IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS, VOL. 34, NO. 9, SEPTEMBER 2023

5629

Distributed Optimization for Second-Order
Discrete-Time Multiagent Systems
With Set Constraints
Yao Zou , Member, IEEE, Kewei Xia , Bomin Huang , and Ziyang Meng , Senior Member, IEEE

Abstract— The optimization problem of second-order discretetime multiagent systems with set constraints is studied in this
article. In particular, the involved agents cooperatively search
an optimal solution of a global objective function summed by
multiple local ones within the intersection of multiple constrained
sets. We also consider that each pair of local objective function
and constrained set is exclusively accessible to the respective
agent, and each agent just interacts with its local neighbors.
By borrowing from the consensus idea, a projection-based distributed optimization algorithm resorting to an auxiliary dynamics
is first proposed without interacting the gradient information of
local objective functions. Next, by considering the local objective
functions being strongly convex, selection criteria of step size and
algorithm parameter are built such that the unique solution to the
concerned optimization problem is obtained. Moreover, by fixing
a unit step size, it is also shown that the optimization result can be
relaxed to the case with just convex local objective functions given
a properly chosen algorithm parameter. Finally, practical and
numerical examples are taken to verify the proposed optimization
results.
Index Terms— Consensus, convex functions, discrete-time systems, distributed optimization, multiagent systems.

I. I NTRODUCTION
ECENT years have witnessed the rapid development
in distributed optimization technique [1] and its extensive applications in networked systems, including large-scale
machine learning [2], economic dispatch of power systems [3],
and sensor network scheduling [4]. In particular, by means
of local information exchange over a particular network, a
cluster of agents are supposed to seek a minimal solution of

R

Manuscript received 12 April 2021; revised 31 August 2021; accepted
20 November 2021. Date of publication 7 December 2021; date of current
version 1 September 2023. This work was supported in part by the National
Natural Science Foundation of China under Grant 62073028, Grant 61873140,
Grant 61833009, and Grant U19B2029; and in part by the Interdisciplinary
Research Project for Young Teachers of University of Science and Technology
Beijing (Fundamental Research Funds for the Central Universities) under
Grant FRF-IDRY-20-027. (Corresponding author: Yao Zou.)
Yao Zou is with the School of Automation and Electrical Engineering and
the Institute of Artificial Intelligence, University of Science and Technology
Beijing, Beijing 100083, China (e-mail: zouyao@ustb.edu.cn).
Kewei Xia is with the Advanced Research Institute of Multidisciplinary
Sciences, Beijing Institute of Technology, Beijing 100081, China (e-mail:
kwxia134@gmail.com).
Bomin Huang is with the School of Control Engineering, Northeastern
University at Qinhuangdao, Qinhuangdao 066004, China (e-mail:
huangbomin01@hotmail.com).
Ziyang Meng is with the Department of Precision Instrument, Tsinghua University, Beijing 100084, China (e-mail: ziyangmeng@mail.tsinghua.edu.cn).
Color versions of one or more figures in this article are available at
https://doi.org/10.1109/TNNLS.2021.3130173.
Digital Object Identifier 10.1109/TNNLS.2021.3130173

a global objective/payoff/cost function summed by multiple
local ones, whereas each agent merely has access to the local
information [5].
There are a variety of distributed continuous-time optimization algorithms in the current studies. For instance,
motivated by the zero-gradient-sum idea, several sorts of distributed continuous-time algorithms were proposed in [6] for
the cooperative solution to the global optimization problem.
By following this idea and further considering issues of
communication delay, finite- or fixed-time convergence,
and event-triggered interaction, several improved distributed
continuous-time algorithms were developed in [7]–[10] for the
same optimization objective. Nonetheless, the zero-gradientsum algorithms in [6]–[10] are based on an indispensable
hypothesis that each local objective function is strongly convex
such that the required Jacobian matrix of the global objective
function is nonsingular. To relax the strong convexity requirement on the local objective functions, two fully distributed
continuous-time optimization algorithms using the adaptive
gain technique were designed in [11] such that the optimization objective was realized with just convex local objective
functions.
On the other hand, from the perspective of distributed
computation, the distributed optimization problem is generally
studied in terms of discrete-time algorithms. For example, a
distributed subgradient discrete-time computation model was
first built in [12] for the multiagent optimization problem
with convex local objective functions over a time-varying
topology. Then, another distributed projected consensus
algorithm was proposed in [13] to complete the global
optimization objective in the case of local set constraints.
Instead of using the relative states as done in [12] and [13],
the distributed discrete-time optimization algorithm proposed
in [14] just relies on their signs. It is also shown in [14] that
the convergence rate is essentially not affected by the loss of
state information. In addition, a consensus-based primal–dual
perturbation method was designed in [15] for the solution to
the distributed optimization problem with both smooth and
nonsmooth constraints. An extended gradient method, called
the push–pull gradient method, was proposed in [16] for the
distributed optimization objective, in which each agent pushed
the gradient information to its neighbors and simultaneously
pulled the decision information from them. By introducing an
event-triggered mechanism in communication, both distributed
continuous-time and discrete-time optimization algorithms

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were designed in [17] via input feedforward passivity.
Besides, the distributed optimization problem associated
with a multiagent system composed of both continuousand discrete-time dynamic agents was studied in [18].
By modifying the distributed primal–dual algorithms
in [19]–[21], a distributed online primal–dual push-sum
algorithm was proposed in [27] without requiring parameter
boundedness and network balance. However, the distributed
discrete-time algorithms in [12]–[27] focus more on the
online optimization computation regardless of the dynamics
of physical objects.
In many practical applications including cooperative source
localization [23], control of network systems [24], and optimal alignment of hydraulic manipulators [28], the considered
problems can be characterized by the distributed optimization
models. These optimization problems are generally solved by
the physical objects with dynamical systems. Toward this end,
it is of more practical significance to study the distributed
optimization problem for dynamical multiagent systems. For
example, distributed optimization algorithms using event- and
time-triggered communication were proposed in [26] and [27]
for a second-order multiagent system. By designing proper
disturbance estimators to compensate for undesirable disturbances, another two distributed optimization algorithms
were developed in [28] and [29] for the disturbed secondorder multiagent system. By considering time-varying objective functions, the proposed distributed algorithms developed
in [30] and [31] also solved the distributed optimization
problem for the second-order multiagent system. Moreover,
by introducing the quasi-zero-gradient-sum idea in [30], the
distributed optimization problem was solved in terms of a
backstepping framework in [32] for high-order nonlinear systems. Besides, by considering the input saturation, another
distributed optimization algorithm with the zero-gradientsum idea was studied in [33] for multi-integrator systems.
In addition, for Euler–Lagrange systems characterizing a large
class of mechanical systems, several distributed optimization
algorithms were designed in [34]–[37]. However, the systems
in [26]–[37] are considered in a continuous-time setting,
and the proposed distributed optimization algorithms in these
references cannot be applied to their discrete-time counterparts
by direct sampling since the step size of sampling has an effect
in convergence performance. Under the discrete-time setting,
a distributed optimization algorithm with properly chosen step
sizes was designed in [38] and [39] for a mixed multiagent
system composed of both first- and second-order agents.
On the other hand, the optimization problems studied
in [26]–[39] do not impose any constraint on the optimal
solutions. To the best of our knowledge, there have been no
works on the distributed optimization for the second-order
discrete-time multiagent system subject to constraints on the
optimal solutions.
Inspired by the above discussions, we are interested in
solving the constrained distributed optimization problem for
the second-order discrete-time multiagent system in this article. In particular, the distributed computation and physical
dynamical constraints are simultaneously considered. To be
more specific, all the agents cooperatively search an optimal

solution of the global objective function within the intersection
of the constrained sets, each of which is allocated to the
respective agent. By introducing an auxiliary dynamics, a
projection-based distributed optimization algorithm is proposed without exchanging the gradient information of local
objective functions. The convergence analysis is based on
the Lyapunov framework from the consensus perspective.
First, by considering strongly convex local objective functions,
selection criteria of step size and algorithm parameter are
formulated such that all the agents come to a consensus at
the unique solution to the constrained optimization problem of
concern. Next, by fixing a unit step size, it is demonstrated that
the distributed optimization algorithm with a properly chosen
algorithm parameter is applicable to a more general case where
the requirement of local objective functions is relaxed into
convexity. Simulation examples are taken to confirm the effectiveness of the proposed distributed optimization algorithm.
The main contributions compared with the previous works are
as follows.
1) Unlike the distributed discrete-time optimization algorithms in [12]–[27] that do not apply to practical systems
with dynamical constraints, we develop a distributed
optimization algorithm for the second-order discretetime multiagent system subject to set constraints. The
second-order system in fact characterizes a number of
mechanical and locomotive dynamics such that the studied optimization problem is of more practical interest.
2) Rather than the distributed optimization algorithms for
dynamical multiagent systems in [26]–[39] without any
constraints, we develop a projection-based distributed
optimization algorithm for the second-order multiagent
system subject to set constraints. The implementation
of the projection operation induces strong nonlinearity
in the closed-loop system, causing great complication
in the convergence analysis. By resorting to the convex
analysis, it is proven based on discrete-time Lyapunov
theory that all the states come to a consensus at the
optimal solution.
3) Compared with the distributed optimization algorithms
in [16] and [30]–[32] that require exchanging the
gradient information of local objective functions, the
proposed one with the help of an auxiliary dynamics just needs the interaction of the agent states and
auxiliary variables, which is of more significance in
preserving privacy. Moreover, in contrast to the distributed optimization algorithms in [6]–[10], [17]–[19], and
[34]–[37] where the strong convexity assumption of
local objective functions is required, the proposed one
allows the local objective functions to be just convex
given a unit step size.
The remaining sections are organized as follows. First,
several mathematical preliminaries are presented in Section II.
Then, the studied constrained optimization problem is stated
in Section III. Next, the distributed optimization algorithm
design and the stability analysis are detailed in Section IV.
Subsequently, two examples are taken in Section V to confirm
the proposed optimization results. Finally, conclusions are
drawn in Section VI.

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ZOU et al.: DISTRIBUTED OPTIMIZATION FOR SECOND-ORDER DISCRETE-TIME MULTIAGENT SYSTEMS

II. M ATHEMATICAL P RELIMINARIES

III. P ROBLEM D ESCRIPTION

A. Convex Properties
A differentiable function f : R → R is called convex if
for any ξ1 , ξ2 ∈ Rn
n

f (ξ1 ) − f (ξ2 ) ≥ ∇ f (ξ2 ) (ξ1 − ξ2 )
T

(1)

or alternatively
(∇ f (ξ1 ) − ∇ f (ξ2 ))T (ξ1 − ξ2 ) ≥ 0

(2)

with ∇ f being the gradient of f . Besides, f is said to be
ω-strongly convex with ω being a positive constant if for any
ξ1 , ξ2 ∈ Rn
f (ξ1 ) − f (ξ2 ) ≥ ∇ f (ξ2 )T (ξ1 − ξ2 ) +

5631

ω
ξ1 − ξ2 2
2

(3)

or alternatively

Consider a multiagent system of the second-order discretetime dynamics
x i [s + 1] = x i [s] + τ v i [s]

(7a)

v i [s + 1] = v i [s] + τ u i [s], i ∈ V

(7b)

where s ∈ Z+ denotes the sampling instant, x i , v i ∈ Rm
denote the system states, u i ∈ Rm denotes the system input,
and τ ∈ R+ denotes the step size of sampling. Every
agent i is exclusively allocated with a local objective function
f i : Rm → R, while other agents cannot have access to it.
Moreover,
we introduce a global objective function f (y) =
n
i=1 f i (y). In the constrained optimization problem, all the
agents cooperatively seek an optimal solution of the global
objective function with set constraints
minm f (y), s.t. y ∈

(∇ f (ξ1 ) − ∇ f (ξ2 )) (ξ1 − ξ2 ) ≥ ωξ1 − ξ2  .
T

2

y∈R

(4)

Moreover, a map F : Rn → Rm is called θ -Lipschitz with
θ being a positive constant if for any ξ1 , ξ2 ∈ Rn
F(ξ1 ) − F(ξ2 ) ≤ θ ξ1 − ξ2 .

(5)

Next, a closed set  ⊆ Rn is said to be convex, if for any
y1 , y2 ∈ Rn and μ ∈ [0, 1], μy1 + (1 − μ)y2 ∈ . Then,
the projection π (y) : Rn →  onto set  characterizes
the unique point within  that is closest to y, i.e., π (y) =
arg minz∈ y − z. Next, Lemma 1 states a basic result about
the projection.
Lemma 1 [40]: Assume that set  ⊆ Rn is closed and
convex. z = π (w) if and only if z ∈  and for any y ∈ ,
(w − z)T (z − y) ≥ 0.
Lemma 2: Given any closed convex set  ⊆ Rn , the
following inequality holds:
π (ξ1 ) − π (ξ2 ) ≤ ξ1 − ξ2  ∀ξ1 , ξ2 ∈ Rn .

(6)

B. Interaction Topology
An undirected graph G = {V, E} is implemented to establish the interaction topology among agents, where V =
{1, 2, . . . , n} and E ⊆ V × V are, respectively, node set and
edge set. (i, j ) ∈ E implies that ( j, i ) ∈ E, which indicates
that nodes i and j can interact mutually, and they are treated
as neighbors. Set Ni = { j ∈ V|( j, i ) ∈ E} incorporates all
the neighbors of node i . A path from node i to node j is a
sequence of nonrepetitive edges. An undirected graph is called
connected if there exists a path between any pair of nodes.
Given an undirected graph G, the definition of its (symmetric)
adjacency matrix A = [ai j ] ∈ Rn×n is that ai j = 1 (or = 0)
if ( j, i ) ∈ E (or ∈
/ E). Also, the definition of its 
(symmetric)
Laplacian matrix L = [li j ] ∈ Rn×n is that lii = nj =1, j =i ai j
and li j = −ai j for j = i .

n


i

(8)

i=1

where the closed and convex set i exclusively belongs to
agent i . Assume that each agent undertakes a local information
exchange with its neighbors. Then, an undirected graph G =
{V, E} is available for the characterization of the internal interaction topology. Before proceeding, three basic assumptions
are first given.
Assumption 1: The underlying undirected graph G is connected.

Assumption 2: ni=1 i = ∅.
Assumption 3: Each local objective function f i is differentiable, and its gradient ∇ f i is θ -Lipschitz for a constant θ > 0.
IV. M AIN R ESULTS
We next concentrate on the search of an optimal solution
to the constrained optimization problem (8) of the multiagent
system (7) using the distributed methodology. In particular,
in terms of an equivalent constrained optimization form, a
distributed algorithm resorting to an auxiliary dynamics is
first synthesized. Next, by postulating that each local objective
function is strongly convex, selection criteria of step size and
algorithm parameter are formulated such that the synthesized
distributed algorithm successfully evaluates the unique optimal
solution to the constrained optimization problem (8). Finally,
by fixing a unit step size, it is demonstrated that the proposed
distributed algorithm with a proper parameter is also applicable to a relaxed case with just convex local objective functions.

A. Distributed Optimization Algorithm
Motivated by [41], the constrained optimization problem
(8) can be equivalently transformed to an alternative form
associated with local variables. With such a transformation,
the concerned constrained optimization problem can be solved
from a consensus perspective. In particular, Lemma 3 formally
establishes the transformation result.

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IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS, VOL. 34, NO. 9, SEPTEMBER 2023

Lemma 3 [41]: Given Assumptions 1–3, the constrained
optimization problem (8) can be equivalently written as
minmn f¯( y) =

y∈R

n


f i (yi )
(9)

where y = col(y1 , y2 , . . . , yn ), L denotes the Laplacian matrix
defined in Section II-B, ⊗ denotes the Kronecker product, and
 = 1 × 2 × · · · × n with × being the Cartesian product.
Given Assumption 1, (L ⊗ Im ) y = 0 indicates y1 = y2 =
· · · = yn . Based on [42], y∗ = 1m ⊗ y ∗ is an optimal solution
to the transformed optimization problem (9) if and only if
y ∗ is an optimal solution to the constrained optimization
problem (8), where 1m is a vector of m-dimensional entries 1.
Therefore, we alternatively focus on the transformed optimization problem (9) in the following.
In addition, to facilitate the distributed algorithm design,
Lemma 4 presents a KKT condition to characterize the optimal
solution to the transformed optimization problem (9).
Lemma 4 (See [42], [43]): Suppose that each local objective function f i is convex. y∗ = col(y1∗ , y2∗ , . . . , yn∗ ) is
an optimal solution to the transformed optimization problem (9), if and only if there is an optimal multiplier α ∗ =
col(α1 , α2 , . . . , αn ) such that


  
y∗ − π y∗ − β ∇ f¯ y∗ + (L ⊗ Im )α ∗ = 0 (10a)
(L ⊗ Im ) y∗ = 0 (10b)
where β is a positive parameter. Equation (10) is called the
KKT condition.
Besides, in terms of Lemma 1, (10a) is equivalent to


y − y∗

T 

 
∇ f¯ y∗ + (L ⊗ Im )α ∗ ≥ 0

(11)

for any y ∈ , which is called the variational inequality
condition.
In order to obtain an optimal solution satisfying (10), we
propose the following distributed algorithm:
u i [s]
= −x i [s] − 2v i [s]
⎡

αi [s + 1]

⎡

= αi [s] + τ πi ⎣x i [s] + v i [s]

i=1

s.t. (L ⊗ Im ) y = 0 and y ∈ 

following dynamics:

⎛
−β ⎝∇ f i (x i [s] + v i [s])
+

+





ai j x i [s] + v i [s] − x j [s] − v j [s]

j ∈Ni

⎞⎤

ai j αi [s] − α j [s] ⎠⎦.





(13)

j ∈Ni

Note from (12) and (13) that, to implement the proposed
distributed algorithm, each agent interacts its state variables x i
and v i and auxiliary variable αi with its neighbors. Moreover,
rather than the distributed optimization proposed in [16], [30],
and [32], by resorting to the auxiliary dynamics (13), it is
not necessary to exchange the private information of the local
objective function f i as well as individual constrained set i .
Remark 1: The intuition behind the proposed distributed
algorithm (12) is to guarantee that all v i converge to zero and
all x i reach the optimal solution that satisfies the KKT condition 10. According to Lemma 4, this solves the constrained
optimization
problem (8). For this purpose, the interaction

a
item
j ∈Ni i j (x i [s] + v i [s] − x j [s] − v j [s]) in the proposed
distributed algorithm (12) plays a role in driving all x i + v i to
reach a consensus, while the remaining items are responsible
for coordinating each v i to zero and each x i to a solution
to (10a). These thus guarantee that all x i [s] reach a consensus
that satisfies (10b).
Next, to facilitate the subsequent analysis, we introduce a
composite variable ri = x i + v i for each agent i . By virtue
of (7) and (12), the dynamics of each ri is given by
ri [s + 1] = (1 − τ )ri [s] + τ πi
⎡
⎛
×⎣ri [s] − β ⎝∇ f i (ri [s]) +



ai j (ri [s]

j ∈Ni



−r j [s] +

⎛



⎞⎤

ai j αi [s]−α j [s] ⎠⎦.


j ∈Ni

−πi ⎣x i [s] + v i [s] − β ⎝∇ f i (x i [s]+v i [s])
+



(14)


ai j x i [s] + v i [s]−x j [s]

j ∈Ni

⎞⎤
  

−v j [s] +
ai j αi [s]−α j [s] ⎠⎦
j ∈Ni

Also, with the composite variable ri , the auxiliary dynamics
(13) becomes
αi [s + 1] = αi [s] + τ πi
⎡
⎛
×⎣ri [s] − β ⎝∇ f i (ri [s])



ai j ri [s]−r j [s]

j ∈Ni

(12)
where the algorithm parameter β is defined identically to that
in (10), the adjacency matrix A = [ai j ] has been defined in
Section II-B, and αi is an auxiliary variable generated by the



+



⎞⎤


ai j αi [s]−α j [s] ⎠⎦.

j ∈Ni

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(15)

ZOU et al.: DISTRIBUTED OPTIMIZATION FOR SECOND-ORDER DISCRETE-TIME MULTIAGENT SYSTEMS

In addition, by considering (7), (12), and (14), the dynamics
of each v i is further derived as
v i [s + 1] = (1 − τ )v i [s] + ri [s + 1] − ri [s].

(16)

Next, by defining r, α, and v as the column stacks of ri , αi ,
and v i , (14)–(16) are equivalently written as
r[s + 1] = (1 − τ )r[s]


+τ π r[s] − β ∇ f¯(r[s])


+(L ⊗ Im )r[s] + (L ⊗ Im )α[s]

α[s + 1] = α[s]


+τ π r[s] − β ∇ f¯(r[s])
+(L ⊗ Im )r[s] + (L ⊗ Im )α[s]

(17)

(18)
(19)

B. General Case: Unfixed Step Size
We focus on the general multiagent system (7) where the
step size is unfixed. By considering that each local objective
function is strongly convex, it suffices to guarantee the uniqueness of the optimal solution to the constrained optimization
problem (8) [44]. Explicit criteria of step size and algorithm
parameter are next formulated such that this unique optimal
solution is evaluated by the proposed distributed algorithm.
Theorem 1 elaborates this result.
Theorem 1: Consider the multiagent system (7). Assume
that Assumptions 1–3 hold and each local objective
function f i is strongly convex. If the step size τ satisfies
√
√
2
2
<τ <1+
(20)
1−
2
2
and the algorithm parameter β satisfies
−1

1
1
+ 2(θ + λmax (L))τ +
β<
(21)
2ω
2−τ
the proposed distributed algorithm (12) with (13) guarantees
that each state variable x i [s] asymptotically converges to the
optimal solution to the constrained optimization problem (8)
and each state variable v i [s] asymptotically converges to 0.
Proof: To prove Theorem 1, we first specify three positive
definite functions
V1 (v[s]) = v[s]2 ,
V2 (r[s]) = r[s] − r ∗ 2 ,

T


V3 (α[s]) = α[s] − α ∗ (L ⊗ Im ) α[s] − α ∗

In the second place, take V2 (r[s]) in (23) into account.
Define g[s] = π [r[s] − β(∇ f¯(r[s]) + (L ⊗ Im )r[s] + (L ⊗
Im )α[s])]. By virtue of (17), it follows that
V2 (r[s + 1]) − V2 (r[s])
= r[s + 1] − r ∗ 2 − r[s] − r ∗ 2


= (r[s + 1] − r[s])T r[s + 1] + r[s] − 2r ∗



= −r[s + 1] − r[s]2 + 2(r[s + 1] − r[s])T r[s + 1] − r ∗



v[s + 1] = (1 − τ )v[s] + r[s + 1] − r[s].

5633

(22)
(23)
(24)

where r ∗ = y∗ and α ∗ have been introduced in Lemma 4.
We next discuss the convergence performance of these three
functions.
In the first place, consider V1 (v[s]) in (22). In terms of (19),
we have that
V1 (v[s + 1]) − V1 (v[s])
= v[s + 1]2 − v[s]2
= (1 − τ )v[s] + r[s + 1] − r[s]2 − v[s]2
= r[s + 1] − r[s]2 + 2(1 − τ )v[s]T (r[s + 1] − r[s])


(25)
− 2τ − τ 2 v[s]2 .

= −r[s + 1] − r[s]2 + 2τ (g[s] − r[s])T

· (1 − τ )r[s] + τ g[s] − r ∗



= −r[s + 1] − r[s]2 + 2τ 2 (g[s] − r[s])T g[s] − r ∗


+2τ (1 − τ )(g[s] − r[s])T r[s] − r ∗



= −r[s + 1] − r[s]2 + 2τ 2 g[s] − r[s] − β ∇ f¯(r[s])


+(L ⊗ Im )(r[s] + α[s])))]T g[s] − r ∗


+2τ (1 − τ )(g[s] − r[s])T r[s] − r ∗

T 
−2βτ 2 g[s] − r ∗ ∇ f¯(r[s]) + (L ⊗ Im )(r[s] + α[s]) .
(26)
In terms of Lemma 1, we know that [g[s] − (r[s] −
β(∇ f¯(r[s]) + (L ⊗ Im )(r[s] + α[s])))]T (g[s] − r ∗ ) ≤ 0.
By substituting this inequality into (26), it then follows that
V2 (r[s + 1]) − V2 (r[s])



≤ −r[s + 1] − r[s]2 +2τ (1 − τ )(g[s] − r[s])T r[s] − r ∗

T 
−2βτ 2 g[s] − r ∗ ∇ f¯(r[s])+(L ⊗ Im )(r[s] + α[s]) .
(27)

Moreover, by considering (17), it also follows that

T
τ g[s] − r ∗ ∇ f¯(r[s])

T
= r[s + 1] + (τ − 1)r[s] − τ r ∗ ∇ f¯(r[s])


T
= r[s + 1] − r ∗ ∇ f¯(r[s])

T
+(τ − 1) r[s] − r ∗ ∇ f¯(r[s]).

(28)

Since each f i is strongly convex, by virtue of (3), it further
follows that
T

r[s + 1] − r ∗ ∇ f¯(r[s])


= (r[s + 1] − r[s])T ∇ f¯(r[s]) − ∇ f¯(r[s + 1])


+(r[s + 1] − r[s])T ∇ f¯(r[s + 1]) + r[s] − r ∗ ∇ f¯(r[s])


≥ (r[s + 1] − r[s])T ∇ f¯(r[s]) − ∇ f¯(r[s + 1])
  ω
+ f¯(r[s + 1]) − f¯ r ∗ + r[s + 1] − r[s]2
2
ω
∗ 2
+ r[s] − r  .
(29)
2
Hence, substituting (28) and (29) into (27) gives rise to
V2 (r[s + 1]) − V2 (r[s])
≤ −(1 + βτ ω)r[s + 1] − r[s]2 − βτ ωr[s] − r ∗ 2


+2βτ (r[s + 1] − r[s])T ∇ f¯(r[s + 1]) − ∇ f¯(r[s])

T 

−2βτ (1 − τ ) r[s] − r ∗ g[s] − r[s] + ∇ f¯(r[s])

T
−2βτ 2 g[s] − r ∗ (L ⊗ Im )(r[s] + α[s])

  ∗
(30)
+2βτ f¯ r − f¯(r[s + 1]) .

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5634

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Again, by considering the strong convexity of each f i , it then
follows from (3) and (17) that
 
f¯ r ∗ − f¯(r[s + 1])

T
  ω
≤ r ∗ − r[s + 1] ∇ f¯ r ∗ − r[s + 1] − r ∗ 2
2
 ∗
T
  ω
= r +(τ − 1)r[s] − τ g[s] ∇ f¯ r ∗ − r[s + 1] − r ∗ 2
2

T
 ∗
T
 ∗
 
¯
= τ r − g[s] ∇ f r + (τ − 1) r[s] − r ∗ ∇ f¯ r ∗
ω
− r[s + 1] − r ∗ 2
2


 
T
T
≤ τ g[s] − r ∗ (L ⊗ Im )α ∗ + (τ − 1) r[s] − r ∗ ∇ f¯ r ∗
ω
(31)
− r[s + 1] − r ∗ 2
2
where the last inequality considers (11) with r ∗ = y∗ .
Substituting (31) into (30) further yields
V2 (r[s + 1]) − V2 (r[s])
≤ −(1 + βτ ω)r[s + 1] − r[s]2 − 2βτ ωr[s] − r ∗ 2


+2βτ (r[s + 1] − r[s])T ∇ f¯(r[s + 1]) − ∇ f¯(r[s])

T 
+2βτ (1 − τ ) r[s] − r ∗ g[s] − r[s] − ∇ f¯(r[s])
 


T

+∇ f¯ r ∗ − 2βτ 2 g[s] − r ∗ (L ⊗ Im ) r[s]+α[s]−α ∗
≤ −(1 + βτ ω − 2βτ θ )r[s + 1] − r[s]2
−2βωτ (2 − τ )r[s] − r ∗ 2

T
+2βτ (1 − τ ) r[s] − r ∗ (g[s] − r[s])


T

−2βτ 2 g[s] − r ∗ (L ⊗ Im ) r[s] + α[s] − α ∗
= −(1 + βτ ω − 2βτ θ )r[s + 1] − r[s]2
−2βωτ (2 − τ )r[s] − r ∗ 2

T
+2β(1 − τ ) r[s] − r ∗ (r[s + 1] − r[s])


T

−2βτ 2 g[s] − r ∗ (L ⊗ Im ) r[s] + α[s] − α ∗

(32)

where the second inequality considers (4), the Lipschitz condition claimed in Assumption 3, and the strong convexity of
f¯.
In the third place, consider V3 (α[s]) in (24). It follows from
(18) that
V3 (α[s + 1]) − V3 (α[s])

T


= α[s + 1] − α ∗ (L ⊗ Im ) α[s + 1] − α ∗

T


− α[s] − α ∗ (L ⊗ Im ) α[s] − α ∗

T


= α[s] + τ g[s] − α ∗ (L ⊗ Im ) α[s] + τ g[s] − α ∗

T


− α[s] − α ∗ (L ⊗ Im ) α[s] − α ∗



= τ g[s]T (L ⊗ Im ) τ g[s] + 2 α[s] − α ∗
= τ g[s]T (L ⊗ Im )(τ g[s]−2r[s])

T


+2τ g[s]− r ∗ (L ⊗ Im ) α[s] − α ∗ + r[s]
= (r[s + 1]+(τ −1)r[s])T (L ⊗ Im )(r[s + 1] + (τ − 3)r[s])

T


+2τ g[s]− r ∗ (L ⊗ Im ) α[s] − α ∗ + r[s]
= (r[s + 1] − r[s])T (L ⊗ Im )(r[s + 1] − r[s])


+2(τ − 1)(r[s + 1] − r[s])T (L ⊗ Im ) r[s] − r ∗


T

+τ (τ − 2) r[s] − r ∗ (L ⊗ Im ) r[s] − r ∗

T


+2τ g[s] − r ∗ (L ⊗ Im ) α[s] − α ∗ + r[s] .

(33)

By using the positive definite functions given in (22)–(24),
we assign a Lyapunov candidate function as follows:
V (v[s], r[s], α[s]) = βτ V1 (v[s]) + V2 (r[s]) + βτ V3 (α[s]).
(34)
By virtue of (25), (32), and (33), it follows that
V (v[s + 1], r[s + 1], α[s + 1]) − V (v[s], r[s], α[s])
≤ −(1 − βτ + βτ ω − 2βτ θ )r[s + 1] − r[s]2
−βτ 2 (2 − τ )v[s]2 − 2βωτ (2 − τ )r[s] − r ∗ 2
+2βτ (1 − τ )v[s]T (r[s + 1] − r[s])

T
+2β(1 − τ ) r[s] − r ∗ (r[s + 1] − r[s])
+βτ (r[s + 1] − r[s])T (L ⊗ Im )(r[s + 1] − r[s])


+2βτ (τ − 1)(r[s + 1] − r[s])T (L ⊗ Im ) r[s] − r ∗


T

+βτ 2 (τ − 2) r[s] − r ∗ (L ⊗ Im ) r[s] − r ∗

T


r[s + 1] − r[s]
r[s + 1] − r[s]
=−
(R1 ⊗ Im )
v[s]
v[s]

T


r[s + 1] − r[s]
r[s + 1] − r[s]
−β
(R2 ⊗ Imn )
r[s] − r ∗
r[s] − r ∗

T


r[s + 1]− r[s]
r[s + 1]− r[s]
⊗
L
⊗
I
−β
(R
)
3
m
r[s]− r ∗
r[s]− r ∗
(35)
where

⎡
β
In − 2βτ L
−
2βθ
τ
1
−
βτ
−
R1 =⎣
2ω
βτ (τ − 1)In

 1
τ −1
R2 =
2ω
τ − 1 2ωτ (2 − τ )


τ
τ (1 − τ )
R3 =
.
τ (1 − τ ) τ 2 (2 − τ )

⎤
βτ (τ − 1)In ⎦
βτ 2 (2 − τ )In

It is then trivial to show that matrices R2 and R3 are positive
definite given that the step size τ satisfies (20). Based on this
fact, the algorithm parameter β satisfying (21) is sufficient to
guarantee the positive definiteness of matrix R1 . It thus follows
that V (v[s + 1], r[s + 1], α[s + 1]) − V (v[s], r[s], α[s]) ≤ 0.
In terms of the LaSalle invariance principle for discretetime systems [45], v[s], r[s], and α[s] converge to the largest
invariant set of set
= {v[s] ∈ Rmn , r[s] ∈ Rmn , α[s] ∈
mn
R | V (v[s +1], r[s +1], α[s +1])− V (v[s], r[s], α[s]) = 0}
as s tends to infinity. If V (v[s + 1], r[s + 1], α[s + 1]) −
V (v[s], r[s], α[s]) ≡ 0, it then follows from (35) that v[s] ≡ 0
and r[s + 1] ≡ r[s] ≡ r ∗ . By virtue of (17), this implies
that r ∗ − π [r ∗ − β(∇ f¯(r ∗ ) + (L ⊗ Im )α[s])] ≡ 0. Due to
the fact that r ∗ = y∗ and in terms of Lemma 4, this further
implies that α[s] ≡ α ∗ . Thus, the only solution that stays
identically in set is the trivial solution v[s] = 0, r[s] = y∗
and α[s] = α ∗ , where y∗ = 1n ⊗ y ∗ is the optimal solution to
the transformed optimization problem (9). Therefore, in terms
of the LaSalle invariance principle [45], it can be concluded
that lims→∞ r[s] = y∗ and lims→∞ v[s] = 0. In view of
the definition of r, we further have that lims→∞ x[s] =
y∗ . Finally, based on Lemma 3, it therefore follows that

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ZOU et al.: DISTRIBUTED OPTIMIZATION FOR SECOND-ORDER DISCRETE-TIME MULTIAGENT SYSTEMS

lims→∞ x i [s] = y ∗ , i ∈ V, where y ∗ is the optimal solution to
the constrained optimization problem (8).

According to Theorem 1, the proposed distributed
algorithm (12) with (13) in terms of a properly chosen step size
τ and parameter β is capable of seeking an optimal solution
to the constrained optimization problem (8). Next, by fixing
the step size τ = 1, we discuss the effect of the algorithm
parameter β in the convergence rate of r [s].
First, in terms of Lemma 2, it follows from (17) with τ = 1
that

+(L ⊗ Im )(r[s] − r[s − 1])


+(L ⊗ Im )(α[s] − α[s − 1]) .
(36)

By using the mean-value theorem, we have that


 r[s] − r[s − 1] − β ∇ f¯(r[s]) − ∇ f¯(r[s − 1])


+(L ⊗ Im )(r[s] − r[s − 1]) 



≤ max  Imn − β ∇ 2 f¯(r[s − 1] + ϑ(r[s] − r[s − 1]))
0≤ϑ≤1


+(L ⊗ Im ) r[s] − r[s − 1]




 Imn − β ∇ 2 f¯(ε) + (L ⊗ Im ) r[s] − r[s − 1].
≤ max
mn
ε∈R

(37)
Moreover, in terms of (17) and (18) with τ = 1, we further
have that (L ⊗ Im )(α[s] − α[s − 1]) = (L ⊗ Im )r[s]. It thus
follows that



 Imn − β ∇ 2 f¯(ε) + L ⊗ Im 
r[s + 1] − r[s] ≤ max
mn
ε∈R

(38)

Next, since f¯ is ω-strongly convex, it follows that ∇ 2 f¯(ε) ≥
ωImn for any ε ∈ Rmn . We thus have that



 Imn − β ∇ 2 f¯(ε) + L ⊗ Im  ≤ |1 − βω|.
max
mn

ε∈R

(39)

j =0

= es ln ζ r[1] − r[0] +

s


e j ln ζ ξ [s − j ]

j =0
s


e−η j ξ [s − j ]

(41)

where η = − ln ζ > 0. Note that e−η ∈ (0, 1) given η > 0.
Also, in terms of Theorem 1, we know that lims→∞ ξ [s] =
0. Thus, according to [13], the inequality in (41) is well
defined. Therefore, r[s + 1] − r[s] exponentially
converges

to a neighborhood of origin confined by sj =0 e−η j ξ [s − j ].
Furthermore, under the condition of (21) and β < 1/ω,
increasing β helps increase η, which, in terms of (41), leads
to a faster convergence rate as well as a smaller convergence
region.
Based on the above arguments, we next give a proposition
to give the convergence performance of r [k].
Proposition 1: Consider the multiagent system (7) with
τ = 1. Assume that Assumptions 1–3 hold and each local
objective function f i is strongly convex. If the algorithm
parameter β satisfies (21) and β < 1/ω, the proposed
distributed algorithm (12) with (13) guarantees that each state
variable x i [s] asymptotically converges to the optimal solution
to the constrained optimization problem (8) and each state
variable v i [s] asymptotically converges to 0. Simultaneously,
they satisfy the convergence property formulated by (41).
C. Specific Case: Unit Step Size
We next consider the multiagent system (7) with a fixed
step size τ = 1. In such a case, we show that the proposed
distributed algorithm (12) with (13) also solves the constrained
optimization problem (8) given an appropriate algorithm parameter even though each local objective function is relaxed to
be convex.
Similar to (17)–(19), by substituting the proposed distributed algorithm (12) with the auxiliary dynamics (13), the
dynamics of composite variable r, auxiliary variable α, and
state variable v under the case of the unit step size are derived
as


r[s + 1] = π r[s] − β ∇ f¯(r[s]) + (L ⊗ Im )r[s]

+(L ⊗ Im )α[s]
(42)


¯
α[s + 1] = α[s] + π r[s] − β ∇ f (r[s]) + (L ⊗ Im )r[s]
+(L ⊗ Im )α[s]

On the premise of (21), if β further satisfies β < 1/ω, it then
follows from (38) that
r[s + 1] − r[s] ≤ ζ r[s] − r[s − 1] + ξ [s]

r[s + 1] − r[s]
≤ ζ 2 r[s − 1] − r[s − 2] + ζ ξ [s − 1] + ξ [s]
s

≤ · · · ≤ ζ s r[1] − r[0] +
ζ j ξ [s − j ]

j =0

+(L ⊗ Im )α[s])] − π [r[s − 1]

−β ∇ f¯(r[s − 1]) + (L ⊗ Im )r[s − 1]

+(L ⊗ Im )α[s − 1]) 


≤  r[s] − r[s − 1] − β ∇ f¯(r[s]) − ∇ f¯(r[s − 1])

·r[s] − r[s − 1] + β(L ⊗ Im )r[s].

where ζ = 1 − βω ∈ (0, 1) and ξ [s] = β(L ⊗ Im )r[s]. By
repeating the derivation in (40), it finally follows that

= e−ηs r[1] − r[0] +

r[s + 1] − r[s]
 

≤ π r[s] − β ∇ f¯(r[s]) + (L ⊗ Im )r[s]

5635

(40)

v[s + 1] = r[s + 1] − r[s].

(43)
(44)

Theorem 2 next elaborates the convergence result of this case.
Theorem 2: Consider the multiagent system (7) with τ = 1.
Assume that Assumptions 1–3 hold and each local objective

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function f i is convex. If the algorithm parameter β satisfies
β < (1 + 2θ + λmax (L))−1

(45)

the proposed distributed algorithm (12) with (13) guarantees
that each state variable x i [s] asymptotically converges to an
optimal solution to the constrained optimization problem (8)
and each state variable v i [s] asymptotically converges to 0.
Proof: Consider the positive definite functions defined
in (22)–(24) again. By the similar derivation given in (25), it
follows from (44) that V1 (v[s]) satisfies
V1 (v[s + 1]) − V1 (v[s]) = r[s + 1]− r[s] − v[s] . (46)
2

2

Next, by referring to (26) and (27), it follows from (42) that
V2 (r[s]) satisfies

Besides, by following the derivation in (33), it follows
from (42) and (43) that V3 (α[s]) satisfies
V3 (α[s + 1]) − V3 (α[s])

T


= α[s] + g[s] − α ∗ (L ⊗ Im ) α[s] + g[s] − α ∗

T


− α[s] − α ∗ (L ⊗ Im ) α[s] − α ∗



= g[s]T (L ⊗ Im ) g[s] + 2 α[s] − α ∗
= g[s]T (L ⊗ Im )(g[s] − 2r[s])

T


+2 g[s] − r ∗ (L ⊗ Im ) α[s] − α ∗ + r[s]
= r[s + 1]T (L ⊗ Im )(r[s + 1] − 2r[s])

T


+2 g[s] − r ∗ (L ⊗ Im ) α[s] − α ∗ + r[s]
= (r[s + 1] − r[s])T (L ⊗ Im )(r[s + 1] − r[s])
−r[s]T (L ⊗ Im )r[s]

T


+2 g[s] − r ∗ (L ⊗ Im ) α[s] − α ∗ + r[s] .

(52)
V2 (r[s + 1]) − V2 (r[s])


By implementing the positive definite functions given in
= −r[s + 1] − r[s]2 + 2(r[s + 1] − r[s])T r[s + 1] − r ∗


(22)–(24),
we specify a Lyapunov candidate function as
= −r[s + 1] − r[s]2 + 2(g[s] − r[s])T g[s] − r ∗

T
V̄ (v[s], r[s], α[s]) = βV1 (v[s]) + V2 (r[s]) + βV3 (α[s]). (53)
= −r[s + 1] − r[s]2 − 2β g[s] − r ∗

(47) According to (46), (51), and (52), we have that
· ∇ f¯(r[s]) + (L ⊗ Im )(r[s] + α[s]) .
Since each local objective function f i is convex, following
(28) and (29) yields:

T
g[s] − r ∗ ∇ f¯(r[s])

T
= r[s + 1] − r ∗ ∇ f¯(r[s])


≥ (r[s + 1] − r[s])T ∇ f¯(r[s]) − ∇ f¯(r[s + 1])
 
(48)
+ f¯(r[s + 1]) − f¯ r ∗ .
By substituting (48) into (47), it then follows that
V2 (r[s + 1]) − V2 (r[s])
≤ −r[s + 1] − r[s]2 + 2β(r[s + 1] − r[s])T


· ∇ f¯(r[s + 1]) − ∇ f¯(r[s])

T
−2β g[s] − r ∗ (L ⊗ Im )(r[s] + α[s])
  

+2β f¯ r ∗ − f¯(r[s + 1]) .

(49)

By considering the convexity of f¯ and following (31), it
follows that
 
f¯ r ∗ − f¯(r[s + 1])

T
 
≤ r ∗ − r[s + 1] ∇ f¯ r ∗
T
 

= r ∗ − g[s] ∇ f¯ r ∗
T

(50)
≤ g[s] − r ∗ (L ⊗ Im )α ∗ .
Substituting (50) into (49) further yields
V2 (r[s + 1]) − V2 (r[s])
≤ −(1 − 2βθ )r[s + 1] − r[s]2

T


−2β g[s] − r ∗ (L ⊗ Im ) r[s] + α[s] − α ∗ (51)
where the Lipschitz condition claimed in Assumption 3 is
considered.

V̄ (v[s + 1], r[s + 1], α[s + 1]) − V̄ (v[s], r[s], α[s])
= −(1 − β − 2βθ )r[s + 1] − r[s]2 − βv[s]2
+β(r[s + 1] − r[s])T (L ⊗ Im )(r[s + 1] − r[s])
−β r[s]T (L ⊗ Im )r[s]
≤ −(1 − β − 2βθ − βλmax (L))r[s + 1] − r[s]2
−βv[s]2 − β r[s]T (L ⊗ Im )r[s].

(54)

If β is chosen according to (45), it can be derived that V̄ (v[s +
1], r[s + 1], α[s + 1]) − V̄ (v[s], r[s], α[s]) ≤ 0.
Also, motivated by the LaSalle invariance principle for
discrete-time systems [45], v[s], r[s], and α[s] converge to
the largest invariant set of set ¯ = {v[s] ∈ Rmn , r[s] ∈
Rmn , α[s] ∈ Rmn | V̄ (v[s + 1], r[s + 1], α[s + 1]) −
V̄ (v[s], r[s], α[s]) = 0} as s tends to infinity. By virtue
of (44) and (54), given V̄ (v[s + 1], r[s + 1], α[s + 1]) −
V̄ (v[s], r[s], α[s]) ≡ 0, we have that r[s + 1] ≡ r[s] and
v[s] ≡ 0 and (L⊗Im )r[s] ≡ 0 and further that v[s+1] ≡ 0. By
virtue of (42), this implies that r[s] ≡ π [r[s]−β(∇ f¯(r[s])+
(L ⊗ Im )α[s])]. In such a case, according to Lemma 4, this
implies that r[s] ≡ y∗ and α[s] ≡ α ∗ , where y∗ = 1n ⊗ y ∗ is
the optimal solution to the transformed optimization problem
(9). Thus, the only solution that stays identically in set ¯
is the trivial solution v[s] = 0, r[s] = y∗ and α[s] = α ∗ .
Therefore, it can be concluded that lims→∞ r[s] = y∗ and
lims→∞ v[s] = 0. Due to the definition of r, we have that
lims→∞ x[s] = y∗ . Finally, by referring to Lemma 3, it thus
follows that lims→∞ x i [s] = y ∗ , i ∈ V, where y ∗ is the optimal
solution to the constrained optimization problem (8).

Remark 2: This article considers an ideal optimization solution environment without suffering cyber-physical attacks. In
case of the existence of unknown cyber-attacks, according
to [46] and [47], a feasible solution scheme is to detect the
attack and further to exclude the attack effect with resilient
schemes.

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ZOU et al.: DISTRIBUTED OPTIMIZATION FOR SECOND-ORDER DISCRETE-TIME MULTIAGENT SYSTEMS

Fig. 1.

5637

Network topology among six agents (Example 1).

V. S IMULATIONS
In this section, we carry out the validation of the developed
distributed optimization algorithm. In particular, a practical
source localization example with strongly convex local objective functions is first taken to verify the main results in
Theorem 1. Next, given a unit step size, another numerical
example with just convex local objective functions is taken to
verify the main results in Theorem 2.

Fig. 2.

Trajectories of states xi , i = 1, 2, . . . , 6 (Example 1).

Fig. 3.

Trajectories of states v i , i = 1, 2, . . . , 6 (Example 1).

A. Practical Example
Assume that there exist a team of six agents of second-order
discrete-time dynamics (7) with m = 2. They cooperatively
seek a static acoustic source posited at x̄ 0 ∈ R2 on a plane
guided by five cardinal points r j ∈ R2 , j = 1, . . . , 5. However,
the position information of each cardinal point is just available
to a part of agents, while the availability of the cardinal points
to the agents is characterized by a cardinal matrix B = [bi j ] ∈
R6×5 . To be more specific, if cardinal point j is available
to agent i , then di j = 1; otherwise, di j = 0. Accordingly,
each
its local objective function as f i (yi ) =
5 agent i chooses
2
j =1 bi j yi − r j  , meaning that it tries to move as close as
possible to its available cardinal points. The strong convexity
of each local objective function f i can be easily validated.
Besides, assume that each agent i knows beforehand that the
source is situated within an area described by i = {y ∈
R2 |yi − wi  ≤ φi2 }. The objective of all the agents is to
uniformly aggregate
at an optimal point of the global objective
function f (y) = 6i=1 fi (y) within each available area i .
Such an optimal point is the optimal position estimation of
the unavailable source.
Fig. 1 shows the network topology among six agents with
each weight 1. Accordingly, Assumption 1 can be confirmed.
The acoustic source is situated at [−2, 0]T , the cardinal points
are situated at r1 = [3, 4]T , r2 = [1, 1]T , r3 = [−2, 2]T , r4 =
[−3, −6]T and r5 = [2, −5]T , and the cardinal matrix B =
[0, 0, 1, 1, 0; 0, 0, 0, 0, 1; 0, 1, 0, 0, 0; 1, 1, 0, 0, 1; 1, 0, 0, 0, 1;
1, 0, 1, 0, 0] is specified. In addition, the available areas of
the agents are parameterized as w1 = [−2, −3]T , φ1 = 3,
w2 = [1, −1]T , φ2 = 2, w3 = [0, 0]T , φ3 = 2, w4 = [4, −2],
T
T
φ4 = 5, w5 =
[1, 1] , φ5 = 4, and w6 = [−2, 1] , φ6 = 5. It is
trivial that 6i=1 i = ∅, which means that Assumption 2
holds. The system states are initialized as x 1 [0] = [−2, 7]T ,
v 1 [0] = [0, 0]T , x 2 [0] = [8, −2]T , v 2 [0] = [0, 0]T , x 3 [0] =
[9, −4]T , v 3 [0] = [0, 0]T , x 4 [0] = [0, 4]T , v 4 [0] = [0, 0]T ,

x 5 [0] = [5, 0]T , v 5 [0] = [0, 0]T , and x 6 [0] = [2, 3]T ,
v 6 [0] = [0, 0]T . Besides, the step size τ = 0.5 is fixed such
that (20) holds, and the algorithm parameter β = 0.1 is set
such that (21) holds.
Figs. 2 and 3 show the simulation results. It can be observed
that all the states x i come to a consensus at the optimal
solution [0.1886, −0.9482]T asymptotically, and all the states
v i asymptotically converge to zero. Hence, the optimization
results in Theorem 1 are verified.
B. Numerical Example
We next study the case with just convex local objective
functions. In particular, consider six agents of second-order
discrete-time dynamics (6) with m = 3 and unit step size
τ = 1. Their network topology is shown in Fig. 4 with
each weight 1. The local objective functions
allocated to
3
(sin(x j + 1) +
the agents are chosen as f 1 (x) =
j
=1
3
(x j + 1)2 /2), f 2 (x) =
j =1 (cos(x j + 2)/3 + (x j −
3
(2(x
−
1)2 /9 + 4 cos(x j + 1)/9),
2)2 /6), f 3 (x) =
j
j =1
3
3
4
f 4 (x) =
(x − 3) , f 5 (x) =
j =1 (x j + 4), and
3j =1 j
6/5
f 6 (x) =
(x
+
1)
.
By
twice
differentiating
these
j =1 j

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5638

IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS, VOL. 34, NO. 9, SEPTEMBER 2023

optimal solution
[1.929, 1.929, 1.929]T of the global objective
6
function i=1 f i (x) asymptotically. Next, Fig. 6 shows that all
the states v i converge to zero asymptotically. Therefore, the
optimization results in Theorem 2 for the case of convex local
objective functions are validated.
VI. C ONCLUSION

Fig. 4.

Network topology among six agents (Example 2).

Fig. 5.

Trajectories of states xi , i = 1, 2, . . . , 6 (Example 2).

Fig. 6.

Trajectories of states v i , i = 1, 2, . . . , 6 (Example 2).

local objective functions, it is trivial to obtain that they are
just convex. Moreover, designate an identical constrained set
 = {x ∈ R3 | x − [1, 1, 1]T  ≤ 3} for six agents. The
initial system states are chosen as x 1 [0] = [−2, 2, 1]T , v 1 [0] =
[0, 0, 0]T , x 2 [0] = [3, −3, 5]T , v 2 [0] = [0, 0, 0]T , x 3 [0] =
[−3, 4, 0]T , v 3 [0] = [0, 0, 0]T , x 4 [0] = [1, 1, 1]T ,
v 4 [0] = [0, 0, 0]T , x 5 [0] = [5, −5, 0]T , v 5 [0] = [0, 0, 0]T ,
and x 6 [0] = [0, −2, −2]T , v 6 [0] = [0, 0, 0]T . The algorithm
parameter β = 0.01 is chosen according to (45).
Figs. 5 and 6 show the simulation results. First, Fig. 5
exhibits that all the states x i come to a consensus at an

This article proposes a distributed optimization algorithm
for a second-order discrete-time multiagent system subject to
set constraints. The proposed algorithm is projection-based
without exchanging the gradient information of local objective
functions. By imposing the strong convexity requirement on
the local objective functions, it is first shown that all the agents
driven by the proposed distributed algorithm are capable
of reaching a consensus at the unique optimal solution to
the concerned optimization problem given that the step size
and algorithm parameter are specified according to well-built
selection criteria. Next, by fixing a unit step size, it is shown
that the proposed distributed optimization algorithm is also
applicable to a relaxed case with only convex local objective
functions. Practical and numerical examples are both taken to
validate the built main results. In the future, we will focus on
the distributed optimization algorithm developments for the
cases of: 1) cyber-physical attacks; 2) nonuniform step sizes;
3) nonconvex objective functions; and 4) switching directed
topology.
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