij ¼ 0.
Suppose that there is no self-loop in G, i.e. aii ¼ 0. If there
exists a path between every pair of nodes, then the undirected
graph G is connected. Define the set of neighbors of node i as
N i ¼ fj 2 V : ði; jÞ 2 Eg. The Laplacian
matrix of G is
P
defined by L ¼ ½lij NN , where lii ¼ j2N i aij and lij ¼ aij
for i 6¼ j. For an undirected graph G, L is symmetric, semidefinite, satisfies L1N ¼ 0N and 1TN L ¼ 0TN . G is connected if
and only if L has a simple zero eigenvalue and other eigenvalues are all positive. More concepts of graph theory can be
referred to [50].
B. Convex Analysis and Projection
The set V  Rn is convex, if for any x1 ; x2 2 V and m 2
[0,1], there is mx1 þ ð1  mÞx2 2 V. The function f : Rn !
R is convex if for any x1 ; x2 2 Rn and m 2 ½0; 1, there is
fðmx1 þ ð1  mÞx2 Þ mfðx1 Þ þ ð1  mÞfðx2 Þ. The function
f is locally Lipschitz at x if there exist kx ; d > 0, such that
jfðx1 Þ  fðx2 Þj kx kx1  x2 k for 8x1 ; x2 2 Bðx; dÞ. f is
locally Lipschitz on Rn , if f is locally Lipschitz at 8x 2 Rn .
For a locally Lipschitz function f, the generalized gradient is
defined as


@fðxÞ ¼ co lim rfðxi Þ : xi ! x; xi 62 O [ Nf ;
i!1

where cofg is the convex hull, O represents any measure zero
set, and Nf represents the set of non-differentiable points of f.
In addition, @fðxÞ is locally bounded, upper semicontinuous
and takes nonempty, convex, compact values. For a nonsmooth convex function f, @fðxÞ coincides with the set of
subgradients of f at x, which satisfies fðx2 Þ  fðx1 Þ
1T ðx2  x1 Þ and ðx1  x2 ÞT ð1  2 Þ 0, where 1 2 @fðx1 Þ
and 2 2 @fðx2 Þ.
For a nonempty closed convex set V  Rn , the projection
of a vector x 2 Rn onto V is denoted as
P V ðxÞ ¼ arg min kx  zk:
z2V

The basic property of projection operator is
ðx  P V ðxÞÞT ðz  P V ðxÞÞ

0;

8x 2 Rn ; 8z 2 V:

Define hðx; zÞ ¼ 12 ðkx  P V ðzÞk2  kx  P V ðxÞk2 Þ. Then,
1) hðx; zÞ 12 kP V ðxÞ  P V ðzÞk2 ;

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WANG et al.: DISTRIBUTED NETWORK SYSTEM FOR NONSMOOTH COUPLED-CONSTRAINED OPTIMIZATION

2) hðx; zÞ is continuously differentiable and convex with
respect to x on Rn ;
3) rx 
hðx; zÞ ¼ P V ðxÞ  P V ðzÞ.
The normal cone and tangent cone are used together with
projection. The normal cone of V at x is NV ðxÞ ¼ fn 2 Rn :
nT ðz  xÞ 0; 8z 2 Vg, which can also be defined as NV ðxÞ
¼ fn 2 Rn : P V ðx þ nÞ ¼ xg: The tangent cone of V at x is
defined as TV ðxÞ ¼ fv 2 Rn : vT n 0; 8n 2 NV ðxÞg; which
is the polar cone to normal cone NV ðxÞ. More details on convex analysis, projection operators and cones can be found in
[51]–[53].
C. Differential Inclusion Systems
For a set-valued map F : Rn ! BðRn Þ, a differential inclusion system is
_ 2 F ðxðtÞÞ;
xðtÞ

t

0;

(1)

where BðRn Þ is the collection of all subsets of Rn . If x : ½0; T Þ
! Rn is absolutely continuous on any subsets ½t1 ; t2   ½0; T Þ
and satisfies (1) for almost all t 2 ½0; T Þ, then x is a Caratheodory solution to (1). For any initial state, if F is locally
bounded and upper semicontinuous with nonempty, convex,
compact values, then there exists a Caratheodory solution to
(1). Denote Se ¼ fx 2 Rn : 0 2 F ðxÞg as the set of equilibrium points of system (1). For a continuous differentiable
function V : Rn ! R, the set-valued Lie derivative of V
along (1) is defined as LF V ðxÞ :¼ fðrV ðxÞÞT $ : $ 2
F ðxÞg. More results of differential inclusion systems can be
referred to [54].
III. PROBLEM STATEMENT
Consider a multi-agent network consisting of N agents,
where each agent can exchange information locally over a
communication graph G. For i 2 V, agent i has its state variable xi 2 Vi  Rni and local objective function fi : Rni !
R. The common goal of these agents is to collaboratively
solve the following distributed coupled-constrained optimization problem with local set constraints:
min fðx
xÞ ¼
x 2V

s.t. gðx
xÞ ¼

N
X

i¼1

which are different from those consensus optimization
problems in [27]–[35], [49]. Particularly, when xi 2 Rn
and H i ¼ In for 8i 2 V, coupledPequality constraint
is
PN
N
the supply-demand constraint
x
¼
b
in
i¼1 i
i¼1 i
resource allocation problems [36]–[39].
2) The inequality constraints in problem (2) are coupled
among agents. Particularly, when g1 ðx1 Þ ¼ colfk1 ðx1 Þ;
0; . . . ; 0g, g2 ðx2 Þ ¼ colf0; k2 ðx2 Þ; 0; . . . ; 0g;    , gN ðxN Þ
¼ colf0; . . . ; 0; kN ðxN Þg, coupled inequality constraint
gðx
xÞ is transformed into the independent inequality constraint k ðx
xÞ ¼ colfk1 ðx1 Þ; . . . ; kN ðxN Þg
0 considered in [29]–[35].
Before going on, the following assumptions are given.
Assumption 1: The local objective function fi and inequality constraint function gi are both nonsmooth and convex on
the closed convex set Vi , for 8i 2 V.
Assumption 2: The Slater’s constraint qualification
condiPN
~
tion holds:
there
exists
an
x
2
intðVÞ
such
that
g
ð~
i¼1 i xi Þ <
P
PN
~
0, and N
H
¼
b
.
x
i i
i¼1
i¼1 i
Assumption 3: The communication graph G of the multiagent network is undirected and connected.
Remark 2: The convexity of fi , gi and Vi makes problem (2)
be a convex constrained optimization problem. The nonsmoothness of objective functions and constraint functions makes problem (2) cover more practical problems. Slater’s constraint
qualification condition is widely used to ensure the existence of
feasible solutions, which is basic in constrained convex optimization. The connectivity of G guarantees that the eigenvectors corresponding to 0 eigenvalue of Laplacian matrix L are spanf11N g.
Next, we give the necessary and sufficient condition for the
optimal solution to problem (2), which is the well-known Karuch-Kuhn-Tucker condition in [55].
Lemma 1: Assume that Assumptions 1 and 2 hold, x ¼
colfx1 ; . . . ; xN g is an optimal solution to problem (2) if and
 2 Rp such that
only if there exist & 2 Rm and r
8
x Þ
& þ v þ NV ðx
x Þ;
0 2 @fðx
x Þ þ @gðx
>
<
PN
PN
T
0; &
& 0;
i¼1 gi ðxi Þ
i¼1 gi ðxi Þ ¼ 0;
>
PN
: PN
i¼1 H i xi ¼
i¼1 bi ;
; . . . ; H TN r
g.
where v ¼ colfH
H T1 r

fi ðxi Þ;

IV. SYSTEM DESIGN

i¼1
N
X

3693

gi ðxi Þ

0;

N
X
i¼1

H i xi ¼

N
X

bi ;

(2)

p
The classical Lagrangian function L : V  Rm
þ R ! R
of problem (2) can be given as follows

i¼1

where x ¼ colfx1 ; . . . ; xN g is the decision variable and V ¼
Q
N
i¼1 Vi is the set constraint that is necessary due to the computation limitation and communication capacity limitation of
agents. The inequality constraint function gi : Rni ! Rm ,
H i 2 Rpni and bi 2 Rp are held privately by agent i.
Remark 1: Note that problem (2), which contains local set
constraints and heterogeneous coupled constraints, is an
extension of the existing constrained optimization problems:
1) The decision variables in problem (2) are heterogeneous
with different dimensions, i.e. xi 6¼ xj and ni 6¼ nj ,

Þ ¼ fðx
Lðx
x; &; r
xÞ þ

N
X

&T gi ðxi Þ þ

i¼1

N
X

T ðH
H i xi  bi Þ;
r

i¼1

 2 Rp are the Lagrangian multipliers of
where & 2 Rm
þ and r
coupled inequality and coupled equality constraints, respectively. In light of this, the Lagrangian dual problem of problem
(2) is defined as
max
m
p

r2R
&2Rþ ;

N
X

Þ;
fi ð
&; r

i¼1

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3694

IEEE TRANSACTIONS ON NETWORK SCIENCE AND ENGINEERING, VOL. 9, NO. 5, SEPTEMBER-OCTOBER 2022

Algorithm 1 (For Each i 2 V)

Fig. 1. The transformation process of problem (2).

where



Þ ¼ inf fi ðxi Þ þ &T gi ðxi Þ þ r
T ðH
&; r
H i xi  bi Þ :
fi ð
xi 2Vi

 are the common variables couNote that multipliers & and r
pling all agents. In order to decouple, employing the consensus protocol, Lagrangian dual problem (3) is transformed into
max

& 2RNm
r2RNp
þ ;r

’ð&& ; r Þ;

s.t. Lm & ¼ 0; Lp r ¼ 0;

(4)

P
where ’ð&& ; rÞ ¼ N
i¼1 fi ð& i ; ri Þ, & ¼ colf& 1 ; . . . ; & N g, r ¼
colfr1 ; . . . ; rN g, Lm ¼ L  Im and Lp ¼ L  Ip . Based on
the transformation process of problem (2) (see Fig. 1), the augNp
~ : V  RNm
mented Lagrangian function L
 RNm 
þ R
Np
R ! R of problem (2) is
~ x; & ; r ; x ; z Þ
Lðx
1
¼ fðx
xÞ þ & T C ðx
xÞ þ r T ðH
H x  b Þ  x T L m &  & T Lm &
2
1
 z T L p r  r T Lp r ;
2
where C ðx
xÞ ¼ colfg1 ðx1 Þ; . . . ; gN ðxN Þg, H ¼ diagfH
H 1;
   ; H N g, b ¼ colfb1 ; . . . ; bN g, x ¼ colfx1 ; . . . ; xN g and z ¼
colfz1 ; . . . ; zN g are the Lagrangian multipliers.
Further, to avoid the subgradient-based projection, we
employ the projected output feedback of an auxiliary variable
to track the optimal solution, and design the following distributed multi-agent network system:
8
y_ 2 yy þ x  @fðx
xÞ  @C
Cðx
xÞP &þ  H T r; x ¼ P V ðyyÞ;
>
>
>
< &_ ¼ && þ P & ;
þ
(5)
>
_
r
¼
H
x

b
 Lp ðr
r þ z Þ;
>
>
:
x_ ¼ uLm & ; z_ ¼ Lp r;
where y ¼ colfy1 ; . . . ; yN g is the auxiliary variable, u > 0 is
the tunable parameter, and for simplicity P &þ ¼ P RNm ð&& þ
þ
C ðx
xÞ  Lm ð&& þ x ÞÞ. Accordingly, our algorithm for problem
(2) is given in Algorithm 1.
From Algorithm 1, it is clear that system (5) is full distributed since there is no need for a central agent to collect/store
all data and make decisions. Specifically, each agent just
transmits it own information f& i ; ri ; xi ; zi g with its neighbors.
Even better, no private information fxi ; yi ; @fi ðxi Þ;
@gi ðxi Þ; H i ; bi g requires to be transmitted, so the privacy of
problem is also kept.
Remark 3: Compared with [30]–[35], which consider independent inequality and affine equality constraints, in this
paper, consensus constraints Lm & ¼ 0 and Lp r ¼ 0 are introduced to decouple the coupled constraints such that
xÞ ¼
& T C ðx

N
X
i¼1

& Ti gi ðxi Þ; r T ðH
H x  bÞ ¼

N
X
i¼1

rTi ðH
H i xi  bi Þ:

Initialization: yi ð0Þ 2 Rni , & i ð0Þ; xi ð0Þ 2 Rm , ri ð0Þ; zi ð0Þ 2 Rp .
Update flows:
8
xi ¼ P Vi ðyi Þ;
>
>
>
>
&
>
>
y_ 2 yi þ xi  @fi ðxi Þ  @gi ðxi ÞP þi  H Ti ri ;
>
> i
>
< &_ ¼ & þ P &i ;
i
i
þ
>
_
r
¼
H
x

b
i i
i  Sj2N i aij ðri þ z i  rj  z j Þ;
>
i
>
>
>
>
x_ i ¼ u Sj2N i aij ð& i  & j Þ;
>
>
>
: z_ ¼ S
j2N aij ðri  rj Þ;
i
i

P
&
ð& i þ gi ðxi Þ  j2N i aij ð& i þ xi  & j  xj ÞÞ,
where P þi ¼ P Rm
þ
and aij is the communication weight between agents i and j.

Compared with [27], [34], [41], [49], where the subgradientbased projection operators are adopted to handle set constraints, the projection operator in our system depends on the
auxiliary variable. By using the projected output feedback, the
existence of Caratheodory solution can be guaranteed directly.
V. MAIN RESULTS
First, we show the relationship between the equilibrium
point of system (5) and the optimal solution to problem (2).
Theorem 1: Assume that Assumptions 13 hold. If colfyy ;
& ; r ; x ; z g is an equilibrium point of system (5), then x ¼
P V ðyy Þ is an optimal solution to problem (2).
Proof: Let colfyy ; & ; r ; x ; z g be an equilibrium point of
system (5), which satisfies
x ¼ P V ðyy Þ;
y ¼x 
&

(6a)
 h P &þ

T

H r ;

¼ P &þ ;

(6b)
(6c)

0 ¼ H x  b  Lp ðr
r þ z Þ;
0 ¼ Lm & ¼ L p r ;

(6d)
(6e)

where  2 @fðx
x Þ, h 2 @C
Cðx
x Þ and P &þ ¼ P RNm ð&& þ
þ
C ðx
x Þ  Lm ð&& þ x ÞÞ. By projecting (6b) onto V, we have
P V ðyy Þ ¼ P V ðx
x    h P &þ  H T r Þ;
which yields that x ¼ P V ðx
x    h P &þ  H T r Þ from
(6a). By the definition of normal cone, we have  
h P &þ  H T r 2 NV ðx
x Þ, which means that
0 2 @fðx
x Þ þ @C
Cðx
x ÞP &þ þ H T r þ NV ðx
x Þ:

(7)

From (6e) and the connectivity of G, we have
& ¼ 1N  &;

;
r ¼ 1N  r

(8)

 2 Rp . Further, it can be obtained from
where & 2 Rm and r
(6c), (7) and (8) that
x Þ
& þ v þ NV ðx
x Þ;
0 2 @fðx
x Þ þ @gðx
; . . . ; H TN r
g.
where v ¼ colfH
H T1 r

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WANG et al.: DISTRIBUTED NETWORK SYSTEM FOR NONSMOOTH COUPLED-CONSTRAINED OPTIMIZATION

Similarly, from (6c), we have
C ðx
x Þ  Lm ð&& þ x Þ 2 NRNm ð&& Þ:
þ

(10)

PN
Left multiplying (10) by 1TN  Im , there
i¼1 gi ðxi Þ 2
PN is
m
NRm
ð&
Þ,
which
equals
to
P
ð&
þ
g
ðx
Rþ
i
i
i¼1 i i ÞÞ ¼ & i . It
þ
follows from & ¼ 1N  & that
P Rm
ð
&þ
þ

N
X

3695

Denote the set-valued Lie derivative of V1 , V2 and V along
system (5) as Lð5Þ V1 , Lð5Þ V2 and Lð5Þ V , respectively. For any
c1 2 Lð5Þ V1 , there exists  2 @fðx
xÞ and h 2 @C
Cðx
xÞ such that
x  x ÞT ðyy þ x    h P &þ  H T r Þ
c1 ¼ ðx

þ ðr
r  r ÞT ðH
H x  bÞ  Lp ðr
r þ zÞ
þ ðzz  z ÞT Lp r
¼  ðx
x  x ÞT ðyy  xÞ þ ðx
x  x ÞT ðyy  x Þ

gi ðxi ÞÞ ¼ &;

i¼1

x  x ÞT ðh
hP &þ  h P &þ Þ
 ðx
x  x ÞT ð   Þ  ðx

that is,

 ðx
x  x ÞT H T ðr
r  r Þ þ ðr
r  r ÞT H ðx
xx Þ
&

0;

N
X

gi ðxi Þ

0; &T

N
X

i¼1

gi ðxi Þ ¼ 0:

 ðr
r  r ÞT Lp ðr
r  r Þ  ðr
r  r ÞT Lp ðzz  z Þ

(11)

þ ðzz  z ÞT Lp ðr
r  r Þ;

i¼1

Further, left multiplying (6d) by 1TN  Ip , we have
N
X
i¼1

H i xi ¼

N
X

bi :

(12)

where P &þ ¼ P RNm ð&& þ C ðx
xÞ  Lm ð&& þ x ÞÞ. From the propþ
erty of the projection operator and the convexity of f, we have
8
x  x ÞT ðyy  xÞ 0;
>
< ðx
ðx
x  x ÞT ðyy  x Þ 0;
>
:
ðx
x  x ÞT ð   Þ 0:

i¼1

Therefore, (9), (11) and (12) coincide with the optimal condition of problem (2) in Lemma 1. In other words, x ¼ P V ðyy Þ
&
is an optimal solution to problem (2).
Then, we discuss that system (5) is convergent to the optimal solution of problem (2).
Theorem 2: Suppose that Assumptions 13 hold. If u 2
ð0;  1ðLÞÞ, then for any initial state, the output vector x ðtÞ ¼
N
P V ðyyðtÞÞ in system (5) converges to the optimal solution x to
problem (2), where N ðLÞ is the maximum eigenvalue of Laplacian matrix L.
Proof: Let colfyy ; & ; r ; x ; z g be an equilibrium point of
system (5). Consider the following Lyapunov function

It follows that
c1

ðx
x  x ÞT ðh
hP &þ  h P &þ Þ  rT Lp r :

For any c2 2 Lð5Þ V2 , we have
c2 ¼ uð&&  & ÞT ð&& þ P &þ Þ þ uð&&  & ÞT Lm &_
þ uð&&  & ÞT Lm x_ þ uðx
x  x ÞT Lm &_

V ðyy; & ; r; x ; z Þ ¼ u V1 ðyy; r; z Þ þ V2 ð&& ; x Þ;

þ uðx
x  x Þ T Lm & :

where
V1 ðyy; r; z Þ ¼

1
ðkyy  P V ðyy Þk2  kyy  P V ðyyÞk2 Þ
2
1
1
r  r k2 þ kzz  z k2
þ kr
2
2
o
1n
2
kx
x  x k þ kr
r  r k2 þ kzz  z k2
2
0;

and
u
u
k&&  & k2 þ ð&&  & ÞT Lm ð&&  & Þ
2
2
1
T
x  x k2
x  x Þ þ kx
þ uð&&  & Þ Lm ðx
2
1
¼ t T ðP  Im Þtt
2


uðI þ LÞ uL
.
with t ¼ colf&&  & ; x  x g and P ¼
uL
I
1
Since u 2 ð0;  ðLÞÞ, matrix P is positive semidefinite, which
N
means that V2 ð&& ; x Þ 0.

Dealing with the first item of c2 , we have
ð&&  & ÞT ð&& þ P &þ Þ
¼ ð&&  P &þ þ P &þ  & ÞT ð&& þ P &þ Þ
¼  kP &þ  & k2 þ ðP &þ  & ÞT ðC
Cðx
xÞ  Lm ð&& þ x ÞÞ
þ ðP &þ  & ÞT ðP &þ  &  C ðx
xÞ þ Lm ð&& þ x ÞÞ:

(13)

Similarly, from the property of the projection operator, we
have

V2 ð&& ; x Þ ¼

ðP &þ  & ÞT ðP &þ  &  C ðx
xÞ þ Lm ð&& þ x ÞÞ

0;

and
ðP &þ  & ÞT ðC
Cðx
x Þ  Lm x Þ
x Þ  Lm ð&& þ x Þ  P &þ Þ
¼ ðP &þ  P &þ ÞT ð&& þ C ðx

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bounded and exists for t 0. By the set-valued LaSalle
Invariance Principle [54], the trajectory of system (5) is convergent to the largest weakly positively invariant set W contained in the set Se ¼ fcolfyy; & ; r ; x ; z g : 0 2 Lð5Þ V g.
Next, we will show that the output vector x ðtÞ ¼ P V ðyyðtÞÞ
related to any state in W is an optimal solution to problem (2).
^
^; x
^ ¼ 0, P &þ
^ ; ^z g 2 W , which implies that Lp r
Let colf^
y; &^ ; r

^ ¼ P V ð^
y Þ. From (14), we have
&^ ¼ 0, Lm &^ ¼ 0 and x
(
^
ðP &þ
 & ÞT ðC
Cðx
x Þ  Lm x Þ ¼ 0;
(15)
^ T
^ T ð^
ðP &þ
Þ ðC
Cð^
xÞ  C ðx
x Þh
x  x ÞÞ ¼ 0:

Therefore, (13) can be simplified to
ð&&  & ÞT ð&& þ P &þ Þ
 kP &þ  & k2 þ ðP &þ  & ÞT ðC
Cðx
xÞ  Lm ð&& þ x ÞÞ
¼  kP &þ  & k2  ðP &þ  & ÞT Lm ð&& þ x Þ
þ ðP &þ  & ÞT ðC
Cðx
xÞ  C ðx
x ÞÞ
þ ðP &þ  & ÞT ðC
Cðx
x Þ  Lm x Þ þ ðP &þ  & ÞT Lm x
 kP &þ  & k2  ðP &þ  & ÞT Lm ð&& þ x  x Þ
þ ðP &þ ÞT ðC
Cðx
xÞ  C ðx
x Þ  h T ðx
x  x ÞÞ
 ð&& ÞT ðC
Cðx
xÞ  C ðx
x Þ  ðh
h ÞT ðx
x  x ÞÞ
þ ðP &þ ÞT h T ðx
x  x Þ  ð&& ÞT ðh
h ÞT ðx
xx Þ
 kP &þ  & k2  ðP &þ  & ÞT Lm ð&& þ x  x Þ
hP &þ  h P &þ Þ;
þ ðx
x  x ÞT ðh

(14)

0 and the
where the last inequality holds due to P &þ 0, &
convexity of C ðx
xÞ. Thus, for any c2 2 Lð5Þ V2 , there exists
h 2 @C
Cðx
xÞ such that
c2

 ukP &þ  & k2  uðP &þ  & ÞT Lm ð&& þ x  x Þ
hP &þ  h P &þ Þ þ uð&&  & ÞT Lm &_
þ uðx
x  x ÞT ðh
x  x ÞT Lm &_
þ uð&&  & ÞT Lm x_ þ uðx
þ uðx
x  x ÞT L m &

^ &^  H T r
^ ¼ uð^
^
^  ^  h
c
x  x ÞT ^
yþx
^  ^ Þ ¼ 0;
¼ uð^
x  x ÞT ð^
yþx
^ 2 @C
where ^ 2 @fð^
xÞ and h
Cð^
x Þ. Due to the convexity of f
and the property of the projection operator, we have
fð^
xÞ  fðx
xÞ

x  x ÞT ðh
hP &þ  h P &þ Þ
 ukP &þ  & k2 þ uðx
 u&_ T Lm &  uð&&  & ÞT Lm &
 u&_ T Lm ðx
x  x Þ  uð&&  & ÞT Lm ðx
xx Þ
þ uð&&  & ÞT Lm &_ þ uð&&  & ÞT Lm x_
þ uðx
x  x ÞT Lm &_ þ uðx
x  x Þ T Lm &
x  x ÞT ðh
hP &þ  h P &þ Þ
 ukP &þ  & k2 þ uðx

 & T ðuL  u2 L2 Þ  Im & ;
where the third inequality holds because Lm & ¼ 0.
Consequently, for any c 2 Lð5Þ V , we have
c ¼ uc1 þ c2


ur
rT Lp r  ukP &þ  & k2  & T ðuL  u2 L2 Þ  Im & :

Since u 2 ð0;  1ðLÞÞ, uL  u2 L2 is a positive semidefinite
N
matrix. Therefore, we have
maxLð5Þ V

^
¼ &^ and Lm &^ ¼ 0, we have &^ T C ð^
xÞ ¼ 0. Similarly,
Since P &þ
T
ð&& Þ C ðx
x Þ ¼ 0. Recalling (15), it can be obtained that
^ &^ ¼ 0. According to r
^_ ¼ H x
^
x Þ ¼ 0 and ð^
x  x ÞT h
&^ T C ðx
^  b  Lp ð^
r þ ^z Þ, we have H x
r þ ^z Þ ¼ 0; otherbb  Lp ð^
^ will go to infinity, which contradicts with the boundedwise r
ness of trajectory. Therefore, the set-valued Lie derivative of
^; x
^ ; ^z Þ along system (5) is
V ð^
y; &^ ; r

0;

which implies that V ðtÞ V ð0Þ. The boundedness of colfx
x;
& ; r ; x ; z g can be obtained from the definition of V . Since fi
and gi are convex, @fðx
xÞ and @C
Cðx
xÞ are bounded, which
means that x  @fðx
xÞ  @C
Cðx
xÞP &þ  H T r is bounded. Denote
the upper bound as Q. Then, according to the expression of y_
in system (5), it follows that y is also bounded, where the
bound can be deduced from kyyðtÞk kyyð0Þket þ ð1 
et ÞQ
kyyð0Þk þ Q. In a word, colfx
x; y; & ; r; x ; z g is

ð^
x  x ÞT ^ ¼ ð^
x  y^Þ
x  x ÞT ð^

0:

On the contrary, since x is the optimal solution to problem
(2), we have fð^
x Þ fðx
x Þ. Consequently, fð^
xÞ fðx
x Þ, i.e.
^ is also an optimal solution to problem (2).
x
Finally, we will prove that output vector x ðtÞ ¼ P V ðyyðtÞÞ
of system (5) is convergent. From Bolzano-Weierstrass theorem and the boundedness of trajectory, there exists a strictly
increasing sequence ftn g such that
lim colfyyðtn Þ; & ðtn Þ; rðtn Þ; x ðtn Þ; z ðtn Þg

n!þ1

; x
 ; z g;
¼ colf
y; & ; r
which means that for any d > 0, 9 tN > 0 such that kyyðtN Þ 
 k d, kx
k d
yk d, k&& ðtN Þ  & k d, kr
rðtN Þ  r
xðtN Þ  x
and kzzðtN Þ  z k d. Based on the aforementioned discus; x
 ; z g 2 W and x
 ¼ P V ð
sions, colf
y; & ; r
yÞ is an optimal
solution of problem (2). Consider V by replacing
; x
 ; z Þ of V . We can obtain that
ðyy ; & ; r ; x ; z Þ with ð
y; & ; r
maxLð5Þ V 0 by a similar augment as before. Since V is a
; x
 ; z Þ ¼ 0,
continuous nonincreasing function and Vð
y; & ; r
then for any  > 0, there 9 d > 0 such that Vðyy; & ; r ; x ; z Þ <
 k d, kx
k
, when kyy  yk d, k&&  & k d, kr
rr
xx
d and kzz  z k d. Hence, when t > tN , we have
u
 k2
kx
xx
2

VðyyðtÞ; & ðtÞ; r ðtÞ; x ðtÞ; z ðtÞÞ
VðyyðtN Þ; & ðtN Þ; rðtN Þ; x ðtN Þ; z ðtN ÞÞ < ;

where the second inequality holds because V is nonincreasing.
 . Since
Consequently, we can deduce that limt!þ1 xðtÞ ¼ x

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3697

Fig. 2. The communication topology of the network with nine users.

 k2 is radially unbounded with respect to x , the
xx
V 2u kx
output vector xðtÞ ¼ P V ðyyðtÞÞ in system (5) is globally con&
vergent to the optimal solution x of problem (2).
Remark 4: Compared with [47]–[49], there is no need to
restrict the initial state to remain in V. In other words, our system is free of the initial values.
System (5) has the benefit of flexibly solving several problems. Considering
problem
PN
PN(2) without the coupled equality
constraint
H
x
¼
i
i
i¼1
i¼1 bi , system (5) is transformed
into the following form
8
xÞ  @C
Cðx
xÞP &þ ; x ¼ P V ðyyÞ;
>
< y_ 2  y þ x  @fðx
(16)
&_ ¼  & þ P &þ ;
>
:
x_ ¼ uLm & ;

Fig. 3. Evolution of the output vector x ¼ P V ðyyÞ, and error function
kx
x  x k.

where u > 0 is the tunable parameter and P &þ ¼ P RNm ð&& þ
þ
C ðx
xÞ  Lm ð&& þ x ÞÞ. In this case, the convergence of system
(16) can be similarly illustrated by the following corollary.
Corollary 1: Suppose that Assumptions 13 hold. If u 2
ð0;  1ðLÞÞ, then for any initial state, the output vector xðtÞ ¼
N
P V ðyyðtÞÞ in system (16) converges to the optimal solution to
problem (2) without considering coupled equality constraint.
VI. SIMULATION RESULTS
The effectiveness of the proposed systems is illustrated in
this section via two different types of practical examples.
A. Network Utility Maximization Problems
Network utility refers to the satisfaction degree of users
after allocating network bandwidth to network users. Network
utility optimization [56] is to allocate network resources so
that the total utility of all users is maximized and the utility of
a single user can make the user satisfied. Due to the limitation
of the available bandwidth of communication networks, it is
of great significance to study network utility maximization
problems to improve the utilization rate of network resources
and enhance the stability and fairness of the network.
Consider the following network utility problem with nine
users:
min Uðx
xÞ ¼ 
x 2V

s.t.

9
X

ak jxk j

1  e 5

;

k¼1
9
X
k¼1

ðg k x2k  kk Þ

0;

9
X
k¼1

s k xk ¼

9
X
k¼1

tk ;

(17)

Fig. 4. Trajectories of the Lagrangian multiplier & of coupled inequality constraint, and the Lagrangian multiplier r of coupled equality constraint.

where V ¼ ½0; 19 , ak ; g k , kk , s k and t k are randomly derived
from ½0; 1. Here, xk is the utilization rate of network resource
k and each
resource has an identical utility function uk ðxk Þ ¼
ak jxk j
1  e 5 . Resource k requires to meet logical link capacities
and local set constraint [0,1]. Note that problem (17) satisfies
Assumptions 1 and 2.
Consider a network of nine users to solve this problem, where
the communication topology with connection weights being 1 is
described in Fig. 2. Let u ¼ 0:2 <  1ðLÞ . The initial state yk ð0Þ
9
is randomly generated from ½1; 0 ~Vk . & k ð0Þ, rk ð0Þ, xk ð0Þ,
and zk ð0Þ are randomly generated from ½0; 1. Simulation results
are shown in Figs. 3 and 4. Fig. 3 depicts the evolution of the output vector x ¼ P V ðyyÞ and error function kx
x  x k. It can be
observed that x converges to x ¼ ½0:8968; 0:0332; 0:9984;
0:1687; 0:9537; 0; 0; 0:3083; 0:4383T , which mean that x is
the optimal solution to problem (17). As shown in Fig. 4,
the Lagrangian multipliers & and r are convergent and
achieve consensus, which are consistent with the theoretical results.

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3698

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Fig. 5. The communication topology of the sensor network.
Fig. 6. Transient behaviors of W 2 R40 .

B. Distributed Estimate Problems in Sensor Networks
In the sensor networks, sensors collaborate to fulfill certain
tasks, where sensors transmit wireless information and process
signals. Compared with the traditional single sensor, each sensor in sensor networks uses not only its own measurement but
also the measurement and estimation information of its neighboring sensors. In practical applications, due to the fault of
sensors or the delay during information transmission, data loss
will occur, which results that the measurement contains noise
signals and does not contain the true measurement values.
Therefore, it is necessary to study distributed estimation problems under data loss in sensor networks.
 i 2 Rd at time t
To be specific, sensor i observes vector W
through a time-varying observation vector zi;t ðÞ 2 Rpi , which
shows the susceptibility of sensors to unknown errors. Sensor i
is modeled as hi ðWi Þ ¼ Pi Wi , where Pi 2 Rpi d is the observation matrix. Combining with the view of a least square point, the
objective is to get the optimal W ¼ colfW1 ; . . . ; WN g that solves the following distributed estimation problem [57], [58]:
N
1X
iÞ 2;
min
hi ðWi Þ  zi;t ðW
W 2V 2
i¼1

s.t.

N
X

jWi j

q;

(18)

i¼1

Q
where V ¼ N
i¼1 Vi and q > 0.
System (16) is applied on the distributed sensor setup (18)
(without coupled equality constraints) for N ¼ 20 sensors.
Fig. 5 describes the network structure, where connection
2i
weights are 1. Let u ¼ 0:1, d ¼ pi ¼ 2, Pi ¼ diagf2i1
20 ; 20g,
q ¼ 10, and Vi ¼ ½0:4; 0:4 for i 2 f1; . . . ; 20g. The obser i Þ ¼ ai W
 i þ bi with bi ¼ ðbi1 ; bi2 ÞT , where
vation vector zi;t ðW
ai , bi1 , bi2 are randomly generated from [1,2], ½ 12 ; 12, ½ 12 ; 12,
respectively. Moreover, each component of the actual value
 ¼ colfW
1; . . . ; W
 20 g is randomly generated from ½ 1 ; 1.
W
2 2
Let yi ð0Þ, & i ð0Þ, and xi ð0Þ be the initial states of system (16),
which are randomly generated from ½1; 1, [0,1], and ½1; 1,
respectively.

Fig. 7. Trajectories of the Lagrangian multiplier & of coupled inequality constraint, error function kW  W k, and the coupled inequality constraint
P
20
i¼1 jWi j  q.

Clearly, yi ð0Þ of some sensors may be outside of their local set
constraints Vi . Simulation results are shown in Figs. 6 and 7,
which illustrate that W converges to the optimal solution W
to problem (18) and the coupled inequality constraint is satisfied asymptotically. In addition, the Lagrangian multiplier &
of coupled inequality constraint converges and reaches
consensus.

VII. CONCLUSION
In this paper, the distributed constrained optimization problems subjected to local set constraints and heterogeneous coupled constraints, including inequality and equality ones, have
been studied. By introducing the projected output feedback
instead of the subgradient-based projection operator, a full distributed multi-agent network system has been proposed based
on the consensus protocol for the Lagrange multipliers. It has
been proved that the output vector of the system is convergent
to the optimal solution to the optimization problem. Compared
with previous results, the proposed system can solve

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WANG et al.: DISTRIBUTED NETWORK SYSTEM FOR NONSMOOTH COUPLED-CONSTRAINED OPTIMIZATION

nonsmooth problems subjected to local set constraints and heterogeneous coupled constraints from any initial state.
Future research may focus on extending the results to finite/
fixed time convergence or considering other communication
topologies, such as state-dependent communication, or communication with attacks.
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Xiaoxuan Wang received the B.S. degree from
Henan Normal University, Xinxiang, China, in 2016,
and was admitted to the School of Mathematics,
Hunan University, Changsha, China, in 2016. Her
research interests include distributed optimization
and multiagent network systems.

Shaofu Yang (Member, IEEE) received the B.S. and
M.S. degrees in applied mathematics from the
Department of Mathematics, Southeast University,
Nanjing, China, in 2010 and 2013, respectively, and
the Ph.D. degree in engineering from the Department
of Mechanical and Automation Engineering, The
Chinese University of Hong Kong, Hong Kong, in
2016. He was a Postdoctoral Fellow with the City
University of Hong Kong, Hong Kong, in 2016. He is
currently an Associate Professor with the School of
Computer Science and Engineering, Southeast University. His research interests include distributed optimization, game theory,
and their applications.

Zhenyuan Guo received the B.S. degree in mathematics and applied mathematics and the Ph.D. degree
in applied mathematics from the School of Mathematics, Hunan University, Changsha, China, in 2004
and 2009, respectively. From 2008 to 2009, he was a
Joint Ph.D. Student with the Department of Applied
Mathematics, University of Western Ontario, London, ON, Canada. He was a Postdoctoral Research
Fellow with the Department of Mechanical and Automation Engineering, The Chinese University of Hong
Kong, Hong Kong. He is a Professor with the School
of Mathematics, Hunan University. His research interests include the theory
of functional differential equations and differential equations with discontinuous right hands, and their applications to the dynamics of neural networks,
memristive systems, and control systems.

Shiping Wen received the M.Eng. degree in control
science and engineering from the School of Automation, Wuhan University of Technology, Wuhan,
China, in 2010, and the Ph.D. degree in control science and engineering from the School of Automation,
Huazhong University of Science and Technology,
Wuhan, China, in 2013. He is currently a professor
with the Australian Artificial Intelligence Institute,
University of Technology Sydney, Ultimo, NSW,
Australia. His research interests include memristorbased neural networks, deep learning, computer
vision, and their applications in medical informatics. Dr. Wen was the General/Publication Chair or a Member of the technical programming committee
for various international conferences. He was the recipient of the 2017 Young
Investigator Award of the Asian Pacific Neural Network Association and the
2015 Chinese Association of Artificial Intelligence Outstanding Ph.D. Dissertation Award. In 2018 and 2020, he was listed as a Clarivate Analytics Highly
Cited Researcher in the Cross-Field, respectively. He was a Leading Guest
Editor of the Special Issues of IEEE TRANSACTIONS ON NETWORK SCIENCE
AND ENGINEERING, Sustainable Cities and Society, and Environmental
Research Letters. He is an Associate Editor for the Knowledge-Based Systems,
IEEE ACCESS, and Neural Processing Letters.

Tingwen Huang (Fellow, IEEE) received the B.S.
degree from Southwest Normal University (now
Southwest University), Chongqing, China, in 1990,
the M.S. degree from Sichuan University, Chengdu,
China, in 1993, and the Ph.D. degree from Texas
A&M University, College Station, TX, USA, in
2002. He is currently a Professor with Texas A&M
University-Qatar (TAMUQ), Doha, Qatar. After
graduation, he was a Visiting Assistant Professor
with Texas A&M University. Then he joined
TAMUQ as an Assistant Professor in August 2003,
then he was promoted to Professor in 2013. His research interests include neural networks based computational intelligence, distributed control and optimization, nonlinear dynamics and applications in smart grids. He has authored or
coauthored more than three hundred peer-review reputable journal papers,
including more than one hundred papers in IEEE Transactions. He is currently
an Associate Editor for four journals, including IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS, IEEE TRANSACTIONS ON CYBERNETICS, and Cognitive Computation.

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