IEEE TRANSACTIONS ON CONTROL OF NETWORK SYSTEMS, VOL. 8, NO. 1, MARCH 2021

413

Projection-Free Distributed Optimization With
Nonconvex Local Objective Functions and
Resource Allocation Constraint
Dewen Li , Ning Li , Member, IEEE, and Frank Lewis , Life Fellow, IEEE

Abstract—We present a novel generalized constrained
convex optimization model for multiagent systems that contains both the local, coupled equality, and inequality constraints, and a global resource allocation constraint. This
model unifies the traditional constrained optimization problem, the resource allocation problem, and the economic
dispatch problem. Unlike the majority of literature where
each local objective function is required to be convex, we
only require a milder condition that the global objective
function is convex. The gradient of the global Lagrangian is
estimated locally by each agent using the dynamic average
consensus protocol. Synchronously, modified primal-dual
dynamics produce the optimal solution via the estimated
gradient. The generalized Lagrange multiplier method is
introduced to avoid the usual positive projections in the
presence of inequality constraints. This leads to smooth
dynamics and a continuous Lyapunov derivative, which enables the exponential stability analysis. Simulation examples support the proposed distributed methods.
Index Terms—Distributed Algorithms, multiagent system, optimization, stability.

I. INTRODUCTION
ISTRIBUTED constrained optimization of multiagent
systems aims to minimize the sum of local objective functions while satisfying local constraints. Each agent’s objective
function and constraints might be unknown by other agents due
to security, privacy, and communication concerns. There is a
large body of literature on distributed constrained optimization

D

Manuscript received May 15, 2020; revised July 22, 2020; accepted September 12, 2020. Date of publication September 29, 2020;
date of current version February 26, 2021. The work of D. Li was
supported by the National Nature Science Foundation of China under Grant 61773260 and Grant 61590925; in part by the Key Research and Development Program of the Ministry of Science, and
Technology under Grant 2018YFB1305902; and in part by Chinese
Scholarship Council. The work of N. Li was supported by the National Nature Science Foundation of China under Grant 61773260
and Grant 61590925; and in part by the Key Research and Development Program of the Ministry of Science, and Technology under Grant
2018YFB1305902. Recommended by Associate Editor I. Shames.
(Corresponding author: Ning Li.)
Dewen Li and Ning Li are with the Department of Automation, Shanghai Jiao Tong University, Shanghai 200240, China (e-mail: lidewensju@sjtu.edu.cn; ning_li@sjtu.edu.cn).
Frank Lewis is with the UTA Research Institute, The University
of Texas at Arlington, Fort Worth, TX 76118-7115 USA (e-mail:
lewis@uta.edu).
Digital Object Identifier 10.1109/TCNS.2020.3027787

techniques. Projection-based distributed algorithms in [1] and
[2] accommodate inequality constraints, but could result in
nonsmooth dynamics and discontinuous Lyapunov derivatives.
An alternative Lagrangian-based distributed method [3] assumes
that constraints are global and known by all agents. In [4] and
[5], local constraints are private, yet simple and independent.
When considering private and coupled local constraints, many
methods exist for both discrete-time [6]–[8] and continuous-time
systems [9]–[11]. The latter case is considered here due to the
well-developed stability theory for continuous-time systems.
Distributed primal-dual dynamics provides an efficient way to
find the saddle point of the Lagrangian, and is increasingly used
in constrained convex optimization problems [12]–[14]. In [12],
a distributed nonsmooth primal-dual dynamics was designed via
an augmented Lagrangian, and its stability is studied through
Lyapunov functions and LaSalle invariance principle. When
considering inequality constraints, the primal-dual dynamics
is generally modified by using positive projection methods to
guarantee the non-negativity of the Lagrange multiplier in the
Karush-Kuhn-Tucker (KKT) conditions. This could make the
dynamics nonsmooth and complicates the stability proof. To
our knowledge, very few continuous-time distributed algorithms
with smooth primal-dual dynamics exist for constrained optimization problems. Based on the general Lagrange multiplier method (GLMM), a distributed algorithm with smooth
dynamics solves the constrained optimization with inequality
constraints [14]. However, the objective function of each agent
is assumed to be independent of other agents’ actions. It must
be noted that the original primal-dual dynamics requires each
constraint to be locally known to each agent. The traditional distributed primal-dual dynamics cannot be directly implemented
when considering the resource allocation constraint, which requires the global information from all the agents and is a crucial
constraint in both the resource allocation problem [15], [16] and
the economic dispatch problem [17], [18].
In this article, we present a novel generalized constrained convex optimization model. It contains the local, coupled equality
and inequality constraints, and a global resource allocation constraint. The proposed model unifies the traditional constrained
optimization problem, the resource allocation problem, and the
economic dispatch problem. Then, we propose a distributed
solution with smooth primal-dual dynamics. The recent paper [19], that studies multicoalition noncooperative games, is
most relevant to our work. There, in each coalition, the agents

2325-5870 © 2020 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.
See https://www.ieee.org/publications/rights/index.html for more information.

Authorized licensed use limited to: XiangTan University. Downloaded on June 05,2026 at 03:34:18 UTC from IEEE Xplore. Restrictions apply.

414

IEEE TRANSACTIONS ON CONTROL OF NETWORK SYSTEMS, VOL. 8, NO. 1, MARCH 2021

aim to cooperatively minimize the sum of the objective functions
with no constraints. Here, we extend [19] to solve a similar
problem but with constraints. The salient contributions of this
article can be summarized as follows:
1) A new generalized constrained optimization model is formulated by adding the resource allocation constraint to the traditional constrained optimization problem. This new model unifies
the traditional constrained optimization problem, the resource
allocation problem, and the economic dispatch problem.
2) Distributed solutions are provided given a prescribed interaction graph topology.
3) A dynamic average consensus protocol estimates the gradient of the global Lagrangian for each agent. Therefore, we only
assume the global objective function to be convex, in contrast
to the majority of literature [1]–[18] where each local objective
function is required to be convex.
4) The original primal-dual dynamics in [20] requires each
constraint to be locally known to each agent, and cannot handle
the resource allocation constraint in a distributed manner. To
confront this issue, we execute two dynamic average consensus
protocols in parallel. One estimates the gradient information,
and the other one calculates the resource mismatch. Then, each
agent implements the primal-dual dynamics using the estimated
information.
5) We use a variant of the primal-dual dynamics in [20] to
develop a new smooth distributed algorithm that deals with the
inequality constraints. This algorithm is based on the generalized
Lagrangian multiplier method (GLMM) and, as such, is smooth,
in contrast to the projection methods in [10] and [12]. This yields
a continuous Lyapunov derivative and enables the exponential
stability analysis for the proposed algorithm.
The rest of this article is organized as follows. The problem
formulation and some preliminaries are introduced in Section II.
Section III presents a distributed algorithm to solve the proposed
optimization problem and considers the distributed solutions to
the reduced problems. These algorithms are supported by the
simulation examples in Section IV. Finally, Section V concludes
this article.
Notations: The following notations are used throughout this
article R is the set of real numbers. IN ×N ∈ RN ×N is an identity
matrix, and 1 ∈ RN is the vector with all entries of one. Diag{}
denotes the diagonal matrix. Qmax {} and Qmin {} represent the
maximum and minimum eigenvalues of a square matrix. [•]vec
builds a vector from its arguments. ⊗ is the Kronecker product. ◦
denotes the Hadamard product. For matrix M , M ◦2 = M ◦ M .
∅ represents the entry-wise division for matrices.
II. PROBLEM FORMULATION AND PRELIMINARIES
In this section, we formulate a generalized optimization model
that includes the traditional constrained optimization problem,
the resource allocation problem, and the economic dispatch
problem. Then, some preliminaries are given.
A. Generalized Constrained Optimization Problem
Consider a group of agents, A = {1, 2,. . .,N }, where each
agent can only interact with its neighbors on a communication

graph. Define x = [x1 , x2 , . . ., xN ]T where xi ∈ R denotes the
action of agent i. Suppose each agent has a local objective function fi (x), and local constraints hi (x) = 0 and gi (x) ≤ 0, where
hi (x) is assumed to be linear, i.e., hi (x) := BiT x − ci with
are endowed with the
Bi ∈ RN and ci ∈ R. Moreover, all agents 
resource allocation constraint hr (x) := N
i=1 (xi − di ) = 0,
where di ∈ R is a constant and only known by agent i. Note
that hr (x) integrates all agents’ information and is unknown by
any individual agent.
The agents’ goal is to cooperatively find the optimal solution
to the following problem:
N


fi (x)

(1a)

s.t. h(x) = 0

(1b)

hr (x) = 0

(1c)

g(x) ≤ 0

(1d)

min f (x) =

i=1

where
h(x) = [h1 (x), . . ., hN (x)]T = Bx − c
with
and
c = [c1 , c2 , . . ., cN ]T .
B = [B1 , B2 , . . ., BN ]T
g(x) = [g1 (x), . . ., gN (x)]T and g(x) ≤ 0 means gi (x) ≤ 0 for
i ∈ {1, 2, . . ., N }.
Remark 1: (1) is a generalized constrained optimization problem. When considering only the constraints in (1b) and (1d),
(1) is a traditional constrained optimization problem. When
fi (x) = fi (xi ) and considering only the constraint in (1c),
(1) becomes the resource allocation problem [15], [16]. This
problem is turned into the economic dispatch problem [17], [18]
if adding the inequality constraints in (1d).
Remark 2: For notational convenience, without loss of generality, we use one equality constraint hi (x) and one inequality
constraint gi (x) to represent the local constraints for each agent.
Remove Bi and ci from h(x) when agent i has no equality constraint, and delete gi (x) if agent i has no inequality constraint.
The following two assumptions are made to satisfy the Slater’s
condition and convexity condition of (1), respectively.
Assumption 1: Problem (1) has at least one finite solution
and the constraints are strictly feasible.
Assumption 2: For i ∈ {1, 2, . . ., N }, fi (x) and gi (x) are
C 2 , gi (x) is convex, and f (x) is strictly convex.
Remark 3: The traditional distributed optimization methods [1]–[18] assume that each fi (x) in (1a) is convex. Our new
optimization problem (1) only requires a milder condition of the
convexity of f (x) in (1a).
B. Graph Theory
The network topology can be described by a graph
G = (A, E), where A = {1, 2, . . ., N } denotes the agents and
E ⊂ N × N is the set of the edges in the graph. Define A =
[aij ]N ×N to be the adjacency matrix of G. In this article, the
graph is supposed to be simple, i.e., aii = 0, aij = 1 if agent
j is a neighbor of agent i; otherwise aij = 0. G is said to
be undirected if and only if aij = 1 ⇔ aji = 1. The in-degree

N
matrix of G is given as D = Diag{ N
j=1 a1j , . . .,
j=1 aN j }.
The Laplacian matrix is defined as L = D − A.

Authorized licensed use limited to: XiangTan University. Downloaded on June 05,2026 at 03:34:18 UTC from IEEE Xplore. Restrictions apply.

LI et al.: PROJECTION-FREE DISTRIBUTED OPTIMIZATION WITH NONCONVEX LOCAL OBJECTIVE FUNCTIONS

415

Definition 1: (Interaction Graph Topology) For the generalized optimization problem in (1), agent j is said to be a neighbor
of agent i if and only if fi /hi /gi explicitly depends on xj or,
equivalently, fj /hj /gj explicitly depends on xi .
Assumption 3: The graph G is undirected and connected.

(4a)

C. Dynamic Average Consensus
For a multi-agent network, consider the dynamics


N
q̇i = −qi − N
j=1 aij (qi − qj ) −
j=1 aij (si − sj ) + ui
N
ṡi = j=1 aij (qi − qj )
(2)
where qi is the state, si is an auxiliary variable, and ui is the
input for i ∈ {1, 2, . . ., N }. Then, the following result can be
derived [21].
Lemma 1: Suppose that the graph is undirected and conT
nected. Then, q → 11
N u exponentially.
III. MAIN RESULTS
In this section, we first deal with (1) with all the constraints
in (1b)–(1d). Then, we consider the distributed solutions to the
reduced problems of (1) when considering specific constraints.
We do not use the positive projection methods [10], [12], [20] to
handle the inequality constraints, since it makes the dynamics
nonsmooth. Rather, we utilize the idea of generalized Lagrange
multiplier method (GLMM) [14] to design a new smooth algorithm that deals with inequality constraints. GLMM results
in a new primal-dual algorithm that is smooth yet satisfies the
inequality constraints. Meanwhile, the continuous Lyapunov
derivative enables the exponential stability analysis for the proposed algorithm.
A. GLMM for the Proposed Problem
The original primal-dual dynamics requires each constraint
to be locally known to each agent and cannot be directly implemented when considering the resource allocation constraint
hr = 0. To handle this issue, we firstly assume that each agent
has an additional local equality constraint hr,i := hr = 0 for i =
1, 2, . . ., N . One can replace the resource allocation constraint
hr = 0 by hr,i = 0 for i = 1, 2, . . ., N in (1c). It can be concluded that the solution remains unchanged after the above modification. Without confusion, we use hr = [hr,1 , hr,2 , . . ., hr,N ]T
for notational simplification. Then, Problem (1) has a generalized Lagrangian,
L(x, λ, λr , ν) = f (x)+

N


(λi hi (x)+λr,i hr,i (x) + νi2 gi (x))

i=1

(3)
where λi , λr,i , νi2 ∈ R are the Lagrange multipliers for the
equality constraint, the resource allocation constraint, and the
inequality constraint, respectively. Note that the generalized
Lagrange multipliers for the inequality constraints are νi2 , for
i = 1, 2, . . ., N , that guarantees the non-negativity without any
positive projection. Define the vectors of the Lagrange multipliers as λ = [λ1 , λ2 , . . ., λN ]T , λr = [λr,1 , λr,2 , . . ., λr,N ]T , and
v = [v1 , v2 , . . ., vN ]T , respectively. Then, the generalized KKT

conditions for (1) are

N 
∗
∂f (x∗ )  ∗ ∂hi (x∗ )
∗
∗2 ∂gi (x )
+
+ λr,i + νi
λi
= 0,
∂x
∂x
∂x
i=1
ν ∗ ◦ ν ∗ ◦ g(x∗ ) = 0,
∗

h(x ) = 0,
∗

hr (x ) = 0,
∗

g(x ) ≤ 0.

(4b)
(4c)
(4d)
(4e)

The generalized primal-dual dynamics under L is


N
N
N

∂f (x)  ∂hi (x) 
2 ∂gi (x)
+
+
λi
λr,i +
νi
ẋ = −
,
∂x
∂x
∂x
i=1
i=1
i=1
(5a)
λ̇ = h(x),

(5b)

λ̇r = hr (x),

(5c)

ν̇ = 2ν ◦ g(x).

(5d)

According to [14], given an initial point (x0 , λ0 , λr,0 , ν0 ) and
ν0 = 0, x(t), λ(t), λr (t), and ν(t) generated by (5) will converge
to (x∗ , λ∗ , λ∗r , ν ∗ ).
Remark 4: [14] focus on the stability analysis of the generalized primal-dual dynamics. When dealing with the distributed
optimization problem, [14] assumes that the local convex objective function only depends on the local action, i.e., fi (x) =
fi (xi ). Moreover, it only considers a single coupled inequality
constraint. Rather, we utilize the idea of GLMM to design a
new smooth algorithm that deals with the coupled objective
functions and constraints. Furthermore, the objective functions
in our problem can be nonconvex.
B. Distributed Algorithm Based on GLMM
Next, based on GLMM and the dynamic average consensus
protocol (2), we propose a distributed solution to (1), as shown
in Algorithm 1.
Remark 5: In (7), ε is a global parameter for each agent.
To achieve distributed implementation, we assume each agent
has a local parameter εi,0 . Before implementing the algorithm,
each parameter reaches consensus by using the dynamic
average
consensus protocol in (6). Based on Lemma 1, εi → N1 N
i=1 εi .
Here, we use ε for each agent for notational simplification.
Remark 6: (8) takes the dynamic average consensus protocol
∂L
for agent i. Meanwhile, (9)
(2) to estimate the gradient N1 ∂x
i
is a second dynamic average consensus protocol
 that estimates
the average resource allocation mismatch N1 N
i=1 (xi − di ) for
each agent.
Remark 7: When considering the economic dispatch problem, as described in Remark 1, it incorporates the resource
allocation constraint (1c) and the inequality constraint (1 d).
Moreover, the local objective function only depends on the
local action, i.e., fi (x) = fi (xi ), and the inequality constraint
.
becomes the capacity constraint, i.e., gi (x) = gi (xi ) ≤ xmax
i
Here, we only take the upper capacity bound for simplicity. Thus,

Authorized licensed use limited to: XiangTan University. Downloaded on June 05,2026 at 03:34:18 UTC from IEEE Xplore. Restrictions apply.

416

IEEE TRANSACTIONS ON CONTROL OF NETWORK SYSTEMS, VOL. 8, NO. 1, MARCH 2021

Algorithm 1: Distributed Solution for Constrained Optimization Problem With Resource Allocation Constraint.
Input: Time constants εi,0 for i = 1, 2, . . ., N .
Output: Solution of (1), x∗ .
1: Unify the time constants εi by the dynamic average
consensus protocol (2):
⎧

⎪
εi − N
⎨ ε̇i = − 
j=1 aij (εi − εj )
(6)
a
(ρi − ρj ) + εi,0
− N
ij
N j=1
⎪
⎩
ρ̇i = j=1 aij (εi − εj )
2:

where ρi for i = 1, 2, . . ., N are the auxiliary variables.
Let ε to be the unified time constant from (6). For each
agent i, the action xi , the Lagrange multipliers λi , λr,i ,
and νi are updated according to the following
dynamics:
⎧
ẋi = −εpii
⎪
⎪
⎨
λ̇i = εhi (x)
(7)
⎪
λ̇r,i = εςi
⎪
⎩
ν̇i = 2εvi gi (x)
where pij and ςi , for i, j ∈ {1, 2, . . ., N } are generated
by
⎧
N
⎪
ij −
m=1 aim (pij − pmj )
⎪ ṗij = −p

⎪
⎨
− N
a
m=1 im (sij − smj )
∂fi (x)
i (x)
i (x)
+( ∂xj + λi ∂h∂x
+ λr,i + νi2 ∂g∂x
)
⎪
⎪
j
j
⎪

⎩
ṡij = N
a
(p
−
p
)
mj
m=1 im ij
(8)
⎧
N
⎪
i−
⎨ ς˙i = −ς
j=1 aij (ςi − ςj )
(9)
a
(ξ − ξj ) + (xi − di )
− N
N j=1 ij i
⎪˙
⎩
ξi = j=1 aij (ςi − ςj )
with auxiliary variables sij and ξi .

the gradient of L, in (3), with respect to xi becomes
∂L/∂xi = ∂fi (xi )/∂xi +

N


λr,j + vi2 ,

j=1



(11)
⎧
N
N
⎨ ς˙i = −ςi − j=1 aij (ςi − ςj ) − j=1 aij (ξi − ξj )
+(x − di )
(12)
 i
⎩˙
ξi = N
a
(ς
−
ς
).
ij
i
j
j=1
C. Discussion on Distributed Solution
Rewrite (7)–(9) in global compact form
⎧
ẋ = −ε[pii ]vec
⎪
⎪
⎨
λ̇ = εh(x)
⎪
λ̇ = ε[ςi ]vec
⎪
⎩ r
ν̇ = 2εv ◦ g(x),
⎧
⎨ ṗ = −(IN ×N + L ⊗ IN ×N )p − (L ⊗ IN ×N )s
+F (x, λ, λr , v)
⎩
ṡ = (L ⊗ IN ×N )p,
ς˙ = −(IN ×N + L)ς − Lξ + Γ(x)
ξ˙ = Lς

(13)

(14)

(15)

where
p = [p11 , p12 , . . ., p1˜N , p21 , . . ., pN N ]T ,
s=
∂f1 (x)
T
[s11 , s12 , . . . s1˜N , s21 , . . ., sN N ] , F (x, λ, λr , v) = [ ∂x1 +
1 (x)
1 (x)
λ1 ∂h∂x
+ λr,1 + v12 ∂g∂x
,
1
1

∂f1 (x)
∂h1 (x)
∂x2 + λ1 ∂x2 + λr,1 +
1 (x)
v12 ∂g∂x
, . . .]T , ς = [ς1 , ς2 , . . ., ςN ]T , ξ = [ξ1 , ξ2 , . . ., ξN ]T ,
2
Γ(x) = [x1 − d1 , . . ., xN − dN ]T . Following [19], let
2
U1 ∈ RN ×N be such that U1T (L ⊗ IN ×N ) = 0. U = [U1 U2 ]
N 2 ×N 2

and
is an orthonormal matrix, with U ∈ R
2
2
U2 ∈ RN ×(N −N ) . Define W1 ∈ RN such that W1T L = 0.
W = [W1 W2 ] is an orthonormal matrix, where W ∈ RN ×N and
W2 ∈ RN ×(N −1) . Replace s = U [q̂ T q T ]T and ξ = W [ζ̂ζ T ]T ,
2
where q̂ ∈ RN , q ∈ RN −N , ζ̂ ∈ R, and ζ ∈ RN −1 , to obtain
˙
˙ 0, ζ̂=
q̂=
0, and
⎧
⎨ ṗ = −(IN ×N + L ⊗ IN ×N )p − (L ⊗ IN ×N )U2 q
+F (x, λ, λr , v)
(16)
⎩
q̇ = U2T (L ⊗ IN ×N )p,

N

and we only need to estimate j=1 λr,j for each agent. In this
case, (7)-(9)
 can be reduced to (10)-(12). Based on Lemma 1,
pi → N1 N
j=1 λr,j . Note that the original primal-dual dynamics
cannot be directly implemented in a distributed manner when
considering global constraints. By using (9), the global resource
allocation constraint in (1c) is transferred to local equality
constraints hr,i in (3) and we can utilize the generalized primaldual dynamics to design the distributed algorithm. Unlike the
leader-following algorithm in [17] and augmented Lagrangian
methods in [12], [14], the proposed algorithm requires no leaderagent and fits the plug-and-play requirement when dealing with
economic dispatch problems.
⎧
∂fi (xi )
+ pi + vi2 )
⎨ ẋi = −ε( ∂x
i
(10)
⎩ λ̇r,i = εςi
ν̇i = 2εvi gi (xi ),



aij (pi − pj ) − N
ṗi = −ςi − N
j=1
j=1 aij (si − sj ) + λr,i
N
ṡi = j=1 aij (pi − pj ),

ς˙ = −(I + L)ς − LW2 ζ + Γ(x)
ζ̇ = W2T Lς.




(17)

Thus, q̂ ≡ q̂(0), ζ ≡ ζ (0) and can be eliminated to obtain
 
  

ṗ
p
F (x, λ, λr , v)
=H
+
,
(18)
q̇
q
0N 2 −N

 
  
ς˙
ς
Γ(x)
(19)
=K
+
ζ
0N −1
ζ̇
where
H = −[IN ×N + L ⊗ IN ×N / − U2T (L ⊗ IN ×N )
× (L ⊗ IN ×N )U2 /0]
and



K = − (I + L)/−W2T LLW2 /0 .

Authorized licensed use limited to: XiangTan University. Downloaded on June 05,2026 at 03:34:18 UTC from IEEE Xplore. Restrictions apply.

LI et al.: PROJECTION-FREE DISTRIBUTED OPTIMIZATION WITH NONCONVEX LOCAL OBJECTIVE FUNCTIONS

are both Hurwitz [19].
Define T = εt. Writing (13), (18), and (19) in the T -time
scale gives
⎧ d
⎪
dT x = −[pii ]vec
⎪
⎪ d
⎨
dT λ = h(x)
(20)
d
⎪
⎪
dT λ = [ςi ]vec
⎪
⎩ d
dT ν = 2v ◦ g(x),


  

d
p
F (x, λ, λr , v)
dT p
ε d
=H
+
,
(21)
0N 2 −N
q
dT q


  

d
ς
Γ(x)
dT ς
ε d
=K
+
.
(22)
ζ
0N −1
ζ
dT

Setting the right side of (21) and (22) to 0, we have the quasisteady states, p̄, q̄, ς¯, and ζ̄ as


 
F (x, λ, λr , v)
p̄
,
(23)
= H −1
0N 2 −N
q̄


 
Γ(x)
ς¯
.
(24)
= K−1
0N −1
ζ̄

∂L
and ς¯i = N1 N
Based on Lemma 1, p̄ii = N1 ∂x
j=1
i
(xj − dj ). Define p̃ = p − p̄, q̃ = q − q̄, ς˜ = ς − ς¯, and
ζ̃ = ζ − ζ̄, we have
⎡
⎤
dx
⎥
⎤⎢
⎡
⎢ dT
⎥
⎡
⎤
∂
p̄
∂
p̄
∂
p̄
∂
p̄
dλ
⎢
⎥


dp̃
⎢
⎥
p̃
⎥
⎢
∂x
∂λ
∂λ
∂ν
dT
⎢
r
⎣
⎦
ε dT = H
− ε ⎣ ∂ q̄ ∂ q̄ ∂ q̄ ∂ q̄ ⎦ ⎢ dλ ⎥
⎥ , (25)
q̃
r
dq̃
⎢
⎥
dT
∂x ∂λ ∂λr ∂ν ⎢
dT ⎥
⎣ dν
⎦
dT
⎤
⎤
⎡
∂
ς
¯
d
 
ς
˜
ς˜
⎢
⎥
⎢ ∂x ⎥ dx
ε ⎣ dT
d ⎦ = K ζ̃ − ε ⎣ ∂ ζ̄ ⎦ dT .
ζ̃
dT
∂x
⎡

(26)

ς T , ζ̃ T ]T →
Letting ε → 0 gives [p̃T , q̃ T ]T → 02N 2 −N and [˜
02N −1 since H and K are Hurwitz. Then p and ς reach the
quasi-steady states and (20) becomes
⎧
d
1 ∂L
⎪
⎪
x=−
⎪
⎪
dT
N
∂x
⎪
⎪
⎪
⎪
d
∂L
⎪
⎨
λ=
dT
∂λ
.
(27)
d
1 ∂L
⎪
⎪
⎪
λ
=
r
⎪
⎪
dT
N ∂λr
⎪
⎪
⎪
∂L
d
⎪
⎩
ν=
dT
∂ν
Note that (27) is exactly the generalized primal-dual dynamics
(5) by which the solution is supposed to be obtained.
D. Convergency Results
In this part, we first present the convergency result of the
distributed algorithm (7)–(9). Then, we consider the solutions
to reduced problems of (1) with special constraints.

417

Theorem 1: Suppose Assumptions 1–3 are satisfied. Given an
initial point (x0 , λ0 , λr,0 , ν0 ) and ν0 = 0, if ε → 0, the solution
of system (20)–(22) approaches to the solution to (1), and the
agents’ actions converge to the solution to (1) under (7)–(9).
See Appendix A for the proof.

Remark 8: According to the proof, ε is supposed to be a small
positive constant. In the implementation, ε can be started with
a small number, then one can increase ε to get fast convergency
before the system gets divergent. Note that during the tuning of
ε, each agent reaches the same value of ε by using the dynamic
average consensus protocol (6).
In the following, we consider reduced problems of (1) when
considering specific constraints in (1b) and (1d). We first deal
with (1) with the local equality constraint (1b). In this case,
problem (1) reduces to problem (28) and Algorithm 1 reduces
to Algorithm 2

min f (x) =

N


fi (x), s.t., h(x) = Bx − c = 0

(28)

i=1

Algorithm 2: Distributed Solution With Equality Contraint.
Input: Time constants εi,0 for i = 1, 2, . . ., N .
Output: Solution of (28), x∗ .
1: Unify the time constant ε as shown in (6).
2: For each agent i, the action xi and the Lagrange
multipliers λi are updated according to the following
dynamics:
ẋi = −εpii
λ̇i = εhi (x)

(29)

where pij , for i, j ∈ {1, 2, . . ., N } are generated by
⎧

⎪
− N
⎨ ṗij = −p
m=1 aim (pij − pmj )
ijN
i (x)
i (x)
− m=1 aim (sij − smj ) + ( ∂f∂x
+ λi ∂h∂x
)
j
j
⎪

⎩
N
ṡij = m=1 aim (pij − pmj )
(30)
with auxiliary variables sij .
It can be concluded from the analysis of Algorithm 1 that
Algorithm 2 is convergent and provides the solution to (28).
Following assumption is made to facilitate the exponential stability analysis of Algorithm 2.
Assumption 4: For the equality constraint h(x) = Bx − c in
(1b), B is full row rank. The global objective function f (x) in
(1a) is u-strictly convex, i.e., for x, x ∈ RN ,


∂f (x) ∂f (x )
2
−
(x − x )T
(31)
≥ μx − x  .
∂x
∂x
In this case, (20)–(21) reduce to
⎧
⎪
⎨ d x = −[pii ]vec
dT
,
⎪
⎩ d λ = h(x) = Bx − c
dT
⎤
⎡
d

  
p
F (x, λ)
p
⎥
⎢
.
ε ⎣ dT ⎦ = H
+
d
q
0N 2 −N
q
dT

Authorized licensed use limited to: XiangTan University. Downloaded on June 05,2026 at 03:34:18 UTC from IEEE Xplore. Restrictions apply.

(32)

(33)

418

IEEE TRANSACTIONS ON CONTROL OF NETWORK SYSTEMS, VOL. 8, NO. 1, MARCH 2021

Corollary 1: Suppose Assumptions 1–4 are satisfied. There
exist a positive constant ε∗ such that (32)–(33) is exponentially
stable for ε ∈ (0, ε∗ ). Hence, the agents’ actions exponentially
converge to the optimal solution to (28) under (29)–(30).
See Appendix B for the proof.

Next, we consider problem (1) with the local, coupled equality
and inequality constraints in (1b) and (1d). In this case, problem
(1) reduces to the following problem (34) and Algorithm 1
reduces to Algorithm 3

min f (x) =

N


Fig. 1.

The Communication Graph of the Five-agent Network.

TABLE I
COMPARISON OF THE STATES COMMUNICATED BETWEEN AGENTS

fi (x), s.t., h(x) = Bx − c = 0, g(x) ≤ 0

i=1

(34)
Algorithm 3: Distributed Solution With Equality and Inequality Contraint.
Input: Time constants εi,0 for i = 1, 2, . . ., N .
Output: Solution of (34), x∗ .
1: Unify the time constant ε as shown in (6).
2: For each agent i, the action xi , the Lagrange
multipliers λi , and νi are updated according to the
following dynamics:
⎧
⎨ ẋi = −εpii
(35)
λ̇ = εhi (x)
⎩ i
ν̇i = 2εvi gi (x)
where pij , for i, j ∈ {1, 2, . . ., N } are generated by
⎧

⎪
ṗij = −pij − N
m=1 aim (pij − pmj )
⎪

⎪
⎨
− N
a
m=1 im (sij − smj )
(36)
∂fi (x)
i (x)
i (x)
+( ∂xj + λi ∂h∂x
+νi2 ∂g∂x
)
⎪
⎪
j
j
⎪

⎩
ṡij = N
m=1 aim (pij − pmj )
with auxiliary variables sij .
It can be concluded from the proof of Theorem 1 that (35)–
(36) is stable and provides solution to (34). Next, we consider the
linear form for the inequality constraint, i.e., g(x) = Bg x − cg .
Assumption 5: For the equality constraint h(x) = Bx − c in
(1b) and inequality constraint g(x) = Bg x − cg in (1d), B =
[B T , BgT ]T is full row rank. The global objective function f (x)
in (1a) is u-strictly convex.
In this case, (20)–(21) reduces to
⎧ d
⎨ dT x = −[pii ]vec
d
(37)
λ = h(x) = Bx − c
⎩ dT
d
v
=
2v
◦
g(x)
=
2v
◦
(B
x
−
c
),
g
g
dT
 d 

  
p
p
F (x, λ, v)
ε dT
.
(38)
=
H
+
d
q
0N 2 −N
dT q
Corollary 2: Suppose Assumptions 1-3 and 5 hold. There
exist a positive constant ε∗ such that (37)–(38) is exponentially
stable for ε ∈ (0, ε∗ ). Then, the agents’ actions exponentially
converge to the optimal solution to (34) under (35)–(36).
See Appendix C for the proof.

Remark 9: The stability of Algorithm 3 is based on a continuous Lyapunov derivative. For the projection-based methods,
it is generally hard to directly verify the stability by only using

Lyapunov functions. For example, in [10], [12], [20], the stability
analysis involves a discontinuous Lyapunov derivative. Moreover, the continuous Lyapunov derivative enables to achieve
exponential convergency.
Remark 10: Based on the proofs, Corollaries 1 and 2 provide
locally exponential convergency, i.e, ε∗ relates to the initial
point. When the objective function f (x) is quadratic, the second
derivatives of f (x) are positive constants, and Corollaries 1 and
2 give the globally exponential convergency.
IV. SIMULATION EXAMPLES
A. Numerical Example on a Five-agent Network
In this part, we consider constrained optimization problems
for a five-agent network with the communication graph shown
in Fig. 1.
1) Nonconvex Local Functions: The non-convex, local
objective functions are f1 (x) = x21 + 4x23 − x24 − x1 x2 + ex1 ,
f2 (x) = 2x22 + 2x1 x2 − x23 − ex1 , f3 (x) = x21 − 2x23 + x2 x3 ,
f4 (x) = −x21 + 3x24 , and f5 (x) = −x22 + x25 . The constraints
are given as h1 (x) = 0.5x1 + x2 + x3 − x4 − 2 = 0, h2 (x) =
x2 +x5 − 1 = 0 and g2 (x) = x2 + x3 − 2 ≤ 0. It can be verified that Assumptions 1-3 hold. Using the centralized primaldual dynamics [20], the optimal solution is [x∗1 , x∗2 , x∗3 , x∗4 , x∗5 ] =
[0.0930, 0.8372, 0.6047, −0.5116, 0.1628]. Initialize all actions
to be 0 and Lagrange multipliers as random in the region of (0,1).
We use the ode45 in MATLAB as the solver. The agents’ actions
generated by Algorithm 1 are shown in Fig. 2(a). All the actions
converge to the optimal solution. We compare Algorithm 1 with
the distributed algorithms in [10] and [12] which produce the
divergent results as shown in Fig. 2(b) and Fig. 2(c). Note that
all the local objective functions are nonconvex. [10] and [12]
fail in this situation. It verifies the milder assumption of the
convexity in the proposed algorithm.
2) Convex Local Functions: To facilitate the comparison of the proposed algorithm with existing algorithms,
we modify the local functions such that they satisfy the
convexity assumptions in [10] and [12]. The local functions are f1 (x) = x1 − x2 − x3 − x4 , f2 (x) = x21 + 2x22 +
(x3 − 2)2 + x24 + x25 + x1 x2 , f3 (x) = x21 + x2 x3 , f4 (x) =
x21 + 3x24 and f5 (x) = x22 + 4x25 . The constraints are the same
in Case I. Table I presents the states communicated between
agents in different algorithms. Therein, the states are represented
by the symbols in the corresponding papers, and N = 5 in this

Authorized licensed use limited to: XiangTan University. Downloaded on June 05,2026 at 03:34:18 UTC from IEEE Xplore. Restrictions apply.

LI et al.: PROJECTION-FREE DISTRIBUTED OPTIMIZATION WITH NONCONVEX LOCAL OBJECTIVE FUNCTIONS

419

Fig. 3. Simulation Results for Agents’ Actions When Considering
Equality Constraint.

Fig. 4. Simulation Results for Agents’ Actions When Considering
Equality and Inequality Constraints.

Fig. 5.
Fig. 2. Simulation Results for Agents Actions in Case I. (a) Response
of actions by Algorithm 1. (b) Response of actions by Algorithm in [10].
(c) Response of actions by Algorithm in [12].

case. Fig. 3 gives the simulation result when only considering
equality constraints. Fig. 4 shows the result when considering
both equality and inequality constraints. As shown, the numbers
of the states communicated in different algorithms are similar.
However, the proposed algorithms provide faster convergency.
This advantage can be explained by the fact that assumptions
1–5 are satisfied in this case and Algorithms 2 and 3 achieve
exponential convergency.
B. Multi-Area Economic Dispatch
In this part, we consider the multi-area economic dispatch
problem (MAEDP) which aims to minimize the total generation
cost while satisfying certain constraints. The system contains
three areas and six agents (generators) as shown in Fig. 5. The
cost functions for the agents are given as quadratic functions
fi (xi ) = σi x2i + τi xi + κi for i = 1, 2, . . ., 6. σi > 0, τi , and
κi are constants and the detailed values are taken from [22].

Communication Graph for the Multi-area System.

The MAEDP must satisfy the resoure allocation constraint in
min
max
(1c) and the generation capacity constraints
6xi ≤ xi ≤ xi .
We assume that the total power load is i=1 di = 1200M W .
Meanwhile, the output powers of agent 1 and agent 2 are endowed with a local constraint, h1 = x1 + x2 = 600M W . The
simulation result is given in Fig. 6. As seen, the ouput powers
for each agent achieve optimal soulution and the constraints are
all satisfied.
V. CONCLUSION
We have formulated a distributed continuous-time solution to
a generalized constrained convex optimization model that unifies
the traditional constrained optimization problem, the resource
allocation problem, and the economic dispatch problem. The dynamic average consensus protocol estimates the global gradient
information and the convexity requirement for each local objective function is relaxed to the convexity requirement of the global
objective function. To incorporate the inequality constraints,
the generalized Lagrange multiplier method modifies the
primal-dual dynamics such that the dynamics is smooth and the

Authorized licensed use limited to: XiangTan University. Downloaded on June 05,2026 at 03:34:18 UTC from IEEE Xplore. Restrictions apply.

420

IEEE TRANSACTIONS ON CONTROL OF NETWORK SYSTEMS, VOL. 8, NO. 1, MARCH 2021

Lemma 2: (Tikhonov’s theor., [23]) When ε → 0, the solution
of (39) approaches to the solution of the degenerate system (40)
if z = ωz (x, t) is a stable root of the adjoined system (41).
According to Tikhonov’s theorem, the adjoined system of
(21)–(22) is

 d 
  
dT p = H p + F (x, λ, λr , v) ,
(42)
d
0N 2 −N
q
dT q

 d 
  
ς
Γ(x)
dT ς
,
(43)
=K
+
d
ζ
0N −1
dT ζ

As mentioned before, H and K are both Hurwitz. Therefore,
(42)–(43) has a stable solution, i.e., the quasi-steady states, p̄,
q̄, ς¯, and ζ̄ defined in (23)–(24). Moreover,
based on Lemma

(x
1, p̄ii = 1/N ∂L/∂xi and ς¯i = 1/N N
j − dj ) for i =
j=1
1, 2, . . ., N . Substituting p̄ii and ς¯ii to (20) gives
⎧ d
x = − N1 ∂L
⎪
∂x
⎪
⎨ dT
d
∂L
dT λ = ∂λ
(44)
d
∂L
λr = N1 ∂λ
⎪
dT
⎪
r
⎩ d
∂L
dT ν = ∂ν .

Fig. 6. Simulation Results of the MAEDP By Using Algorithm 1.
(a) Output Power for each agent. (b) Power mismatches.

Lyapunov derivative is continuous. The stability and optimality
are proven by using Lyapunov functions. Simulation examples
illustrate the efficiency of the proposed methods. The application
on a MAEDP is presented to support the proposed model.
APPENDIX A
PROOF FOR THEOREM 1

Next, we prove the stability of (44). Inspired by [14],
let
V1 = N2 x − x∗ 2 + 12 λ − λ∗ 2 + N2 λr − λ∗r 2 +
2
1
1 ∗◦2
∗ 2
(ln ν − ln ν ∗ ) be the Lyapunov
4 (ν − ν  ) − 2 ν
candidate for (44), and define 0 ln 0 = 0. It can be verified that
V1 > 0. Then,


dV1
1
◦2
= (x − x∗ )T , (λ − λ∗ )T , (λr − λ∗r )T , (ν − ν ∗ ∅ν)T
dT
2
⎡
⎤
∂L
− ∂x
⎢ ∂L
⎥
∂λ
⎥
•⎢
∂L
⎣
⎦
∂λr

2ν ◦ g(x)

Since U and W are orthonormal matrices, it can be concluded
the stability of (7)–(9) is equivalent to that of (20)–(22). Note
(20)–(22) can be regarded as a singular perturbation system in
the T -time scale. The stability can be studied by the Tikhonov’s
theorem [23]. We start with the following Lemma of Tikhonov’s
theorem. Cosider the folowing system
dx
= ϕ(x, z, t),
(39a)
dt
dz
ε
= ω(x, z, t).
(39b)
dt
When ε → 0, the above system becomes the following degenerate system:
dx
= ϕ(x, z, t),
dt

(40a)

z = ωz (x, t)

(40b)

= − (x − x∗ )T

∂L
∂L
∂L
+ (λ − λ∗ )T
+ (λr − λ∗r )T
∂x
∂λ
∂λr

◦2

+ (ν ◦2 − ν ∗ )T g(x).

Note that L(x, λ, λr , e) is strictly convex in x and concave
(explicitly linear) in (λ, λr , e). Hence,
dV1
< L(x∗ , λ, λr , e) − L(x, λ, λr , e)
dT
+ L(x, λ, λr , e) − L(x, λ∗ , λ∗r , e∗ )
= L(x∗ , λ, λr , e) − L(x∗ , λ∗ , λ∗r , e∗ )
+ L(x∗ , λ∗ , λ∗r , e∗ ) − L(x, λ∗ , λ∗r , e∗ ) < 0.
∗

dz
= ω(x, z, t).
(41)
dt
Then, the stability of (39) can be studied by the following
theor.

(46)

∗

Thus, (44) is stable at (x , λ , λ∗r , e∗ ). Based on Lemma 2, the
conclusion is obtained.

where ωz (x, t) is the solution of ω(x, z, t) = 0 when regarding
x and t as constants. Define the adjoined system as

(45)



APPENDIX B
PROOF FOR COROLLARY 1
η = [p̃T , q̃ T ]T
Define
χ = [xT , λT ]T ,
∗ T T T
ϑ= [(χ − χ ) ,η ] . Following (25), we have
 ∂ p̄ ∂ p̄ 
dχ
dη
∂λ
ε
= Hη − ε ∂x
.
∂ q̄ ∂ q̄
dT
dT
∂x ∂λ

Authorized licensed use limited to: XiangTan University. Downloaded on June 05,2026 at 03:34:18 UTC from IEEE Xplore. Restrictions apply.

and

(47)

LI et al.: PROJECTION-FREE DISTRIBUTED OPTIMIZATION WITH NONCONVEX LOCAL OBJECTIVE FUNCTIONS

Moreover,
d
1 ∂f (x)
T
dT x = − N ( ∂x + B λ) − [p̃ii ]vec
d
dT λ = h(x) = Bx − d.

Following the derivation of (23), it can be concluded that
p̄ and q̄ are linear in the elements of F (x, λ). Let D1
be a compact set that contains the origin. Then, using Assumption 2, for i, j ∈ {1, 2, . . ., N }, the partial derivatives
∂ p̄
i (x)
i (x)
+ λi ∂h∂x
are bounded for ϑ ∈ D1 . Hence,  ∂x
,
of ∂f∂x
j
j

(48)

Let Vg = (1−θ)V1 + θV2 be the Lyapunov candidate for the
system in (32)–(33), where 0 < θ < 1, V1 = N2 x − x∗ 2 +
1
1
∗ 2
2
2 λ − λ  , and V2 = 2 η . We have

∂ q̄
 ∂∂λp̄ ,  ∂x
, and  ∂∂λq̄  are all bounded for ϑ ∈ D1 . Moreover,
(x)
since f (x) is a C 2 function, ∂f∂x
is Lipschitz. Then, Ω ≤
α2 η2 +ε(α3 χ − χ∗ η + α4 η2 ) for some positive constants α3 , and α4 . Note that α2 = −Qmax (H) > 0. Therefore,

∂V1 dχ
θ ∂V2 dη
dVg
= (1 − θ)
+
dT
∂χ dT
ε ∂η dT


= (1 − θ) N (x − x∗ )T , (λ − λ∗ )T
 1 ∂f (x)

− N ( ∂x + B T λ) − [p̃ii ]vec
•
Bx − c

 ∂ p̄ ∂ p̄ 

dχ
θ T
∂x
∂λ
+ η
Hη − ε ∂ q̄ ∂ q̄
ε
dT
∂x ∂λ
θ
:= (1 − θ)Ψ + Ω.
ε
Thus



dVg
θ
= (1 − θ)Ψ + Ω
dT
ε
≤ − (1 − θ)α1 χ − χ∗ 2 + (1 − θ)N χ − χ∗  η
θ
+ α2 η2 +θα3 χ − χ∗  η + θα4 η2
ε
= − ϑT Θϑ
(49)



(x)
+ B T λ) − N [p̃ii ]vec
−( ∂f∂x
Bx − c
  ∂f (x)

+
B T λ)
−(
T
T
∂x
= (x − x∗ ) , (λ − λ∗ )
Bx − c 


−N
[p̃ii ]vec
.
+ (x − x∗ )T , (λ − λ∗ )T
0
(50)
Considering the first term of the the second line in (50), we
have

  ∂f (x)

−( ∂x + B T λ)
∗ T
∗ T
(x − x ) , (λ − λ )
Bx − c

  ∂f (x) ∂f (x∗ )

− ∂x + B T (λ − λ∗ )
∗ T
∗ T
∂x
= − (x − x ) , (λ − λ )
−B(x−x∗ )



x − x∗
−Diag{μ} −B T
≤ (x − x∗ )T , (λ − λ∗ )T
B 
0
λ − λ∗
 

∗
x−x
:= (x − x∗ )T , (λ − λ∗ )T M
λ − λ∗
= (χ − χ∗ )T M (χ − χ∗ ).
(51)
Note that the inequality in the above formulation is derived
based on the μ-strictly convex property of f (x) in Assumption
4. Since B is full row rank and M is Hurwitz. Let α1 =
−Qmax (M ). We have α1 > 0 and

Ψ=

(x − x∗ )T , (λ − λ∗ )T

Ψ ≤ −α1 χ − χ∗ 2 − N (x − x∗ )T [p̃ii ]
≤ −α1 χ − χ∗ 2 + N x − x∗  p̃
≤ −α1 χ − χ∗ 2 + N χ − χ∗  η .

421

(52)

For Ω in (49), we have

 ∂ p̄ ∂ p̄ 

dχ
∂λ
Ω = η T Hη − ε ∂x
∂ q̄ ∂ q̄
dT
∂x ∂λ


 ∂ p̄ ∂ p̄    ∂f (x)
T
−
+
B
λ
−
[p̃
]
ii vec
∂λ
∂x
= η T Hη − εη T ∂x
∂ q̄ ∂ q̄
Bx − c.
∂x ∂λ
(53)

where


Θ=



(1 − θ)α1
− 12 (1 − θ)N − 12 θα3

(54)

− 12 (1 − θ)N − 12 θα3
θ( αε2 − α4 )


.

Setting the determinant of matrix Θ to 0 gives
ε∗ =

α1 α2
.
1
α1 α4 + 4θ(1−θ) [(1 − θ)N + θα3 ]2

d
Then, for all ε ∈ (0, ε∗ ), Θ is positive-definite and dT
Vg ≤
2
−Qmin (Θ)ϑ < 0.
Therefore, (32)–(33) is exponentially stable for ε ∈ (0, ε∗ ),
and the agents’ actions exponentially converge to the solution
to (28) under (29)–(30).


APPENDIX C
PROOF FOR COROLLARY 2
Following the proof for Corollary 1, define χ =
[xT , λT , eT ]T , η = [p̃T , q̃ T ]T and ϑ= [(χ − χ∗ )T ,η T ]T .
Let Vg = (1−θ)V1 + θV2 be the Lyapunov candidate for the
system in (37)–(38), where 0 < θ < 1, V1 = N2 x − x∗ 2 +
2
1
1
1 ∗◦2
∗ 2
∗ 2
(ln ν − ln ν ∗ ), and
2 λ − λ  + 4 (ν − ν  ) − 2 ν
1
2
V2 = 2 η . Following the derivation of (51), we have
⎡ ∂f (x)
⎤
 −( ∂x + B T λ + BgT e)

T
T
T
⎦
(x − x∗ ) , (λ − λ∗ ) , (e − e∗ ) ⎣
Bx − c
Bg x − cg


= − (x − x∗ )T , (λ − λ∗ )T , (e − e∗ )T
⎡ ∂f (x) ∂f (x∗ )
⎤
T
∗
T
∗
∂x − ∂x + B (λ − λ ) + Bg (e − e )
⎦
• ⎣ −B(x − x∗ )
−Bg (x − x∗ )


≤ (x − x∗ )T , (λ − λ∗ )T , (e − e∗ )T
⎤
⎡
 x − x∗

T
T
−Diag{μ} −[B , Bg ] ⎣
λ − λ∗ ⎦
•
0
[B T , BgT ]T
e − e∗

Authorized licensed use limited to: XiangTan University. Downloaded on June 05,2026 at 03:34:18 UTC from IEEE Xplore. Restrictions apply.

422

IEEE TRANSACTIONS ON CONTROL OF NETWORK SYSTEMS, VOL. 8, NO. 1, MARCH 2021



:= (x − x∗ )T , (λ − λ∗ )T , (e − e∗ )T



= (χ − χ∗ )T M (χ − χ∗ ).

⎡

⎤
x − x∗
M ⎣ λ − λ∗ ⎦
e − e∗
(55)

Based on assumption 5, M is Hurwitz. The rest proof can
refer to Appendix B and is omitted for brevity.


[20] D. Feijer and F. Paganini, “Stability of primal–dual gradient dynamics
and applications to network optimization,” Automatica, vol. 46, no. 12,
pp. 1974–1981, 2010.
[21] A. Menon and J. S. Baras, “Collaborative extremum seeking for welfare optimization,” in Proc. 53rd IEEE Conf. Decis. Control, 2014,
pp. 346–351.
[22] Zwe-Lee Gaing, “Particle swarm optimization to solving the economic
dispatch considering the generator constraints,” IEEE Trans. Power Syst.,
vol. 18, no. 3, pp. 1187–1195, Aug. 2003.
[23] A. N. Tikhonov, “Systems of differential equations containing small
parameters in the derivatives,” Matematicheskii sbornik, vol. 73, no. 3,
pp. 575–586, 1952.

REFERENCES
[1] E. Nozari, P. Tallapragada, and J. Cortës, “Differentially private distributed
convex optimization via functional perturbation,” IEEE Trans. Control
Netw. Syst., vol. 5, no. 1, pp. 395–408, Mar. 2018.
[2] Y. Lou, G. Shi, K. H. Johansson, and Y. Hong, “Approximate projected
consensus for convex intersection computation: Convergence analysis
and critical error angle,” IEEE Trans. Autom. Control, vol. 59, no. 7,
pp. 1722–1736, Jul. 2014.
[3] M. Zhu and S. Martinez, “On distributed convex optimization under
inequality and equality constraints,” IEEE Trans. Autom. Control, vol. 57,
no. 1, pp. 151–164, Jan. 2012.
[4] J. Chen and A. H. Sayed, “Diffusion adaptation strategies for distributed
optimization and learning over networks,” IEEE Trans. Signal Process.,
vol. 60, no. 8, pp. 4289–4305, Aug. 2012.
[5] B. Johansson, T. Keviczky, M. Johansson, and K. H. Johansson, “Subgradient methods and consensus algorithms for solving convex optimization
problems,” in Proc. 47th IEEE Conf. Decis. Control, 2008, pp. 4185–4190.
[6] D. Yuan, Y. Hong, D. W. Ho, and G. Jiang, “Optimal distributed stochastic
mirror descent for strongly convex optimization,” Automatica, vol. 90,
pp. 196–203, 2018.
[7] Q. Liu, S. Yang, and Y. Hong, “Constrained consensus algorithms with
fixed step size for distributed convex optimization over multiagent networks,” IEEE Trans. Autom. Control, vol. 62, no. 8, pp. 4259–4265,
Aug. 2017.
[8] T. Chang, A. Nedić, and A. Scaglione, “Distributed constrained optimization by consensus-based primal-dual perturbation method,” IEEE Trans.
Autom. Control, vol. 59, no. 6, pp. 1524–1538, Jun. 2014.
[9] P. Lin, W. Ren, and J. A. Farrell, “Distributed continuous-time optimization: Nonuniform gradient gains, finite-time convergence, and convex
constraint set,” IEEE Trans. Autom. Control, vol. 62, no. 5, pp. 2239–2253,
May 2017.
[10] Y. Zhu, W. Yu, G. Wen, G. Chen, and W. Ren, “Continuous-time distributed
subgradient algorithm for convex optimization with general constraints,”
IEEE Trans. Autom. Control, vol. 64, no. 4, pp. 1694–1701, Apr. 2019.
[11] S. Liu, L. Xie, and D. E. Quevedo, “Event-triggered quantized
communication-based distributed convex optimization,” IEEE Trans. Control Netw. Syst., vol. 5, no. 1, pp. 167–178, Mar. 2018.
[12] P. Yi, Y. Hong, and F. Liu, “Distributed gradient algorithm for constrained
optimization with application to load sharing in power systems,” Syst. &
Control Lett., vol. 83, pp. 45–52, 2015.
[13] Q. Liu and J. Wang, “A second-order multi-agent network for boundconstrained distributed optimization,” IEEE Trans. Autom. Control,
vol. 60, no. 12, pp. 3310–3315, Dec. 2015.
[14] M. Li, “Generalized Lagrange multiplier method and KKT conditions with
an application to distributed optimization,” IEEE Trans. Circuits Syst. II:
Exp. Briefs, vol. 66, no. 2, pp. 252–256, Feb. 2019.
[15] S. Liang, X. Zeng, and Y. Hong, “Distributed sub-optimal resource allocation over weight-balanced graph via singular perturbation,” Automatica,
vol. 95, pp. 222–228, 2018.
[16] J. Xu, S. Zhu, Y. C. Soh, and L. Xie, “A dual splitting approach for
distributed resource allocation with regularization,” IEEE Trans. Control
Netw. Syst., vol. 6, no. 1, pp. 403–414, Mar. 2019.
[17] G. Wen, X. Yu, Z. Liu, and W. Yu, “Adaptive consensus-based robust
strategy for economic dispatch of smart grids subject to communication
uncertainties,” IEEE Trans. Ind. Informat., vol. 14, no. 6, pp. 2484–2496,
Jun. 2018.
[18] A. Cherukuri and J. Cortés, “Distributed generator coordination for initialization and anytime optimization in economic dispatch,” IEEE Trans.
Control Netw. Syst., vol. 2, no. 3, pp. 226–237, Sep. 2015.
[19] M. Ye, G. Hu, and F. L. Lewis, “Nash equilibrium seeking for n-coalition
noncooperative games,” Automatica, vol. 95, pp. 266–272, 2018.

Dewen Li received the B.S. degree from the
Department of Automation, Hunan University,
Changsha, China, in 2015.
He was a Visiting Scholar at The University of Texas at Arlington during 2018-2019.
He is currently pursuing the Ph.D. degree
with the Department of Automation, Shanghai
Jiao Tong University, Shanghai, China. His current research interests include reinforcement
learning, distributed optimization, and machine
learning.

Ning Li (Member, IEEE) received the B.S. and
M.S. degrees in control science and engineering
from Qingdao University of Science and Technology, Qingdao, China, in 1996 and 1999, respectively, and the Ph.D. degree in control science and engineering from Shanghai Jiao Tong
University, Shanghai, China, in 2002.
She is currently a Professor with the Department of Automation, Shanghai Jiao Tong University. Her research interests include modeling
and control of complex systems, artificial intelligence, and big data analysis.
Frank Lewis (Life Fellow, IEEE) received the
bachelor’s degree in physics and electronics engineering and the MSEE from Rice University,
Houston, TX, USA, the M.S. degree in aeronautical engineering from the University of West
Florida, Pensacola, FL, USA, and the Ph.D.
degree from Georgia Institute of Technology,
Atlanta, GA, USA.
He is a UTA Distinguished Scholar Professor, UTA Distinguished Teaching Professor, and
Moncrief-O’Donnell Chair with the University of
Texas at Arlington Research Institute, Fort Worth, TX, USA. He is
the author of seven U.S. patents, numerous journal special issues,
420 journal papers, and 20 books, including Optimal Control (CRC,
1995), Aircraft Control (Wiley, 2003), Optimal Estimation (CRC, 2017),
and Robot Manipulator Control (CRC, 2003), which are used as university textbooks worldwide. His research interests include feedback
control, intelligent systems, cooperative control systems, and nonlinear
systems.
Dr. Lewis was the recipient of the Fulbright Research Award, the NSF
Research Initiation Grant, the ASEE Terman Award, the International
Neural Network Society Gabor Award, U.K., the Institute Measurement
& Control Honeywell Field Engineering Medal, the IEEE Computational
Intelligence Society Neural Networks Pioneer Award, the AIAA Intelligent Systems Award, the Outstanding Service Award from Dallas IEEE
Section, and the Texas Regents Outstanding Teaching Award 2013. He
was selected as the Engineer of the year by Ft. Worth IEEE Section.
He was listed in Ft. Worth Business Press Top 200 Leaders in Manufacturing. He is a Founding Member of the Board of Governors of
the Mediterranean Control Association. He is a Fellow of the National
Academy of Inventors, IFAC, AAAS, U.K. Institute of Measurement &
Control, PE, Texas, U.K. He is a Chartered Engineer.

Authorized licensed use limited to: XiangTan University. Downloaded on June 05,2026 at 03:34:18 UTC from IEEE Xplore. Restrictions apply.

