2023 62nd IEEE Conference on Decision and Control (CDC)
December 13-15, 2023. Marina Bay Sands, Singapore

Initialization-free distributed constrained optimization algorithms with
a pre-specified time

2023 62nd IEEE Conference on Decision and Control (CDC) | 979-8-3503-0124-3/23/$31.00 ©2023 IEEE | DOI: 10.1109/CDC49753.2023.10384064

Xiasheng Shi, Chaoxu Mu, Changyin Sun, Lu Ren, and Yanxu Su
Abstract— The distributed constrained optimization problem
over an undirected communication topology is investigated in
this study. It focuses on addressing a global coupled equality
constraint that applies to all agents. To tackle this problem,
a distributed approach with arbitrary initialization is developed by virtue of the aperiodic sampling control idea and
the consensus-based multi-agent system(MAS) technology. This
approach is developed to address constrained optimization
problems within a pre-specified time. In addition, this predefined time is freely defined by users and irrelevant to the initial
states, control coefficients, and network structure of systems.
The Lyapunov stability theory completes the convergence proof
of the developed method. Then, the developed method is extended to handle distributed nonlinear constrained optimization
problems. Finally, The availability of two developed methods is
demonstrated through two simulation examples.
Index Terms— distributed constrained optimization, prespecified time, nonlinear, non-periodical sampling

I. INTRODUCTION
Recently, the field of distributed optimization problems
over MASs has gained significant attention due to its applicability in various practical examples. These examples
include process operational optimization in chemical industries, spread control of epidemic diseases, source localization in UAV systems, and more. The primary objective
of distributed optimization problems is to develop efficient
methods that enable agents to collaborate and minimize the
global objective function by leveraging local communication
among themselves [1].
Therefore, to handle optimization problems in a distributed
way, various discrete-time algorithms have been developed
by using consensus-based MAS technology and gradient dependent strategy. For instance, in the case of distributed convex optimal coordination problem over time-varying jointly
connected undirected networks, a well-known subgradientbased distributed method with diminishing step size is created in [2]. For time-varying cost function-based distributed
Corresponding author: Yanxu Su suyanxu0616@gmail.com.
This work was partly supported by National Natural Science Foundation
of China (nos. 61921004, 62236002, 62022061, 62203001) and partly by
the Key Laboratory of Intelligent Control and Optimization for Industrial
Equipment, Ministry of Education, Dalian University of Technology (no.
LICO2022TB02).
Xiasheng Shi, Chaoxu Mu, Changyin Sun, Lu Ren, and Yanxu Su are
with with the School of Artificial Intelligence, Anhui University, Hefei,
China; Engineering Research Center of Autonomous Unmanned System
Technology, Ministry of Education, Anhui University, Hefei, China; Anhui Provincial Engineering Research Center for Unmanned System and
Intelligent Technology, Anhui University, Heifei, China; Xiasheng Shi is
also with the Key Laboratory of Intelligent Control and Optimization for
Industrial Equipment, Ministry of Education, Dalian University of Technology; Chaoxu Mu is also with the School of Electrical and Information
Engineering, Tianjin University, Tianjin, China.

979-8-3503-0124-3/23/$31.00 ©2023 IEEE

optimization problem over an undirected communication
topology, a quantized distributed method utilizing random
quantization operation is introduced [3]. In the realm of
the distributed constrained optimization, a proximal primaldual method with diminishing and nonsummable step size is
proposed [4]. However, the above diminishing step size leads
to a lower convergence rate, i.e., a sublinear convergence
rate. To address this limitation, an exact first-order method
with constant control step size is designed in [5] to reach a
linear convergence rate. Additionally, a Nesterov accelerated
optimization method is suggested in [6] to further enhance
the convergence performance for problems over the undirected communication topology. For distributed optimization
over an unbalanced directed network, a gradient trackingbased distributed method is proposed in [7] to achieve
linear convergence. In addition, a surplus-based accelerated
method with an uncoordinated step size is provided in [8]
for achieving a linear convergence rate. A general unified
framework for distributed optimization is proposed in [9].
Compared with discrete-time algorithms, continuous-time
distributed methods have attracted significant attention because of their real-time solutions and the realization for hardware implementation, which guarantees the more straightforward performance of the proposed methods in physical
systems. Furthermore, more robust convergence properties
can be ensured through Lyapunov stability, such as the
fixed time convergence theory [10]. Similar to discrete-time
algorithms, the linear(or exponential) convergence of the distributed algorithm is designed first. For example, a distributed
primal-dual distributed method combined by fixed control
parameters is designed for solving distributed optimization
problems over weight-balanced digraphs [11]. For reducing the communication consumption, a distributed method
with on demanded communication scheme is introduced in
[12]. However, these linear protocol-based approaches handle
distributed optimization problems with an exponential or
asymptotical convergence rate, meaning that the methods
achieve the optimal result as time goes to infinity. Nevertheless, there is a need to address the finite-time requirement
optimization problems in various applications.
To address this requirement, an adaptive distributed gradient optimization algorithm with scale sign consensus scheme
is proposed [13], enabling the establishment of optimal
solutions within a finite time. Considering the unknown
disturbance of agent, the finite-time result is guaranteed using
a sliding mode control scheme [14]. For the distributed
optimization over an unbalanced directed communication
topology, by using the nonsmooth theory, a distributed

3592

Authorized licensed use limited to: XiangTan University. Downloaded on June 05,2026 at 03:48:32 UTC from IEEE Xplore. Restrictions apply.

finite-time optimization method is created [15]. However,
the setting time of these finite-time distributed methods is
usually influenced by the primary state value, which may be
unavailable and hard to off-line pre-assign the setting time.
Therefore, to overcome this restriction, for the time-varying
cost function-based distributed optimization problem, the
optimal result is obtained within fixed time using three scale
sign terms [16]. In view of the zero-gradient-sum framework
and the scale sign function, the fixed-time result is obtained
for the quadratic optimization problems [17]. For distributed
optimization problems coupled with an equality constraint,
the fixed-time convergence is obtained via communicating
local gradient information among agents [18]. In more complex cases involving global equality and local inequality
constraints, a projection-gradient-based method is developed
in [19].
Existing finite- and fixed-time distributed algorithms are
known to exhibit setting times that are significantly correlated with algorithm parameters and communication topologies, such as the agent number and its Laplacian matrix
eigenvalues. Therefore, studying predefine-time approaches
for distributed optimization problems is meaningful. Thus,
some excellent distributed methods have been designed recently. One approach that has been explored is the the timebased generator scheme, which has been used to derive many
predefined-time distributed methods for distributed convex
optimization problems. However, it is notable that the timebased generator scheme can merely get its underutilized
solution. In order to obtain the exact most favorable solution,
a new class of predefined-time distributed methods has been
designed based on the sign function. These methods have
been applied to solve optimal coordination problems [20]
and resource allocation problems [21]. Nevertheless, sign
function leads to frequent trembling behavior. Currently, the
specified-time theory has gained considerable attention. For
instance, a new out-degree-based distributed method with
pre-specified time is designed in [22], where the convergence time has nothing to do with other parameters. a
novel predefined time coordination algorithm is described
in [23] utilizing a multi-step planning, which requires the
initialization of the initial state of all agents. However,
an efficient method is still lacking to achieve exact most
favorable solutions within a pre-specified time without the
need for initializing all agent states.
Motivated by the aforementioned discussions, this study
further researches the distributed optimization problem,
specifically focusing on cases where a global coupled equality constraint exists. To address this problem, a novel
initialization-free distributed method with pre-specified time
is designed, leveraging a non-periodic sampling control strategy. The contributions of this study can be summarized in
the following two folds: 1) Different from the existing results
[22], [23], our developed methods eliminate the requirement
for initialization, enhancing its practicality and efficiency;
2) Moreover, this study takes into account the presence of
a global nonlinear constraint, expanding the scope of the
problem addressed.

This study is organized as below. Section II provides
several preliminaries and presents the problem formulation.
Section III gives the main results achieved in this study.
Section IV illustrates the validity of our created approaches
through illustrative examples. Section V concludes the whole
work, highlighting potential avenues for future research.
II. P RELIMINARIES AND PROBLEM FORMULATION
Notations: Let the set R indicate all real numbers. For
a differentiable function f (z), its gradient function and
Hessian matrix are denoted by ∇f (z) and ∇2 f (z) in turn.
Here, z represents the input variable. For a positive integer
n, the symbols 0n and 1n respectively represent the all-zero
and all-one vector. Additionally, the notation span{1n } refers
to the vector space spanned by the all-one vector 1n .
A. Graph theory
Given a MAS consisting of n n agents. Let G(V, E, A)
denote its communication topology. Particularly, V =
{1, 2, ..., n} represents the agent set, E ⊂ V × V denotes
its communication edge set, and A := [aij ] ∈ Rn×n is the
weight adjacency matrix. If the information from agent j
can be acquired by agent i, i.e., (i, j) ∈ E, then agent j
is considered a neighbor of agent i and thus aij > 0, and
aij = 0 otherwise. Additionally, we define aii = 0 even
if the information of i can be used by itself. This implies
that the graph G is assumed to have no self-loop. If at least
one neighbor agent exists for all agents in graph G, then G
is a connected graph. For the graph G, the corresponding
Laplacian matrix L := [lij ]P∈ Rn×n is defined as lij =
−aij , ∀i 6= j ∈ V and lii = nj=1 aij , ∀i ∈ V. It is evident
that L1n = 0. Furthermore, we have 1Tn L = 0 and the
ordered eigenvalues 0 = ρ1 (L) < ρ2 (L) ≤, ..., ≤ ρn (L)
when G is undirected.
B. Aperiodic sampling
Different from the periodic sampling and the eventtriggered sampling control, theP
aperiodic sampling is defined
6T
k
as {tk }k=1,2,...,∞ with tk = p=1 (πp)f2 , where Tf is the
P
1
π2
pre-specified time. Based on the fact ∞
k=1 k2 = 6 , then
we have limk→∞ tk = Tf .
C. Problem formulation
The dynamic of agent i, ∀i ∈ V, is given by
ẋi (t) = ui (t),

(1)

in which xi (t) ∈ RN and ui (t) ∈ RN respectively denote
their state variable and control input. For simplicity, let N =
1 in the following context. The case N > 1 can be similarly
deduced by the Kroncker product.
Let x = [x1 , x2 , ..., xn ]T ∈ Rn denotes the stack vector of
variables x1 , x2 , ..., xn . The goal of all agents is to minimize
the total cost function f (x) while satisfying a global coupled

3593
Authorized licensed use limited to: XiangTan University. Downloaded on June 05,2026 at 03:48:32 UTC from IEEE Xplore. Restrictions apply.

equality constraint. Mathematically, the following distributed
constrained optimization problem is provided:
min f (x) =
x

n
X
i=1

xi =

n
X

i=1
n
X

fi (xi ),
(2)
di ,

i=1

where fi (xi ) and di represent its local cost function and
demand resource, respectively. To ensure the following convergence analysis, the convex assumption is provided as
below.
Assumption 1: For agent i, ∀i ∈ V, let the differentiable
function fi (xi ) satisfy 0 < ∇2 fi (xi ) ≤ mi (mi > 0).
Remark 1: The aforementioned Assumption 1 ensures the
uniqueness of the optimal solution for problem (2), which has
been employed in various existing methods [22], [23]. Unlike
the strong convexity requirements in most existing literature
[5]–[7], [9], the above smooth function requirement is more
practical. For instance, the exponential function is convex
but not strongly convex.
Based on Assumption 1 and the Karush-Kuhn-Tucker
conditions [22], the optimal result x∗ ∈ Rn needs to satisfy
that
∇f (x∗ ) ∈ span{1n },
n
n
X
X
di .
x∗i =
i=1

(3)

where ǫ is a small positive constant. It implies that the
periodic sampling will stand in for the aperiodic sampling
when the sampling interval is relatively small. As proved in
[22], the convergence precision can be adjusted by choosing
proper ǫ.
B. Convergence analysis

i=1

III. M AIN RESULTS

A. Algorithm design
From (3), to acquire the optimal result x∗ , it is necessary
to achieve the consensus on gradient function and the global
coupled constraint should be also satisfied. Thus, in view of
the aperiodic sampling control approach and the consensus
scheme in multi-agent systems, we design the following
distributed optimization method:
1
ei (tl )
tl+1 − tl
n
X
aij (∇fi (xi (tl )) − ∇fj (xj (tl ))),
−α

R t Pn
integral term 0 j=1 aij (∇fi (xi (tl )) − ∇fj (xj (tl )))dτ is
designed to balance the real state value xi (t) and the demand
state di . The initial value of xi can be randomly selected.
Moreover, let ei (0) = xi (0) − di .
Remark 2: Different from the time-based generator and
sign function method [20], [21], our proposed algorithm (4)
utilize the aperiodic sampling control approach to address the
problem (2) with pre-specified time. In addition, the designed
algorithm not merely obtain the exact most favorable result
but avoids chattering behavior as well. A distinctive feature
of our approach is that it allows for the random selection
of the initial state, which distinguishes it from the existing
work [22], [23].
From the definition of aperiodic sampling in subsection
II-B, the pre-specified time is ensured when k goes to
infinity. However, Zeno’s behavior will occur. To avoid
Zeno’s behavior, the sampling interval sequence is modified
as below:
( 6T
6Tf
f
(πl)2 ,
(πl)2 > ǫ
,
(5)
tl = tl−1 +
6Tf
ǫ,
(πl)2 ≤ ǫ

ẋi (t) = −

Theorem 1: Consider that Assumption 1 is satisfied, and
the graph G is undirected and connected, the developed distributed method (4) achieves the optimal result of (2) within
pre-specified time 2Tf . In addition, the control parameter h
satisfies 0 < h < mρ22(L) .
Proof: the above result is proved by the following two
steps. In step I, the global coupled equality constraint is
guaranteed with pre-specified time; In step II, x(t) achieve
the most favorable result within pre-specified time.
Step I. if t ∈ [0, Tf ], one has
ėi (t) = −

(4a)

j=1

ei (t) =xi (t) − di
Z tX
n
aij (∇fi (xi (tl )) − ∇fj (xj (tl )))dτ
+α
0 j=1

(4b)

where t ∈ [tl , tl+1 ) and α is the control parameter. We define
α = tl+1h−tl with h > 0. The sampling time tl is defined as
( Pl
6Tf
0 ≤ t ≤ Tf
p=1 (πl)2 ,
P
tl =
. Without loss
6T
Tf + lp=1 (πl)f2 , Tf ≤ t ≤ 2Tf
of generality, let t0 = 0. From the aforementioned definition,
the proposed method (4) can be divided into two stages. In
stage 1 (0 ≤ t ≤ Tf ), the global equality constraint is solved.
In stage 2 (Tf ≤ t ≤ 2Tf ), the optimality is guaranteed. The

1
ei (tl ), t ∈ [tl , tl+1 ).
tl+1 − tl

(6)

Integrating (6) over time [tl , tl+1 ), it can be obtained that
Z tl+1
1
ei (tl )dτ
ei (tl+1 ) = ei (tl ) −
tl+1 − tl
tl
(7)
= ei (tl ) − ei (tl )
= 0.
It is notable that ei (tl ) = 0, ∀l ≥ 1. it followings this result
and (4b) that
n
X
i=1

ei (tl ) =

n
X

(xi (tl ) − di ) = 0,

(8)

i=1

in which the resultP
1Tn L = 0 is
Pused. Namely, the coupled
equality limitation ni=1 xi = ni=1 di is guaranteed within
pre-specified time Tf .

3594
Authorized licensed use limited to: XiangTan University. Downloaded on June 05,2026 at 03:48:32 UTC from IEEE Xplore. Restrictions apply.

Step II. When t ∈ [Tf , 2Tf ], we have
ẋi (t) =

h
L∇f (x(tl )), t ∈ [tl , tl+1 ).
tl+1 − tl

extend the proposed algorithm (4) to the global nonlinear
constrained distributed optimization problem shown below:
(9)
min f (x) =
x

Integrating (9) over time [tl , tl+1 ) yields
x(tl+1 ) = x(tl ) − hL∇f (x(tl )).

Let V (l) = f (x(tl )) − f (x∗ ), in which x∗ is the theoretical
optimal result of problem (2). Since the global coupled
equality limitation holds the whole time, V (l) is radially
unbounded and semi-positive. The difference of V (l) along
system (10) is provided by
∆V (l) = V (l + 1) − V (l).

(11)

As f (x) is convex, then one has
f (y) =f (z) + ∇T f (z)(y − z)
1
+ (y − z)T ∇2 f (ȳ)(y − z),
2

(12)

where gi (xi ) is a nonlinear function. For instance, the local
function gi (xi ) is defined as gi (xi ) = xi −di −Bi x2i with loss
coefficient Bi . Generally, Bi is very small. The following
distributed optimization approach is created to obtain the
optimal solution of the problem (16) within a pre-specified
time.
1
ẋi (t) = −
ei (tl )
(tl+1 − tl )∇gi (xi (t))
n
X
α
aij (∇fi (xi (tl )) − ∇fj (xj (tl ))),
−
∇gi (xi (t)) j=1

(17a)

ei (t) =gi (xi )
Z tX
n
aij (∇fi (xi (tl )) − ∇fj (xj (tl )))dτ
+α
0 j=1

1
(x(tl+1 ) − x(tl ))T ∇2 f (ȳ)
2
· (x(tl+1 ) − x(tl ))
T

+ ∇ f (x(tl ))(x(tl+1 ) − x(tl ))
h2 T
∇ f (x(tl ))LT ∇2 f (ȳ)L∇f (x(tl ))
=
2
− h∇T f (x(tl ))L∇f (x(tl )).

(17b)

(13)

We have used the fact (10) in the second equality of (13). It
follows Assumption 1 that
∆V (l) ≤

h2 m T
∇ f (x(tl ))LT L∇f (x(tl ))
2
− h∇T f (x(tl ))L∇f (x(tl )),

(14)

in which m = maxi mi . It follows the undirected and
connected assumption for graph G that LT L ≤ ρn (L)L with
the largest eigenvalue ρn (L). It follows (14) that
∆V (l) ≤ −h(1 −

hm T
)∇ f (x(tl ))L∇f (x(tl )).
2

(16)

gi (xi ) = 0,

i=1

in which ȳ ∈ (y, z). Let y = x(tl+1 ) and z = x(tl ),
substituting the result (12) into (11) yields
∆V (l) =

fi (xi ),

i=1

n
X

(10)

n
X

(15)

If 0 < h < mρn2(L) , we have ∆(l) ≤ 0. Additionally, in view
of the definition of Laplacian matrix L, ∇f (x(tl )) achieves
consensus iff L∇f (x(tl )) = 0. Thus, in view of (3), the
most favorable result is achieved within pre-specified time
due to the fact liml→∞ tl = 2Tf when t ∈ [Tf , 2Tf ].
C. Applications to economic dispatch problems
The global coupled constraint is considered linear in the
problem (2). However, the nonlinear term is more general
in applications, such as the transmission line loss in the
economic dispatch problem of energy internet. Next, we

where t ∈ [tl , tl+1 ) and α is the control parameter, which is
defined the same with algorithm (4). To obtain the optimal
solution of (16), we provide following assumption.
Assumption 2: For each agent i, the condition ∇gi (xi ) >
0 holds.
By similar convergence proof in Theorem 1, we establish the
following conclusion.
Corollary 1: Consider that Assumptions 1 and 2 hold,
and the graph G is undirected and connected, the proposed
distributed method (17) achieves the optimal result of (16)
within pre-specified time 2Tf . In addition, the control pai ∇gi (xi )
rameter h satisfies 0 < h < 2 max
.
mρ2 (L)
IV. S IMULATION
In the following, we will provide some cases to indicate
the accuracy for our designed algorithm.
Example 1. The economic dispatch problem without
transmission line loss over smart grids in [23] is devoted
to illustrating the algorithm (2). For each generator i, let
fi (xi ) = ai x2i + bi xi + ci with coefficients ai , bi , ci , where
xi indicates its generation power output. The coefficients
are shown in Table. I. The economic dispatch problem aims
to design a generation plan for some distributed generators
while minimizing the total generation cost function. Let
the initial state be x(0) = [60, 0, 0, 0, 0, 0]T ∈ R6 . It can
be seen that the global coupled equality constraint does
not hold. Thus, existing results [22], [23] fail to solve the
economic dispatch problem under the same initial states.
2
=
From Theorem 1, it can be calculated that 0 < h <= mρ
2
2
=
1.77.
Let
the
control
parameter
h
be
h
=
1,
2×0.105∗5.7
the detailed trajectories of variable xi and ei , ∀i ∈ V, are

3595
Authorized licensed use limited to: XiangTan University. Downloaded on June 05,2026 at 03:48:32 UTC from IEEE Xplore. Restrictions apply.

shown in Figs. 1 and 2. From Fig. 1, the most favorable
result x∗ = [23.47, 16.09, 15.22, 10.40, 18.12, 16.70]T ∈ R6
is obtained within setting time 2Tf . Additionally, the setting
time 2Tf is independent on the constrained optimization
problem and its network knowledge. From 2, the error
variable ei (t) reaches zero within a given time Tf = 5s,
meaning that the coupled equality limitation is ensured by a
pre-specified time. Additionally, the error is zero after one
iteration from the analysis of the theorem. After one iteration,
the coupling constraint in constraint optimization problem (2)
holds and remains unchanged.
TABLE I
T HE COEFFICIENTS OF EACH GENERATOR .

Generation output xi

1
2
3
4
5
6

ai

bi

ci

di

0.096
0.072
0.105
0.078
0.078
0.090

1.22
3.41
2.53
4.02
2.90
2.72

51
31
78
42
67
49

20
20
20
20
10
10

60

Generation output xi

coefficients
generator i

[51, 31, 78, 42, 62]T ∈ R5 , d = [20, 20, 20, 20, 40]T ∈ R5
and B = [0.0021, 0.0031, 0.0011, 0.0022, 0.0041]T ∈ R5 .
The considered network is an undirected
ring topology.

1,
t ∈ [0, Tf ]
Let the control parameter be h =
0.1, t ∈ [Tf , 2Tf ]
and the initial state be x(0) = [60, 0, 0, 0, 0]T ∈ R5 , the
detailed trajectories of variables xi and ei , ∀i ∈ V,
are depicted in Figs. 3 and 4. From (16), the
algorithm P
is executed in two steps. The consensus term
n
α
j=1 aij (∇fi (xi (tl )) − ∇fj (xj (tl ))) with big
∇gi (xi (t))
control parameter h always working. Therefore, the optimal
solution can be achieved far earlier than the pre-specified
time 2Tf . It has been verified in Fig. 3. Thus, the developed
algorithms are valid for distributed constrained optimization
problems with pre-specified time.

x1

x4

50

x2

x5

40

x3

30
20
10

60

0

40

0

2

4

6
Time (s)

8

10

12

20

Transient behavior of xi , ∀i ∈ V, in Example 2.

Fig. 3.
0
-20

x1

x4

x2

x5

x3

x6

-40
2

4

6
Time (s)

8

10

Transient behavior of xi , ∀i ∈ V, in Example 1.

Fig. 1.

e1

e4

20

e2

e5

10

e3

12

error ei

0

30

0
-10
-20

60

e1

e4

e2

e5

e3

e6

-30
-40
0

2

4

6
Time (s)

8

10

12

error ei

40

Fig. 4.

Transient behavior of ei , ∀i ∈ V, in Example 2.

20

V. CONCLUSIONS AND FUTURE WORKS

0
-20
0

Fig. 2.

2

4

6
Time (s)

8

10

12

Transient behavior of ei , ∀i ∈ V, in Example 1.

Example 2. For the economic dispatch problem
shown in [24], whose coefficients are given
by a
=
[0.094, 0.078, 0.105, 0.082, 0.074]T
∈
5
R , b = [1.22, 3.41, 2.53, 4.02, 3.17]T ∈ R5 , c =

This paper introduces two distributed algorithms for solving distributed constrained optimization problems over undirected MASs. In view of the aperiodic sampling control
approach, our developed methods converge to the optimal
result within a pre-specified time. The developed optimization approaches have been proved to be convergent using
the Lyapunov stability theory. Furthermore, the efficiency
illustration is completed by economic dispatch problems. In
future work, we plan to incorporate the consideration of local
constraints for each agent in the design of pre-specified-time

3596
Authorized licensed use limited to: XiangTan University. Downloaded on June 05,2026 at 03:48:32 UTC from IEEE Xplore. Restrictions apply.

distributed optimization problems. Besides, the exact value
of gradient function is required to develop the optimization
approaches, which heavily increased the system bandwidth
consumption. Consequently, we will also study the quantized
optimization approaches within pre-specified time.
R EFERENCES
[1] J. Lei, P. Yi, J. Chen, et al, ”Distributed variable sample-size stochastic
optimization with fixed step-sizes,” IEEE Transactionns on Automatic
Control, vol. 67, no. 10, pp. 5630-5637, Oct. 2022.
[2] A. Nedić, A. Ozdaglar, ”Distributed subgradient methods for multiagent optimization,” IEEE Transactions on Automatic Control, vol. 54,
no. 1, pp. 48-61, Jan. 2009.
[3] D. Yuan, B. Zhang, D. W. C. Ho, et al, ”Distributed online bandit
optimization under random quantization,” Automatica, 2022, in press,
doi: 10.1016/j.automatica.2022.110590.
[4] X. Li, G. Feng, L. Xie, ”Distributed proximal algorithms for multiagent optimization with coupled inequality constraints,” IEEE Transactions on Automatic Control, vol. 66, no. 3, pp. 1223-1230, March
2021.
[5] W. Shi, Q. Ling, G. Wu, et al, ”EXTRA: An exact first-order
algorithm for decentralized consensus optimization,” SIAM Journal
on Optimization, vol. 25, no. 2, pp. 944-966, May 2015.
[6] G. Qu, N. Li, ”Accelerated distributed nesterov gradient descent,”
IEEE Transactions on Automatic Control, vol. 65, no. 6, pp. 25662581, June 2020.
[7] C. Xi, V. S. Mai, R. Xin, et al, ”Linear convergence in optimization
over directed graphs with row-stochastic matrices,” IEEE Transactions
on Automatic Control, vol. 63, no. 10, pp. 3558-3565, Oct. 2018.
[8] D. Wang, Z. Wang, J. Lian, et al, ”Surplus-based accelerated algorithms for distributed optimization over directed networks,” Automatica, 2022, in press, doi: 10.1016/j.automatica.2022.110569.
[9] J. Xu, Y. Tian, Y. Sun, et al, ”Distributed algorithms for composite
optimization: Unified framework and convergence analysis,” IEEE
Transactions on Signal Processing, vol. 69, pp. 3555-3570, June 2021.
[10] Z. Zuo, Q. L. Han, B. Ning, et al, ”An overview of recent advances
in fixed-time cooperative control of multiagent systems,” IEEE Transactions on Industrial Informatics, vol. 14, no. 6, pp. 2322-2334, June
2018.
[11] B. Gharesifard, J. Cortés, ”Distributed continuous-time optimization
on weight-balanced digraphs,” IEEE Transactions on Automatic Control, vol. 59, no. 3, pp. 781-786, March 2014.
[12] S. S. Kia, J. Cortés, S. Martı́nez, ”Distributed convex optimization via
continuous-time coordination algorithms with discrete-time communication,” Automatica, vol. 55, pp. 254-264, May 2015.
[13] P. Lin, W. Ren, J. A. Farrell, ”Distributed continuous-time optimization: Nonuniform gradient gains, finite-time convergence, and convex
constraint set,” IEEE Transactions on Automatic Control, vol. 62, no.
7, pp. 2239-2253, May 2017.
[14] Z. Feng, G. Hu, C. G. Gassandras, ”Finite-time distributed convex
optimization for continuous-time multiagent systems with disturbance
rejection,” IEEE Transactions onn Control of Network Systems, vol.
7. no. 2, pp. 686-698, June 2020.
[15] X. Shi, J. Cao, G. Wen, et al, ”Finite-time consensus of opinion dynamics and its applications to distributed optimization over digraphs,”
IEEE Transactions on Cybernetics, vol. 49, no. 10, pp. 3767-3779,
Oct. 2019.
[16] B. Ning, Q. L. Han, Z. Zuo, ”Distributed optimization for multiagent
systems: An edge-based fixed consensus approach,” IEEE Transactions
on Cybernetics, vol. 49, no. 1, pp. 122-132, Jan. 2019.
[17] Z. Wu, Z. Li, J. Yu, ”Designing zero-gradient-sum protocols for
finite-time distributed optimization problem,” IEEE Transactions on
Systems, Man, and Cybernetics: Systems, vol. 52, no. 7, pp. 45694577, July 2022.
[18] G. Chen, Z. Li, ”A fixed-time convergent algorithm for distributed
convex optimization in multi-agent systems,” Automatica, vol. 95, pp.
539-543, Sep. 2018.
[19] G. Chen, Z. Guo, ”Initialization-free distributed fixed-time convergent
algorithms for optimal resource allocation,” IEEE Transactions on
Systems, Man, and Cybernetics: Systems, vol. 52, no. 2, pp. 845-854,
Feb. 2022.
[20] T. Zhou, H. Wu, J. Cao, ”Distributed optimization in predefinedtime for multi-agent systems over a directed network,” Information
Sciences, vol. 615, pp. 743-757, Nov. 2022.

[21] W. T. Lin, Y. W. Wang, C. Li, et al, ”Predefined-time optimization for
distributed resource allocation,” Journal of the Franklin Institute, vol.
357, no. 16, pp. 11323-11348, Nov. 2020.
[22] J. Zhou, Y. Lv, C. Wen, et al, ”Solving specified-time distributed
optimization problem via sampled-data-based algorithm,” IEEE Transactions on Network Science and Engineering, vol. 9, no. 4, pp. 27472758, July 2022.
[23] Z. Su, Y. Liu, C. Xian, et al, ”Pre-specified-time coordination algorithm for convex optimization problems over weight-unbalanced
networks,” in Proceedings of the 36th Youth Academic Annual Conference of Chiese Association of Automation, pp. 156-161, 2021.
[24] Z. Wang, G. Chen, H. Li, ”An efficient distributed algorithm for
economic dispatch considering communication asynchrony and time
delays,” Energy Conversion and Economics, vol. 3, pp. 214-226, Aug.
2022.

3597
Authorized licensed use limited to: XiangTan University. Downloaded on June 05,2026 at 03:48:32 UTC from IEEE Xplore. Restrictions apply.

