IEEE TRANSACTIONS ON CYBERNETICS, VOL. 52, NO. 6, JUNE 2022

4451

Distributed Robust Optimization Algorithms
Over Uncertain Network Graphs
Zizhen Wu

and Zhongkui Li , Member, IEEE

Abstract—This article investigates the robustness issues of a set
of distributed optimization algorithms, which aim to approach
the optimal solution to a sum of local cost functions over an
uncertain network. The uncertain communication network consists of transmission channels perturbed by additive deterministic
uncertainties, which can describe quantization and transmission
errors. A new robust initialization-free algorithm is proposed for
the distributed optimization problem of multiple Euler–Lagrange
systems, and the explicit relationship of the feedback gain of
the algorithm, the communication topology, the properties of
the cost function, and the radius of the channel uncertainties
is established in order to reach the optimal solution. This result
provides a sufficient condition for the selection of the feedback
gain when the uncertainty size is less than the unity. As a special
case, we discuss the impact of communication uncertainties on
the distributed optimization algorithms for first-order integrator
networks.
Index Terms—Communication channel, cooperative control,
distributed optimization, robustness.

I. I NTRODUCTION
REMENDOUS advances in computing and communication technology have enriched the application
scenarios of large-scale networked control theory and
distributed cooperation strategies [1]–[3], such as databased learning networks [4]–[6]; unmanned vehicle or
robotic networks [7]–[9]; sensor estimation or monitoring
networks [10], [11]; and smart power allocation and management networks [12]–[14]. Such application domains often
have some technical indicators or cost functions that can be
characterized as a global optimization problem with the related
objective function. How to solve these optimization problems in a cyber-physical system through a distributed strategy
instead of a classical central strategy has received a lot of
attention [15], [16].
Among the existing distributed optimization strategies, the
consensus-based algorithm and its expansions, benefiting from

T

Manuscript received March 25, 2020; revised June 30, 2020; accepted
September 13, 2020. Date of publication November 4, 2020; date of current version June 16, 2022. This work was supported in part by the Beijing
Natural Science Foundation under Grant JQ20025, and in part by the National
Natural Science Foundation of China under Grant 61973006 and Grant
U1713223. This article was recommended by Associate Editor K. You.
(Corresponding author: Zhongkui Li.)
The authors are with the State Key Laboratory for Turbulence and
Complex Systems, Department of Mechanics and Engineering Science,
College of Engineering, Peking University, Beijing 100871, China (e-mail:
zizhenwu@pku.edu.cn; zhongkli@pku.edu.cn).
Color versions of one or more figures in this article are available at
https://doi.org/10.1109/TCYB.2020.3027415.
Digital Object Identifier 10.1109/TCYB.2020.3027415

their convenient implementation and scalability nature, have
become a classic optimization scheme [17]–[20]. Considering
practical applications with complicated agent dynamics, some
distributed optimization protocols are designed for multiagent
systems with high-order node dynamics [21]–[24]. In addition to these distributed unconstrained optimization problems,
there is also some literature that further studies the constrained
optimization strategies in networked systems [25], [26]. In
studying these algorithms, it is common to presume that the
participators interact over an ideal communication network,
just like in traditional consensus problems. However, in practice, both the internal constraint of the network and the external
environment of the participators will damage the communication quality more or less [27]. Hence, it is of both theoretical
and practical significance to study distributed optimization
algorithms with communication uncertainties.
In the literature of distributed optimization problems
(DOPs), only a few preliminary results consider the distributed
optimization algorithms with external perturbations [28]–[31].
In [28], the robustness against the continual Gaussian white
noise is studied for a set of distributed PI protocols for a
single integrator network, and it is shown that the resulting
stochastic dynamic is noise-to-state exponentially stable in the
second moment. Yang et al. [29] considered the case where
the communication delays exist in the distributed optimization
algorithm. Based on the internal model approach, a family of distributed protocols with disturbance rejection ability
is designed to address the DOP in heterogeneous nonlinear systems [30]. To achieve the optimal solution in a finite
time, a class of distributed optimization protocols combined
with an integral sliding-mode scheme is proposed in [31],
while the bounded disturbances are suppressed. In addition
to these external disturbances or measurement noises, some
discrete communication technologies used in continuous-time
systems will actively introduce some communication errors,
such as event-based or/and quantized communication mechanisms [20], [32]–[34]. Since the uncertain network model
can be used to represent the networked systems with uncertain dynamics and quantized communication channels, the
distributed continuous-time optimization problem with communication noises or discrete communication mechanisms can
be formulated as a more general problem [35].
This article mainly considers the convex optimization
problem in the Euler–Lagrange (EL) multiagent system over
an uncertain communication network. This consideration is
motivated by the fact that most of the existing results are
focusing on integrator-type networks, instead of more practical

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4452

IEEE TRANSACTIONS ON CYBERNETICS, VOL. 52, NO. 6, JUNE 2022

and complex node dynamics models [24]. EL systems, as
one of the typical dynamic models, can be used to portray various mechanical systems, such as rigid bodies, mobile
robots, and autonomous vehicles. Benefitting from these rich
application scenarios, DOPs of multiple EL systems have
drawn increasing attention [22], [23]. However, for both the
integrator-type and EL networks, most of these existing protocols are dependent on ideal communication networks. Hence,
the object of this article is two-fold. First, we design a novel
robust distributed algorithm for the optimization problem of
multiple EL systems with uncertain networks. Second and also
meaningful, we investigate the robustness of the classic distributed optimization protocols for integrator-type networks
and provide corresponding analyses.
Our contributions are summarized below. In order to
remove the initialization condition of the optimization algorithm proposed in [22], an initialization-free distributed
optimization algorithm is first designed for the EL networked
system in the ideal case without communication uncertainties. Considering the uncertain communications, we then
derive an explicit expression for the relationship of the feedback gain of the optimization algorithm, the eigenvalues of
the Laplacian matrix, the Lipschitz constant and convexity
constant of the cost function, and the radius of the channel uncertainties. This relationship highlights the constraint
which the optimization algorithm, the cost function, and the
uncertain network should satisfy in order to achieve the
optimal solution. As extensions, we re-examine a set of distributed optimization protocols proposed for single-integrator
networks in the case of uncertain networks and present similar results. Furthermore, it is worth claiming that all edge
weights must remain positive in the existence of uncertainties. These results may provide a reference for the selection
of quantization intervals or triggering conditions in hybrid
systems.
The remainder of this article is organized as follows.
Section II establishes the model of the uncertain network and
formulates the continuous-time DOP. Then, the initializationfree algorithm for EL multiagent systems and its robustness are
presented in Section IV. Section V researches the robustness
of the distribution optimization algorithms for integrator-type
networks. Numerical simulations are drawn in Section VI and
conclusions are provided in Section VII.

gradient of the function f (·). If it is ε-strictly convex, then
(p − q)T (∇f (p) − ∇f (q)) ≥ εp − q2 [15].
B. Graph Theory and Uncertain Communication Channels
The information flow among the communication network is
depicted by an undirected graph G = (V, E) with node set V =
1, . . . , n, and edge set E ⊆ V × V. An edge (i, j) ∈ E means
that the node i can exchange information with its neighbor
j ∈ Ni . If there exists a path between each pair of nodes,
then the undirected graph G is connected. In this article, we
consider a connected communication network.
The graph G merely characterized the network topology of
the information flow. In practice, due to channel constraints,
time delay, or other fault conditions, the information exchanges
over a communication network may be subject to transmission
errors and communication uncertainties. Besides, in order to
reduce communication consumption in a continuous system
by discretizing information transmissions, event-triggered or
quantized communication strategies in these hybrid systems
will generate certain errors. In these cases, each transmission
channel can be modeled as an ideal communication network
perturbed by additive deterministic uncertainties [35]. Hence,
the relative state information obtained from its neighbors will
contain interference through the uncertain channels. The ideal
communication graph can be denoted by an adjacency matrix
A, whose (i, j)th entry is defined as aij = 1 if (i, j) ∈ E
and aij = 0 otherwise. The perturbation associated with the
edge (i, j) is denoted by ωij , which satisfies the following
assumption.
Assumption 1: For any (i, j) ∈ E, each ωij subjects to a
given uncertainty radius ij > 0, that is, ωij  ≤ ij .
Since the edges are bidirectional in an undirected graph, it
is natural to assume ωij = ωji if aij = aji = 0. The
 Laplacian
matrix of G, denoted by L, is defined as Lii = j=i aij and
Lij = −aij , i = j. Assume that the undirected graph G contains
k communication edges and each edge has a head and a tail.
The incidence matrix D = [dij ] ∈ Rn×k is defined as dij = 1 if
i is the head of (i, j), dij = −1 if i is the tail of (i, j), and dij = 0
otherwise. The Laplacian matrix also can be represented as
L = DDT . In light of the fact that the sum of each column
of D is 0, then the Laplacian matrix L has a simple zero
eigenvalue and the corresponding left and right eigenvectors
are 1Tn and 1n , respectively, where 1n represents the column
vector of size n with all entries equal to 1 [1], [36].

II. P RELIMINARIES
A. Notations and Nomenclature
AT ∈ RM×N denotes the transpose of an N ×M-dimensional

matrix A. For a symmetric matrix A = AT ∈ RN×N , A > 0
(resp., A ≥ 0) means that A is a positive (resp., non-negative)
matrix.√IN ∈ RN×N denotes the N-dimensional identity matrix.
v = vT v denotes the Euclidean norm of an N-dimensional
vector v ∈ RN .
A function l(·) : RN → RN is Lipschitz with a constant
α > 0 over a compact set S ⊂ RN if l(p) − l(q) ≤ αp − q
for p, q ∈ S. A differentiable function f (·) : RN → R is
strictly convex over a convex set C ⊆ RN if (p − q)T (∇f (p) −
∇f (q)) > 0 ∀p, q ∈ C and p = q, where ∇f (·) represents the

III. P ROBLEM F ORMULATION
Consider a networked system consisting of n heterogeneous
agents and each of them can be described as an EL system
Mi (pi )p̈i + Ci (pi , ṗi )ṗi + Gi (pi ) = νi , i = 1, . . . , n

(1)

where pi ∈ RN is the position vector and its derivative
ṗi ∈ RN denotes the velocity vectors, Mi (pi ) ∈ RN×N is the
positive-definite inertial matrix, Ci (pi , ṗi )ṗi ∈ RN represents
the Coriolis and centripetal forces, Gi (pi ) ∈ RN is the gravity
force, and νi ∈ RN is the control input to be designed.
In this EL multiagent system, assign each agent a local cost
function fi (pi ) : RN → R so that the global cost function can

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WU AND LI: DISTRIBUTED ROBUST OPTIMIZATION ALGORITHMS OVER UNCERTAIN NETWORK GRAPHS


be defined as F(p) = ni=1 fi (p). These agents via cooperation
attempt to solve the following optimization problem:
min F(p).

(2)

p∈RN

Hence, we need to design a distributed algorithm such that
as t → ∞ the states of all agents approach to the global minimum point p∗ ∈ arg minp∈RN F(p), that is, limt→∞ pi (t) → p∗
and limt→∞ ṗi (t) → 0N for any initial condition (pi (0), ṗi (0)).
In this article, the cost functions are assumed to satisfy the
following properties.
Assumption 2: The global cost function F(p) is differentiable and ε-strictly convex over RN and each local cost
function fi (p) is differentiable and its gradient ∇fi (p) is
α-Lipschitz on RN .

4453

Lemma 1: Under Assumption 2, the equilibrium of the
network (5) is the optimal solution of the DOP (2) over an
undirected graph.
Proof: The equilibrium (μ̄, ϕ̄, )
¯ of the network (5) can be
derived as
0 = ϕ̄
0 = −γ ϕ̄ − ∇f (μ̄) − (L ⊗ IN )μ̄ − (L ⊗ IN )¯

(6a)
(6b)

0 = (L ⊗ IN )(μ̄ + ϕ̄).

(6c)

We next certify that the equilibrium is the unique solution
of the problem (2). By considering (6), it follows that the
equilibrium satisfies:
μ̄i = μ∗ , i =, 1, 2, . . . , n

(7)

n



 
aij ¯ i − ¯ j = −
∇fi μ∗ .

(8)

and


IV. M AIN R ESULTS
In this section, we analyze the DOP with uncertain channels
in two steps. First, we consider in Section IV-A the case where
an optimization algorithm is processed over an ideal communication network and pinpoint the convergence condition for
the closed-loop system. Next, we investigate the robustness
of the distributed optimization algorithm under an uncertain
network in Section IV-B.

j∈Ni

i=1

Since the network topology is undirected and connected,
according to the definition of L, we can verify the property
that 1Tn L = 0n . Due to this fact, we can obtain that
0 ≡ (1n ⊗ IN ) (L ⊗ IN )¯ =
T

n


 
∇fi μ∗ .

(9)

i=1

A. Initialization-Free Distributed Optimization Algorithm for
EL Networks
To avoid the initialization requirement in [22], we propose
a modified distributed coordination optimization algorithm
νi = Ci (pi , ṗi )ṗi + Gi (pi ) − γ Mi (pi )ṗi
 

− Mi (pi )∇fi (pi ) − Mi (pi )
aij pi − pj


− Mi (pi )
˙ i =



aij i − j

j∈Ni

(3a)

j∈Ni




 

aij pi − pj + ṗi − ṗj

(3b)

j∈Ni

where γ > 0 is the feedback gain to be determined.
Under the algorithm (3), the EL multiagent system (1) can be
rewritten as
 

p̈i = −γ ṗi − ∇fi (pi ) −
aij pi − pj
−





aij i − j

j∈Ni

(4a)

j∈Ni

˙ i =




 

aij pi − pj + ṗi − ṗj .

(4b)

j∈Ni

Let pi = μi and ṗi = ϕi . By using the aggregate vector a =
[aT1 , aT2 , . . . , aTn ]T , the networked system can be written in a
compact form as
μ̇ = ϕ
ϕ̇ = −γ ϕ − ∇f (μ) − (L ⊗ IN )μ − (L ⊗ IN )
˙ = (L ⊗ IN )(μ + ϕ).

(5)

To proceed further, the following lemma needs to be
certified first.

n ∇f (μ̄ ) = 0 that the optimality
Then, it follows from i=1
i i
∗
condition ∇F(μ ) = 0 along with the system (5) is satisfied.
In light of the strict convexity of F(·), we have the fact that
¯ is a global
μ̄ = 1n ⊗μ∗ is uniquely determined. This is, (μ̄, ϕ̄, )
minimizer of (2). Note that if (μ̄, ϕ̄, )
¯ is a solution of (2), so
is (μ̄, ϕ̄, ¯ + 1n ⊗ ρ), ρ ∈ RN . This completes the proof.
Based on the aforementioned equilibrium (μ̄, ϕ̄, )
¯ of (5),
we can define the tracking errors as μ̂i = μi − μ̄i , ϕ̂i = ϕi − ϕ̄i ,
and ˆ i = i − ¯ i . Besides, under the definition of the Laplacian
matrix L, it is well known that there exits a unitary matrix U
such that U T LU = = diag{0, λ2 , . . . , λn }, where 0 < λ2 ≤
λ3 ≤ · · · ≤ λn are the nonzero eigenvalues of L [36]. Then,
we can introduce a variable substitution: μ̃ = (U ⊗IN )μ̂, ϕ̃ =
ˆ
(U ⊗ IN )ϕ̂, and ˜ = (U ⊗ IN ).
Based on (5), we have
μ̃˙ = (U ⊗ IN )ϕ
ϕ̃˙ = −γ (U ⊗ IN )ϕ − (U ⊗ IN )∇f (μ)

− (U ⊗ IN )(L ⊗ IN )μ − (U ⊗ IN )(L ⊗ IN )
˙˜ = (U ⊗ IN )(L ⊗ IN )(μ + ϕ).
(10)
Since the orthogonal matrix U can be partitioned as U =
[U1 , U2:n ], where U1 is the first column of matrix U and U2:n is
a n×(n−1)-dimensional matrix composed by the left columns
of U, then we can obtain an aggregate vector ã = col(ã1 , ã2:n ).
Now, it is ready to prove the following theorem.
Theorem 1: Under an undirected and connected graph, the
distributed algorithm (3) can achieve the unique optimal
solution of the DOP (2) asymptotically, provided that
Assumption 2 holds and γ ≥ (α 2 /4ε) + α + (λn /4) + 2.
Proof: By constructing a new variable ζ = col(ϕ̃, μ̃, ˜ 2:n ),
we can establish a quadratic function
V1 = ζ T ( ⊗ IN )ζ

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(11)

4454

IEEE TRANSACTIONS ON CYBERNETICS, VOL. 52, NO. 6, JUNE 2022

⎡1

2 IN
where  = ⎣ 12 IN

1
2 IN
γ
2 IN

0
0

⎤

(13)

where γ ≥ β + (λn /4) + 2 and σ = (γ 2 /λ2 ) − γ + λn +
(1/4). By taking 0 <  < min{([4λ2 ε2 ]/[α 2 (α + 4ε)(1 +
λ2 )]), (1/σ )}, it follows from (20) that V̇3 ≤ −θ1 ζ 2 , where
θ1  min{γ − β − 1, 1 − σ , [(4ε2 )/(α + 4ε)] − α 2 ([1 +
λ2 ]/λ2 ), (λ2 /2)} > 0. Therefore, limt→∞ ζ (t) → 0, which
implies the conclusion.
Remark 1: In the proof of Lemma 1, we can note that the
initialization condition in [22] about the auxiliary variable 
is immune in this algorithm. This is, if the operating environment or network configurations is changed, the algorithm
in this article can avoid reinitialization. Compared with the
existing result, we do not have new requirements for controller parameters. Besides, it should be mentioned that the
structure of this algorithm facilitates the subsequent analysis
of the robust optimization problem in Section IV-B.

(14)

B. Distributed Robust Optimization Algorithm Under
Uncertain Networks

(15)

In this section, we consider the case where the EL systems
coordinate via the uncertain network discussed in Section II.
On the basis of (3), in this case, we propose the following
distributed robust optimization algorithm:

⎦. Evidently, V1 is positive

1
0
0
2 IN−1
definite. The derivative of (11) is given as follows:
 T

V̇1 = −(γ − 1)ϕ̃2 − μ̃T2:n U2:n
LU2:n ⊗ IN μ̃2:n
 T


T
T
U2:n LU2:n ⊗ IN μ̃2:n − ϕ̂ + μ̂ g (12)
− ϕ̃2:n

in which g = ∇f (μ) − ∇f (μ̄). Next, by using the well-known
Young’s inequality [37], the third item in (12) satisfies the
following inequality:
 T

 T

T
U2:n LU2:n ⊗ IN μ̃2:n ≤ μ̃T2:n U2:n
− ϕ̃2:n
LU2:n ⊗ IN μ̃2:n

1 T  T
U2:n LU2:n ⊗ IN ϕ̃2:n .
+ ϕ̃2:n
4

Under Assumption 2, it follows that:
μ̂T g ≥ εμ̂2 .
Since ∇fi (·) is α-Lipschitz, we have
α2

1
− ϕ̂ T g ≤ βϕ̂2 + 4β
g2 ≤ βϕ̂2 + 4β μ̂2

where β is a positive constant with β = (4αε + α 2 )/4ε.
Substituting (13)–(15) into (12), we can obtain that
4ε2
μ̃2
V̇1 ≤ −(γ − β − 1)ϕ̃2 −
α + 4ε

1 T  T
U2:n LU2:n ⊗ IN ϕ̃2:n
+ ϕ̃2:n
4
4ε2
μ̃2
≤ −(γ − β − 1)ϕ̃1 2 −
α + 4ε
λn
ϕ̃2:n 2 .
− γ −β −1−
4
Consider another positive-definite quadratic function
1
V2 = (ϕ̃2:n + ˜ 2:n )2 .
2
The derivative of (17) along (5) satisfies
 T

T
T
U2:n LU2:n ⊗ IN ϕ̃2:n
ϕ̃2:n + ϕ̃2:n
V̇2 = −γ ϕ̃2:n


T
T
T
− γ ϕ̃2:n
˜ 2:n − ˜ 2:n
LU2:n ⊗ IN ˜ 2:n
U2:n


T
− (ϕ̃2:n + ˜ 2:n )T U2:n
⊗ IN g
2
1
λ2
γ
ϕ̃2:n 2 − ˜ 2:n 2
− γ + λn +
≤
λ2
4
2
1
+
λ
2
μ̃2 .
+ α2
λ2

j∈Ni

− Mi (pi )
˙ i =

(17)








aij 1 + ωij i − j + Gi (pi ) (21a)

j∈Ni



 

aij 1 + ωij pi − pj + ṗi − ṗj .

(21b)

j∈Ni

(16)

Under the robust optimization algorithm (21), it follows
from (1) that:
 


aij 1 + ωij pi − pj
p̈i = −∇fi (pi ) − γ ṗi −
−



j∈Ni




aij 1 + ωij i − j

(22a)

j∈Ni

˙ i =





 

aij 1 + ωij pi − pj + ṗi − ṗj .

(22b)

j∈Ni

Using the same variables in Theorem 1, the compact form of
the networked system can be written as
(18)

Here, we use Young’s inequality multiple times to obtain the
inequality in (18).
Now, we can formulate a Lyapunov function as
V3 = V1 + V2

νi = Ci (pi , ṗi )ṗi − γ Mi (pi )ṗi − Mi (pi )∇fi (pi )
 


aij 1 + ωij pi − pj
− Mi (pi )

(19)

where  is a positive constant to be determined later. From (16)
and (18), it follows that:
V̇3 = V̇1 +  V̇2
≤ −(γ − β − 1)ϕ̃1 2 − (1 − σ )ϕ̃2:n 2
1 + λ2
4ε2
λ2
μ̃2 −  ˜ 2:n 2
− α 2
−
α + 4ε
λ2
2
(20)

μ̇ = ϕ
ϕ̇ = −γ ϕ − ∇f (μ)



− (L ⊗ IN )μ − (D ⊗ IN ) DT ⊗ IN μ


− (L ⊗ IN ) − (D ⊗ IN ) DT ⊗ IN 

˙ = (L ⊗ IN )(μ + ϕ)


+ (D ⊗ IN ) DT ⊗ IN (μ + ϕ)

(23)

∀(i, j) ∈ E. Since 1T D = 0, Lemma 1

where  = diag{ωij }
still holds here.
The following theorem designs the distributed optimization
algorithm with uncertain communications.
Theorem 2: Under an uncertain communication network
with an undirected and connected graph, the robust distributed algorithm (21) can solve the DOP (2) asymptotically,

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WU AND LI: DISTRIBUTED ROBUST OPTIMIZATION ALGORITHMS OVER UNCERTAIN NETWORK GRAPHS

if Assumption 2 holds, γ ≥ (α 2 /4ε)+α +([(1+ ω̄)λn ]/4)+2,
and the upper bound ω̄ of the communication uncertainties is
less than the unity.
Proof: Using both (23) and the unitary matrix U in
Theorem 1, it follows from (11) that:
 T

LU2:n ⊗ IN μ̃2:n
V̇1 = −(γ − 1)ϕ̃2 − μ̃T2:n U2:n
 T


T
T
U2:n LU2:n ⊗ IN μ̃2:n − ϕ̂ + μ̂ g
− ϕ̃2:n




T
− μ̃T2:n U2:n
D ⊗ IN  DT U2:n ⊗ IN μ̃2:n

 

T
T
− ϕ̃2:n
D ⊗ IN  DT U2:n ⊗ IN μ̃2:n .
(24)
U2:n
Applying Young’s inequality again, we can obtain that
 T
 

T
−ϕ̃2:n
U2:n D ⊗ IN  DT U2:n ⊗ IN μ̃2:n




T
≤ μ̃T2:n U2:n
D ⊗ IN  DT U2:n ⊗ IN μ̃2:n
 

1 T  T
U2:n D ⊗ IN  DT U2:n ⊗ IN ϕ̃2:n . (25)
+ ϕ̃2:n
4
Since there are k communication channels in the network,
then the block-diagonal matrix  can be ordered as  =
diag{ω1 , ω2 , . . . , ωk }. Let ψ = (DT U2:n ⊗ IN )ϕ̃2:n . It is not
difficult to check that
 T
 

T
ϕ̃2:n
U2:n D ⊗ IN  DT U2:n ⊗ IN ϕ̃2:n
= ψ T ψ ≤

k


ωs ψs2

s=1

≤ ω̄

k


 T

T
ψs2 = ω̄ϕ̃2:n
U2:n LU2:n ⊗ IN ϕ̃2:n

(26)

s=1

where the upper bound ω̄  max{ij |(i, j) ∈ E}.
Substituting (25) and (26) into (24) and recalling (16), it gives
4ε2
μ̃2
α + 4ε

1 + ω̄ T  T
+
ϕ̃2:n U2:n LU2:n ⊗ IN ϕ̃2:n
4
4ε2
μ̃2
≤ −(γ − β − 1)ϕ̃1 2 −
α + 4ε
(1 + ω̄)λn
ϕ̃2:n 2 .
− γ −β −1−
4

Substituting (27) and (28) into (19), we can obtain that
V̇3 = V̇1 +  V̇2
≤ −(γ − β − 1)ϕ̃1 2 − (1 − σ )ϕ̃2:n 2
4ε2
1
−
μ̃2
− α 2 1 +
α + 4ε
(1 − ω̄)λ2
(1 − ω̄)λ2
˜ 2:n 2
−
(29)
2
where γ ≥ β + ([(1 + ω̄)λn ]/4) + 2 and σ = (γ 2 /[(1 −
ω̄)λ2 ]) − γ + (1 + ω̄)λn + (1/4). Besides, to ensure that
the coefficient before each variable is negative, 1 − ω̄ should
greater than 0. This is, 1 > ω̄. Next, by choosing 0 <  <
min{([4(1 − ω̄)λ2 ε2 ]/[α 2 (α + 4ε)(1 + (1 − ω̄)λ2 )]), (1/σ )}
in (29), it follows that V̇3 ≤ −θ2 ζ 2 , where θ2  min{γ −
β −1, 1−σ, [(4ε2 )/(α+4ε)]−α 2 (1+[1/(1− ω̄)λ2 ]), ([(1−
ω̄)λ2 ]/2)} > 0. Hence, we can obtain that limt→∞ ζ (t) → 0.
This completes the proof.
Remark 2: In the presence of uncertain communication
channels, the feedback gain of the robust algorithm is not
only related to convexity constant ε and Lipschitz constant
α of the cost function but also to the radius ω̄ of the channel
uncertainties and the largest eigenvalue λn of the Laplacian
matrix. It can be clearly analyzed that the larger the bound
of the uncertainties, the greater the required feedback gain.
Moreover, it is highlighted that the bound of the uncertainties
cannot be unbounded, which should be less than the unity.
This result is consistent with the condition in [35] and indicates that the proposed distributed optimization protocol in
this article is robust for the uncertain communications with
bounded uncertainties. Since the result in [35] considers only
linear systems, there are different challenges in the proposed
optimization algorithm due to its nonlinear nature.

V̇1 ≤ −(γ − β − 1)ϕ̃2 −

V. E XTENSION

(27)

Correspondingly, the derivative of (17) along (23) is
 T

T
T
V̇2 = −γ ϕ̃2:n
U2:n LU2:n ⊗ IN ϕ̃2:n
ϕ̃2:n + ϕ̃2:n
 T

T
T
U2:n LU2:n ⊗ IN ˜ 2:n
− γ ϕ̃2:n
˜ 2:n − ˜ 2:n
 T
 

T
U2:n D ⊗ IN  DT U2:n ⊗ IN ϕ̃2:n
+ ϕ̃2:n
 T
 

T
U2:n D ⊗ IN  DT U2:n ⊗ IN ˜ 2:n
− ˜ 2:n


T
− (ϕ̃2:n + ˜ 2:n )T U2:n
⊗ IN g
≤

4455

γ2
1
ϕ̃2:n 2
− γ + (1 + ω̄)λn +
4
(1 − ω̄)λ2
(1 − ω̄)λ2
˜ 2:n 2
−
2
1
μ̃2 .
+ α2 1 +
(28)
(1 − ω̄)λ2

When we consider an optimization problem in the domain
of cooperative control, the node dynamics play a key role
in the design of a distributed controller. Nevertheless, from
the perspective of distributed computing, the crucial objective of the optimization problem is to find the global optimal
solution without paying attention to the dynamics of the
nodes. In these cases, there are various distributed optimization
algorithms presented for the networked system comprised
of first-order integrators [15]–[20]. Hence, it is necessary
to further analyze the robustness of such algorithms against
uncertain communications.
Here, we choose the two classic consensus-based algorithms
proposed in [19] and [20] for first-order integrator networks.
The first is the distributed optimization algorithm in [19]
 

aij pi − pj
ṗi = −∇fi (pi ) −
−



j∈Ni



aij i − j

(30a)

j∈Ni

˙ i =





aij pi − pj .

j∈Ni

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(30b)

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Under the same uncertain communication considered in the
previous sections, the algorithm (30) can be revised as
 


aij 1 + ωij pi − pj
ṗi = −∇fi (pi ) −
−



algorithm can be written as
ṗi = −γ1 ∇fi (pi ) − i
 


− γ2
aij 1 + ωij pi − pj

j∈Ni




aij 1 + ωij i − j

j∈Ni

(31a)

˙ i = γ1 γ2

j∈Ni

˙ i =






aij 1 + ωij pi − pj .

(31b)



˙ˆ = γ1 γ2 (L ⊗ IN )p̂ + γ1 γ2 (D ⊗ IN ) DT ⊗ IN p̂.

(37b)

(32b)

The equilibrium (p̄, )
¯ of the closed-loop system (37) remains
to be the optimal solution of the 
optimization problem (2) with
the initialization condition that N
i=1 i (0) = 0.
The robustness of the distributed optimization
algorithm
(36)
is
discussed
in
the
following
corollary.
Corollary 2: Under Assumption 2, given the convexity constant ε and Lipschitz constant α of the cost function, we can
choose an auxiliary parameter δ to satisfy δ + 1 > 4α and
design controller gains γ1 and γ2 to ensure that the following
inequality:

1 2
p̂ + ˆ 2 .
(33)
V4 =
2
Based on the description of (32), its derivative satisfies
V̇4 = p̂T p̂˙ + ˆ T ˙ˆ
= −p̂T g − p̂T (L ⊗ IN )p̂


− p̂T (D ⊗ IN ) DT ⊗ IN p̂
(34)

When ω̄ < 1, by using the property of convex functions
and the LaSalle invariance principle [38], we can prove that
Corollary 1 holds.
Remark 3: Note that there is no longer a specific requirement for the gain in the robust optimization algorithm (30),
which is different from the optimization algorithm (21) for
EL systems. It is worth mentioning that all edge weights must
remain positive in the presence of uncertainties. Hence, we
require that ω̄ < 1.
To reduce the communication burden, the following modified consensus-based optimization algorithm is proposed
in [20]:
 

aij pi − pj − i
(35a)
ṗi = −γ1 ∇fi (pi ) − γ2
˙ i = γ1 γ2

j∈Ni



aij pi − pj

(37a)

(32a)

where p̂  p − p̄ and ˆ   − ¯ denote the tracking error.
Similarly to Lemma 1, it is easy to see that the equilibrium (p̄, )
¯ of the closed-loop system (32) is also the optimal
solution of the optimization problem (2).
Now, we present a corollary about the robustness of the
distributed optimization algorithm (30).
Corollary 1: The distributed algorithm (30) with undirected
connected graphs can achieve the unique optimal solution of
the DOP (2) asymptotically, if Assumption 2 holds and ω̄ < 1.
Proof: For brevity, we omit some tedious details. First, a
Lyapunov function is assigned as



(36b)

The compact form of (36) is
p̂˙ = −γ1 g − ˆ − γ2 (L ⊗ IN )p̂


− γ2 (D ⊗ IN ) DT ⊗ IN p̂

Then, it follows from (31) that:

≤ −p̂T g − (1 − ω̄)p̂T (L ⊗ IN )p̂.




aij 1 + ωij pi − pj .

j∈Ni

j∈Ni

p̂˙ = −g − (L ⊗ IN )p̂ − (L ⊗ IN )ˆ


− (D ⊗ IN ) DT ⊗ IN p̂


− (D ⊗ IN ) DT ⊗ IN ˆ


˙ˆ = (L ⊗ IN )p̂ + (D ⊗ IN ) DT ⊗ IN p̂



(36a)

(35b)

j∈Ni

where γ1 , γ2 > 0 are controller gains to be designed. In
the presence of uncertain channels, the corresponding robust

σ = γ12 ((δ + 1) − 4α)ε
+ 9δγ1 γ2 λ2 (1 − ω̄) − 4γ12 (δ + 1)2 > 0

(38)

holds, when ω̄ < 1. Then, the robust optimization
algorithm (36) can solve the optimization problem (2)
over 
an undirected connected network with the initialization N
i=1 i (0) = 0.
Proof: Using the orthogonal transform matrix U for variables p̂ and ˆ again, we can construct a Lyapunov function as
follows:
γ1 (δ + 1) T
δγ1 T
p̃1 p̃1 +
p̃ p̃2:n
V5 =
18
2 2:n
1
+
(39)
(γ1 p̃2:n + ˜ 2:n )T (γ1 p̃2:n + ˜ 2:n ).
2γ1
The derivative of (39) satisfies
γ 2 (δ + 1) T
7 T
p̂ g − ˜ 2:n
V̇5 = − 1
˜ 2:n
9
16


T
− δγ1 γ2 p̃T2:n U2:n
LU2:n ⊗ IN p̃2:n


− δγ1 γ2 p̃T2:n (D ⊗ IN ) DT ⊗ IN p̃2:n
 2
4γ12
4γ 2  T
⊗ IN g
(δ + 1)2 p̃T2:n p̃2:n + 1  U2:n
9
9
2


 2γ1


3
2γ
1
T

U2:n ⊗ IN g + ˜ 2:n 
−
(δ + 1)p̃2:n +
 .
3
3
4
+

By using the ε-strongly convexity of f and the α-Lipschitzness
of ∇f , we can obtain that
γ 2 ((δ + 1) − 4α)ε T
7 T
p̃ p̃ − ˜ 2:n
˜ 2:n
V̇5 ≤ − 1
9
16

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WU AND LI: DISTRIBUTED ROBUST OPTIMIZATION ALGORITHMS OVER UNCERTAIN NETWORK GRAPHS

4457

4γ 2
− δγ1 γ2 λ2 (1 − ω̄)p̃T2:n p̃2:n + 1 (δ + 1)2 p̃T2:n p̃2:n
9
2


 2γ1
 T

3
2γ
1

U2:n ⊗ IN g + ˜ 2:n 
−
(δ + 1)p̃2:n +
 .
3
3
4
In light of δ + 1 > 4α and the condition (38), we have the
inequality


7 σ
V̇5 ≤ −min
,
ζ̃ 2
16 9
where ζ̃ = col(p̃, ˜ 2:n ). Now, we can apply [38, Th. 4.10] to
complete the proof.
Remark 4: It was shown in [20] that in order to let (35)
solve the optimization problem, it demands that δ̂+1 > 4α̂ and
σ̂ = γ̂12 ((δ̂+1)−4α̂)ε̂+9δ̂ γ̂1 γ̂2 λ̂2 −4γ̂12 (δ̂+1)2 > 0. Note there
always exist γ̂1 and γ̂2 satisfying the condition, for example,
any γ̂2 > 4γ̂1 (δ̂ +1)2 /(9δ̂ λ̂2 ). As an extension, here, we derive
a corresponding result where the selection of γ1 and γ2 is also
related to the upper bound ω̄ of the radius of the uncertainties.
Since the uncertainties occur on the communication network,
the upper bound ω̄ affects the term related to the λ2 in the
expression of σ . Therefore, when ω̄ < 1, we always can find
the controller gains γ1 and γ2 to satisfy the inequality γ2 >
4γ1 (δ + 1)2 /(9δλ2 (1 − ω̄)).

Fig. 1.

Communication graph.

Fig. 2.

Trajectories of (pi,x , pi,y ) over an ideal communication network.

VI. S IMULATIONS
To illustrate the effectiveness of the distributed robust
optimization algorithm (21), we construct a multiagent
network consisted of EL systems, where each node’s dynamics
can be described as the model in [22]
 
  
Mi,11 Mi,12 p̈i,x
νi,x
=
νi,y
Mi,21 Mi,22 p̈i,y
  


G
Ci,11 Ci,12 ṗi,x
+ i,11
(40)
+
Ci,21 Ci,22 ṗi,y
Gi,21
where Mi,11 = ϑ1i + 2ϑ2i cos pi,y , Mi,12 = Mi,21 = ϑ3i +
ϑ2i cos pi,x , Mi,22 = ϑ3i , Ci,11 = −2ϑ2i sin pi,y ṗi,y , Ci,12 =
Ci,21 = −ϑ2i sin pi,x ṗi,x , Ci,22 = 0, and Gi,11 = Gi,21 = 0.
Let ϑ1i = 2.613, ϑ2i = 0.174, and ϑ3i = 0.389. The network
topology among these agents is depicted in Fig. 1.
Consider a global optimization problem, where each agent
only can access its local cost function. Here, the cost function
for each agent i is assigned as


f1 pi,x , pi,y = p2i,x + 0.16p4i,y



4

2
f2 pi,x , pi,y = 0.1 pi,x − 1 + 1.5 pi,y − 2


f3 pi,x , pi,y = 0.36p2i,x ln 1 + p2i,x + 0.12p4i,y + p2i,y



f4 pi,x , pi,y = 0.5p2i,x + p2i,y / p2i,y + 1


f5 pi,x , pi,y = 0.4p2i,x / ln 1 + p2i,x + 0.6p2i,y / ln 1 + p2i,y .
The initial coordinates (pi,x , pi,y ) and their derivatives are
randomly generated in [−5, 5], and there are no special initialization conditions for the auxiliary variables . The feedback
gain is selected as γ = 2.5 and the simulation time is t = 50 s.
First, we apply the distributed optimization algorithm (3)
over an ideal communication network. The trajectories of each

Fig. 3. Trajectories of (pi,x , pi,y ) over an uncertain communication network.

agent are shown in Fig. 2. In the partially enlarged view,
we can see that the communication information without any
interference. This implies that the ideal initialization-free distributed algorithm can solve the optimization problem under
the conditions provided in Theorem 1.
Next, we apply the robust distributed optimization algorithm (21) over an uncertain communication network. The
communication uncertainty for each edge (i, j) is described
as ωij = (i + j/20) sin([i + j/10]t + [i + j/5]π ). The trajectories
of each agent are shown in Fig. 3. In the partially enlarged
view, we can see that the communication information subject
to some interference. This implies that the robust algorithm can
solve the optimization problem under the conditions provided
in Theorem 2. This result confirms our theoretical analysis.
VII. C ONCLUSION
In this article, the robust DOPs have been systematically
researched from the perspective of an uncertain communication network. The model of the distributed optimization
protocol with unreliable transmission channels has been established for multiple EL systems. An initialization-free distributed robust optimization algorithm has been designed and

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a sufficient condition for the selection of the algorithm’s gain
has been presented. The robust optimization problem of firstorder integrator networks with uncertain communications has
been accordingly revisited.
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Zizhen Wu received the B.S. degree in automatic
control from the Beijing University of Chemical
Technology, Beijing, China, in 2016. He is currently
pursuing the Ph.D. degree with the Department
of Mechanics and Engineering Science, College of
Engineering, Peking University, Beijing.
His research interests include the distributed
optimization, adaptive control, and robust control.

Zhongkui Li (Member, IEEE) received the B.S.
degree in space engineering from the National
University of Defense Technology, Changsha, China,
in 2005, and the Ph.D. degree in dynamics and control from Peking University, Beijing, China, in 2010.
Since 2013, he has been with the Department
of Mechanics and Engineering Science, College
of Engineering, Peking University, where he is
currently a Tenured Associate Professor. He has
authored the book entitled Cooperative Control of
Multi-Agent Systems: A Consensus Region Approach
(CRC Press, 2014) and has published a number of journal papers. His current research interests include cooperative control of multiagent systems,
networked control systems, and control of autonomous unmanned systems.
Dr. Li was a recipient of the State Natural Science Award of China (Second
Prize) in 2015, the Yang Jiachi Science and Technology Award in 2015, and
the National Excellent Doctoral Thesis Award of China in 2012. His coauthored papers received the IET Control Theory and Applications Premium
Award in 2013, and the Best Paper Award of Journal of Systems Science and
Complexity in 2012. He was selected into the Changjiang Scholars Program
(Young Scholar), Ministry of Education of China, in 2017. He serves as
an Associate Editor for IEEE T RANSACTIONS ON AUTOMATIC C ONTROL,
Nonlinear Analysis: Hybrid Systems, and the Conference Editorial Board of
IEEE Control Systems Society.

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