IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 62, NO. 5, MAY 2017

2239

Distributed Continuous-Time Optimization:
Nonuniform Gradient Gains, Finite-Time
Convergence, and Convex Constraint Set
Peng Lin, Wei Ren, Fellow, IEEE, and Jay A. Farrell, Fellow, IEEE

Abstract—In this paper, a distributed optimization problem with general differentiable convex objective functions
is studied for continuous-time multi-agent systems with
single-integrator dynamics. The objective is for multiple
agents to cooperatively optimize a team objective function
formed by a sum of local objective functions with only local
interaction and information while explicitly taking into account nonuniform gradient gains, finite-time convergence,
and a common convex constraint set. First, a distributed
nonsmooth algorithm is introduced for a special class of
convex objective functions that have a quadratic-like form.
It is shown that all agents reach a consensus in finite time
while minimizing the team objective function asymptotically. Second, a distributed algorithm is presented for general differentiable convex objective functions, in which the
interaction gains of each agent can be self-adjusted based
on local states. A corresponding condition is then given
to guarantee that all agents reach a consensus in finite
time while minimizing the team objective function asymptotically. Third, a distributed optimization algorithm with statedependent gradient gains is given for general differentiable
convex objective functions. It is shown that the distributed
continuous-time optimization problem can be solved even
though the gradient gains are not identical. Fourth, a distributed tracking algorithm combined with a distributed estimation algorithm is given for general differentiable convex objective functions. It is shown that all agents reach a
consensus while minimizing the team objective function in
finite time. Fifth, as an extension of the previous results, a
distributed constrained optimization algorithm with nonuniform gradient gains and a distributed constrained finite-time
optimization algorithm are given. It is shown that both algorithms can be used to solve a distributed continuous-time
optimization problem with a common convex constraint set.
Numerical examples are included to illustrate the obtained
theoretical results.
Manuscript received March 4, 2016; revised June 24, 2016; accepted
August 14, 2016. Date of publication August 30, 2016; date of current
version April 24, 2017. This work was supported by the National Science
Foundation under Grant ECCS-1611423, the National Natural Science
Foundation of China under Grants 61203080, 61573082, 61528301, and
the State Key Laboratory of Intelligent Control and Decision of Complex
Systems of Beijing Institute of Technology. Recommended by Associate
Editor Prof. Z. Sun.
Peng Lin is with the School of Information Science and Engineering, Central South University, Changsha 410083, China (e-mail:
lin_peng0103@sohu.com).
Wei Ren and Jay A. Farrell are with the Department of Electrical and
Computer Engineering, University of California, Riverside, CA 92521,
USA (e-mail: ren@ee.ucr.edu; jay.farrell@ucr.edu).
Color versions of one or more of the figures in this paper are available
online at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TAC.2016.2604324

Index Terms—Consensus, convex set constraint, distributed optimization, finite-time convergence, multi-agent
systems, nonuniform gradient gains.

I. INTRODUCTION
ISTRIBUTED optimization problems for multi-agent systems have received significant attention in the control
community [1]–[16]. The objective is to solve an optimization problem cooperatively in a distributed manner where the
team objective function is composed of a sum of local objective
functions, each of which is known to only one agent. Earlier
work about distributed optimization problems mostly concentrated on discrete-time algorithms. For example, article [1] gave
a discrete-time projection algorithm where each agent is required to lie in a closed convex set and showed that all agents
reach an optimal point in the intersection of all the convex
sets even when the communication topologies are dynamically
changing as long as the edge weight matrix is doubly stochastic. Moreover, articles [4]–[8] addressed distributed optimization problems with inequality-equality constraints or using other
discrete-time algorithms and derived conditions to ensure that
all agents converge to the optimal point or its neighborhood.
Recently, some researchers turned their attention to distributed
optimization problems for multi-agent systems with continuoustime dynamics. For example, article [9] proposed a continuoustime zero-gradient-sum algorithm. Article [10] and its extension
[16] studied a continuous-time version of the work in [1]. Article [11] studied the continuous-time distributed optimization
problem for undirected graphs and derived explicit expressions
for a lower bound on the algorithm’s convergence rate. Article
[12] proposed a novel distributed continuous-time algorithm for
distributed convex optimization by introducing a dynamic integrator. Founded on the work of [12], articles [13], [14] studied
the distributed continuous-time optimization problem for general strongly connected balanced directed graphs and gave the
estimate of the convergence rate of the algorithm.
Although excellent work has been presented in [1]–[16] to
solve the distributed optimization problem, there are still issues that need to be addressed, in particular when nonuniform
gradient or subgradient gains and finite-time convergence are
taken into account. For example, in [5], a distributed optimization problem with nonuniform subgradients was studied from a
view point of stochastic theory, but by taking the mathematical
expectation, it can be seen that the given algorithm uses uniform

D

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subgradient gains in nature. In reality, it is a difficult task to keep
the gradient or subgradient gains uniform for different agents
all the time, in particular when the number of agents is huge. It
is important and necessary to design algorithms for distributed
optimization with nonuniform gradient gains. However, due to
coexistence of the nonuniformity of the gradient gains and the
nonlinearity of the gradients of the local objective functions, the
convergence rates of the local objective functions are no longer
uniform for the distributed optimization problem with nonuniform gradient gains and hence the existing approaches cannot be
applied directly. Besides this aspect, most of the existing works
on the distributed optimization problem (e.g., [1]–[16]), studied
only the asymptotical stability of the algorithm and rare results
are concerned about the finite-time convergence of the algorithms. Due to the existence and the nonlinearity of the objective
functions, the existing approaches for the distributed finite-time
consensus problem (e.g., [17], [18]) cannot be extended directly
to the distributed finite-time optimization problems. Though
some results have been obtained in our previous works in [19],
[20] for the distributed finite-time optimization problem, they
are limited to a special class of convex objective functions that
have a quadratic-like form and the approaches cannot be applied
to more general convex objective functions. It is meaningful
and challenging to study the distributed finite-time optimization
problem for more general convex objective functions.
In this paper, we are interested in solving the distributed
optimization problem with general differentiable convex objective functions for continuous-time multi-agent systems with the
consideration of nonuniform gradient gains, finite-time convergence, and a common convex constraint set. First, a distributed
nonsmooth algorithm is introduced for a special class of convex
objective functions that have a quadratic-like form. It is shown
that all agents reach a consensus in finite time while minimizing
the team objective function asymptotically. Second, an adaptive
distributed algorithm is presented where the interaction gains of
each agent can be self-adjusted based on local states. It is shown
that the distributed continuous-time optimization problem can
be solved when general differentiable convex local objective
functions are taken into account. Third, to relax the synchronization requirement of the system on the gains of the gradients,
a distributed algorithm with state-dependent gradient gains is
given. It is shown that the optimization problem can be solved
even though the gradient gains are not identical. After that, a
distributed tracking algorithm combined with a distributed estimation algorithm is given. It is shown that all agents reach
a consensus while minimizing the team objective function in
finite time. Finally, as an extension of the previous results, a distributed constrained optimization algorithms with nonuniform
gradient gains and a distributed constrained finite-time optimization algorithm are given. It is shown that both algorithms
can be used to solve a distributed continuous-time optimization
problem with a common convex constraint set.

set {1, . . . , n}; si denotes the ith component of the vector s;
Aij denotes the ijth entry of the matrix A; sT and AT denote,
respectively, the transpose of the vector s and the matrix A;
d
and ∂∂s
||s|| denotes the Euclidean norm of the vector s; ds
denote, respectively, the differential operator and partial differential operator with respect to s; ∇f (s) denotes the gradient of
; the matrix ∇2 f (s)
the function f (s) at s with [∇f (s)]i = ∂∂f s(s)
i
denotes the Hessian or second-order partial derivative matrix of
2
the function f (s) at s with [∇2 f (s)]ij = ∂∂ s if∂(s)
s j ; sgn(s) denotes
a component-wise sign function of s; the symbol / denotes the
division sign; Y − X denotes the relative complement set of
X in Y for any two sets X and Y ; and PX (s) denotes the
projection of the vector s onto the closed convex set X, i.e.,
PX (s) = arg mins̄∈X s − s̄.
Let G(V, E, A) be a graph of order n, where V = {1, . . . , n} is
the set of nodes, E ⊆ V × V is the set of edges, and A = [aij ] ∈
Rn ×n is the weighted adjacency matrix. An edge of (i, j) ∈ E
denotes that agent i can obtain information from agent j. The
weighted adjacency matrix A is defined as aii = 0 and aij = 1
if (i, j) ∈ E and aij = 0 otherwise. The graph G is undirected if
aij = aj i for all i, j. The set of neighbors of node i is denoted
by Ni = {j ∈ V : (i, j) ∈ E}. A path is a sequence of edges
of the form (i1 , i2 ), (i2 , i3 ), . . ., where ij ∈ V. The graph G is
connected, if there is a path from every node to every other node.
Lemma 1: [22] Let f0 (s) : Rm → R be a differentiable convex function. f0 (s) is minimized if and only if ∇f0 (s) = 0.
Lemma 2: [23] Suppose that Y = ∅ is a closed convex set
in Rm . The following two statements hold:
(a) For any y ∈ Rm , y − PY (y) is continuous with respect to y and ∇ 12 y − PY (y)2 = y − PY (y).
(b) For any y, z ∈ Rm and all ξ ∈ Y, [y − PY (y)]T (y −
ξ) ≥ 0, PY (y) − ξ2 ≤ y − ξ2 − PY (y) − y2
and PY (y) − PY (z) ≤ y − z.

II. NOTATION AND PRELIMINARIES
We adopt the following notation throughout this paper: Rm
denotes the set of all m dimensional real column vectors; Rm ×n
denotes the set of all m × n real matrices; I denotes the index

III. DISTRIBUTED CONTINUOUS-TIME OPTIMIZATION
WITHOUT CONSTRAINTS
Consider a multi-agent system consisting of n agents. Each
agent is regarded as a node in an undirected graph G(t), and
each agent can interact with only its neighbors. Suppose that the
agents satisfy the continuous-time dynamics
ẋi (t) = ui (t),

i ∈ I,

(1)

where xi (t) ∈ R is the state of agent i, and ui (t) ∈ R is the
control input of agent i. Our objective is to design ui (t) using
only local interaction and information, such that all agents cooperatively find the optimal state s∗ that solves the optimization
problem

minimize ni=1 fi (s) subject to s ∈ Rm ,
m

m

where fi (s) : Rm → R denotes the local objective function of
agent i, which 
is convex, differentiable, and known only to
agent i. Clearly, ni=1 fi (s) is also a differentiable function. The
problem described above is equivalent to the problem that all
agents reach
 a consensus while minimizing the team objective
function ni=1 fi (xi ), i.e.,

minimize ni=1 fi (xi ) subject to xi = xj ∈ Rm .
(2)

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LIN et al.: DISTRIBUTED CONTINUOUS-TIME OPTIMIZATION: NONUNIFORM GRADIENT GAINS, FINITE-TIME CONVERGENCE, AND CONVEX

 

Assumption 1: Each set Xi  s∇fi (s) = 0 is nonempty
and bounded.
To illustrate Assumption 1, we consider the convex function
fi (s) : R2 → R:

0,
if s ≤ 1,
fi (s) =
0.5(s − 1)2 , otherwise.
By simple calculations, we have

[0, 0]T , if s ≤ 1,
∇f (s) =
s
= [0, 0]T , otherwise.
(s − 1) s
Clearly, Xi = {s|s ≤ 1} and hence fi (s) satisfies
Assumption 1.
In Assumption 1, we only make assumptions
on each fi (s)
n
f
(s)
because the
rather than the team objective
function
i
i=1

team objective function ni=1 fi (s) is a global information for
all agents and cannot
 be used by each agent in a distributed way.
Define X  s ni=1 ∇fi (s) = 0 . From
 Lemma 1, all Xi
and X are the minimum sets of fi (s) and ni=1 fi (s) for all i.
Lemma 3: Let f (s) : Ξ → R be a differentiable convex
function and Y be its minimum set in Ξ, where Ξ ⊆ Rn is
a closed convex set. Suppose that Y is bounded and closed. For
any z = λPY (y) + (1 − λ)y with λ ∈ (0, 1),
Y (y )
T y −P Y (y )
0 < ∇f (z)T yy −P
−P Y (y ) ≤ ∇f (y) y −P Y (y )

for any y ∈ Ξ − Y .
Proof: Let
z = λPY (y) + (1 − λ)y
with λ ∈ (0, 1) for any y ∈ Ξ − Y . Clearly, z ∈ Ξ − Y . From
the convexity of the function f (s), we have f (y) − f (z) ≥
∇f (z)T (y − z) and f (z) − f (y) ≥ ∇f (y)T (z − y). Thus,

T
0 = f (y) − f (z) + f (z) − f (y) ≥ ∇f (z) − ∇f (y) (y −
z). Note that y = z, y = PY (y) and z = PY (y). Hence
y −P Y (y )
z −P Y (y )
y −z
Y (y )
and ∇f (y)T yy −P
y −z  = y −P Y (y ) = z −P Y (y ) ,
−P Y (y )

T y −z
T y −P Y (y )
= ∇f (y)T yy −z
−z  ≥ ∇f (z) y −z  = ∇f (z) y −P Y (y ) , i.e.,

−P Y (y )
Y (y )
≥ ∇f (z)T yy −P
∇f (y)T yy −P
−P Y (y ) . Furthermore, since
Y (y )
PY (y) ∈ Y and Y is the minimum set, from the convexity of
the function f (s), we have 0 > f (PY (y)) − f (z) ≥ ∇f (z)T
Y (y )
(PY (y) − z). That is, ∇f (z)T zz −P
−P Y (y ) > 0. Recalling that
y −P Y (y )
z −P Y (y )
T y −P Y (y )
y −P Y (y ) = z −P Y (y ) , we have ∇f (z) y −P Y (y ) > 0.


Lemma 4: Under Assumption 1, the following two statements hold:
(1) limy →+∞ f
i (y) = +∞ for all i and accordingly
limy →+∞ ni=1 fi (y) = +∞.
(2) All Xi and X are nonempty closed bounded convex sets
for all i.

Proof: From the convexity of the functions fi (s), ni=1 fi (s)
is convex and hence all Xi and X are also convex. Under
Assumption 1, each Xi is nonempty and bounded. Now, we
prove that all Xi are closed sets. If this is not true, there exists an integer ie such that Xi e is an open set. Then there
/ Xi e and a vector sequence {s̃k ∈
must exist a vector se ∈
Xi e } such that se = limk →+∞ s̃k . Clearly, fi e (s̃k ) = ρi e for
all k, where ρi e denotes the minimum value of the function
fi e (s). From the continuity of the function fi e (s), we have

2241

fi e (se ) = limk →+∞ fi e (s̃k ) = ρi e . This implies that se ∈ Xi e ,
which yields a contradiction. Therefore, all Xi are closed sets.
Let Yi be a closed bounded convex set such that
Xi ⊂ Yi and miny ∈/ Y i y − PX i (y) > 0. Clearly, maxy ∈Y i
y − PX i (y) < 0 for some positive constant 0 > 0.
From the property of a continuous function on a closed
bounded set and Lemma 3, we have 1  miny ∈∂¯Y i
n

T y −P X i (y )
and 2  maxy ∈∂¯Y i ni=1
i=1 ∇fi (y) y −P X (y ) > 0
i

y −P (y )
¯ i denotes the boundary
∇fi (y)T y −P XX i (y ) > 0, where ∂Y
i
of the set Yi . Integrating ∇fi (s) along the line from PY i (y)
to y, from Lemma 3, we have fi (y) − fi (PX i (y)) =
y −P X i (y )
y
T
∇fi (PX i (y) +
P X (y ) ∇fi (s) ds = 0
i

y −P X i (y )
T y −P X i (y )
y −P X i (y ) s) y −P X i (y ) ds ≥ 1 y − PY i (y) − 2 0 .

It follows that limy →+∞ fi (y) = +∞. Thus,
n

lim

y →+∞

fi (y) = +∞.

(3)

i=1

On the other hand, since each fi (y) is lower bounded,

n
i=1 fi (y) is lower bounded and hence its infimum exists, denoted by ω2 . From (3), for any sufficiently large
 constant ω3 >
ω2 , there exists a constant hl > 0 such that ni=1 fi (y) > ω3
for any y > hl . Let Ỹ = {y ≤ hl }. It is clear that if X is
nonempty, X ⊂ Ỹ . Note that Ỹ is a closed bounded set. Since

n
i=1 fi (y) is continuous with respect to y, from the property
of a continuous function
on a closed bounded set, we have the

minimum set of ni=1 fi (y) in Ỹ is nonempty. That is, X is
nonempty. Then by using the same analysis approach as for Xi ,
it can be proved that X is bounded and closed.

Assumption 2: The length of the time interval between any
two contiguous switching times is no smaller than a given constant, denoted by dw .
Arbitrary switching of the graph G(t) might lead to the Zeno
behavior. Hence Assumption 2 is imposed to prevent the system
from exhibiting the Zeno behavior. Throughout this paper, our
analysis is founded on Assumption 2. For simplicity, this will
not be repeatedly mentioned except when it is necessary.
A. Distributed Gradient Optimization
In this subsection, we design ui (t) for (1) to solve the convex optimization problem (2). In particular, all agents are driven
to reach a consensus in finite time while minimizing the team
objective function as t → +∞. We propose the following algorithm
sgn xj (t) − xi (t) − γ∇fi (xi (t)),

ui (t) = α

(4)

j ∈N i (t)

where α > 0
and γ > 0 are two constants. In (4), the role of the
first term, α j ∈N i (t) sgn xj (t) − xi (t) , is to drive all agents
to reach a consensus, while the second term, −γ∇fi (xi (t)),
is a weighted negative gradient of fi (xi (t)) playing a role in
minimizing fi (xi (t)).
Remark 1: As our algorithms discussed in this paper contain
sign functions that is piecewise differentiable, the solution of the
system (1) would be considered in the sense of Filippov [24].

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2242

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Assumption 3: Let ∇fi (s) = σs + φi (s), where σ ≥ 0 and
φi (s) < g for a certain positive number g and all s ∈ Rm .
In [2], the subgradients of the local objective functions were
assumed to be bounded and the most common quadratic convex
functions were not considered. Under Assumption 3, when σ =
0, the gradient of each local objective function is bounded, and
when σ > 0, the gradient of each local objective function contains a linear term and a bounded term, which includes the scenarios of [2] and the quadratic convex functions as special cases.
Proposition 1: Suppose that the graph G(t) is undirected and
connected for all t and Assumptions 2 and 3 hold. For system (1)
with algorithm (4), if α/γ > 2ng, all agents reach a consensus
in finite time. That is, there exists a positive number T such that
xj (t) = xi (t) for all t > T and all i, j ∈ I.
Proof: Consider the Lyapunov function candidate
n

V (t) =

n

1
1
2
xi (t) −
xj (t) .
2 i=1
n j =1

(5)

It is clear that when V (t) = 0, xi (t) = xj (t) for all i, j. Calculating V̇ (t) along the solutions of system (1) with (4), we have

n

=

i=1

[xi (t) −


× α

j ∈N i (t)

1
n

n
k =1

xk (t)]T

sgn(xj (t) − xi (t))


n
1
ẋj (t)
− γ∇fi (xi (t)) −
j =1
n
T

n
n
1
=
xk (t)
xi (t) −
i=1
k =1
n

× α
sgn(xj (t) − xi (t))
j ∈N i (t)

− [γ∇fi (xi (t)) − γσxi (t) + γσxi (t)]

n
γσ
+
xj (t)
(6)
j =1
n
n
where
equality holds 
because
i=1 [xi (t) −
n the second

n
n
1
1
1
T
x
(t)]
ẋ
(t)
=
0
×
ẋ
(t)
=
0
and
k =1 k
n
n j =1 j

n n j =1 j
n
n
1
1
T 1
[xi (t) − n k =1 xk (t)] n j =1 xj (t) = 0 × n
i=1
n
j =1 xj (t) = 0. Since the graph G(t) is undirected, then
(i, j) ∈ E if and only if (j, i) ∈ E. Thus,


i=1

1
xi (t) −
n

×α
=

× sgn(xj (t) − xi (t)).

n
i=1

j ∈N i (t)

n
k =1

T
xk (t)

sgn(xj (t) − xi (t))


1
α xi (t) −
j ∈N i (t)
n

× sgn(xj (t) − xi (t))

n
α
=
i=1
j ∈N i (t)
2

n
k =1

T
xk (t)

(7)

Under Assumption 3, ∇fi (xi (t)) = σxi (t) + φi (xi (t)), σ >
0 and φi (xi (t)) < g. Since γ > 0, it follows from (6) and (7)
that
V̇ (t)
α

n

=

V̇ (t)

n

T

n
1
xk (t) sgn(xj (t) − xi (t))
× xi (t) −
k =1
n

T

n
α
1
+
xk (t) sgn(xi (t) − xj (t))
xj (t) −
k =1
2
n
 
n
n
α
1
=
xk (t)
xi (t) −
i=1
k =1
j ∈N i (t)
2
n
T
n
1
xk (t) sgn(xj (t) − xi (t))
− xj (t) +
k =1
n
n
α
[xi (t) − xj (t)]T
=
i=1
j ∈N i (t) 2

j ∈N i (t) 2

i=1

[xi (t) − xj (t)]T

× sgn(xj (t) − xi (t))

n
1
−
xi (t) −
i=1
n
n

− γσ
≤

i=1
n
i=1

n
j =1

T
xj (t) γφi (xi (t))

n
1
2
xj (t)
j =1
n
α
[xi (t) − xj (t)]T
j ∈N i (t) 2

xi (t) −

× sgn(xj (t) − xi (t))
+

n
i=1

xi (t) −

1
n

n
j =1

xj (t)γg.

(8)


Consider the quantity xi (t) − n1 nj=1 xj (t) for all i ∈ I. Let
i0 , j0 ∈ I be the integers such that xi 0 (t) − xj 0 (t) =
maxi,j ∈I x
i (t) − xj (t) at time t. It is clear that
xi (t) − n1 nj=1 xj (t) ≤ n1 nj=1 xi (t) − xj (t) ≤
xi 0 (t) − xj 0 (t). Since G(t) is connected, there must
exist a path (i0 , i0 ), (i0 , i1 ), . . . , (ih−1 , ih ), (ih , j0 ) that connects nodes i0 and j0 . Note that xi 0 (t) − xj 0 (t) ≤ xi 0 (t) −
h
x
+ k =1 xi k −1 (t) − xi k (t) + xi h (t) − xj 0 (t) ≤
i 0n(t)
x (t) − xj (t).
Therefore,
xi 0 (t) −
j ∈N
i=1
i (t) i
xj 0 (t) ≤ ni=1 j ∈N i (t) xi (t) − xj (t). Also, note that
xi (t) − xj (t) ≤ −[xi (t) − xj (t)]T sgn(xj (t) − xi (t)) for
all i, j ∈ I from the relations of operator norms. It follows that
xi (t) −

1
n

n
j =1

xj (t)

≤ xi 0 (t) − xj 0 (t)
≤ −

n
i=1

j ∈N i (t)

[xi (t) − xj (t)]T

× sgn(xj (t) − xi (t)).

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(9)

LIN et al.: DISTRIBUTED CONTINUOUS-TIME OPTIMIZATION: NONUNIFORM GRADIENT GAINS, FINITE-TIME CONVERGENCE, AND CONVEX

If α/γ > 2ng, it follows from (8) and (9) that
n

V̇ (t) ≤

[xi (t) − xj (t)]T

α
− γng ≤ 0.
× sgn(xj (t) − xi (t))
2
This implies that V (t) and hence xi 0 (t) − xj 0 (t) are bounded.
When
0, i.e., xi 0 (t) − xj 0 (t) = 0, noting from (9)
 V (t) = 
that V (t) ≤ n2 xi 0 (t) − xj 0 (t), it follows that
n 
T
V̇ (t)
i=1
j ∈N i (t) [xi (t) − xj (t)] sgn(xj (t) − xi (t))

≤
√
√1
nxi 0 (t) − xj 0 (t)
V (t)
2

α
√
√
− γng ≤ −(α/ 2n − 2nγg) < 0.
×
2
Integrating both sides of this inequality, we have


√
√
2 V (t) − 2 V (0) < −(α/ 2n − 2nγg)t.
(10)
i=1

j ∈N i (t)

It is clear that V (t) converges to zero in finite time. Namely, all
agents reach a consensus in finite time.

Theorem 1: Suppose that the graph G(t) is undirected and
connected for all t and Assumptions 1, 2 and 3 hold. For system
(1) with algorithm (4), if α/γ > 2ng, all agents reach a consensus in finite time and minimize the team objective function (2)
as t → +∞.
Proof: Define
n

x∗ (t) 

1
xj (t).
n j =1

(11)

Under Assumption 3, Proposition 1 holds. From Proposition 1,
there exists a positive number T such that xi (t) = x∗ (t) for
all t > T and all i ∈ I. Since the graph G(t) is undirected, it
follows that for all t > T,
1
n
1
=
n

ẋ∗ (t) =

n
i=1
n
i=1

ẋi (t)

α

j ∈N i (t)

sgn(xj (t) − xi (t))


γ
− γ∇fi (xi (t)) = −
n

n
i=1

∇fi (x∗ (t)). (12)

Consider the Lyapunov function candidate V (t) = 12 x∗ (t) −
of
PX (x∗ (t))2 for t > T . Calculating V̇ (t) along the solutions

(12), it follows from Lemma 2 and the convexity of ni=1 fi (s)
that
V̇ (t) = [x∗ (t) − PX (x∗ (t))]T ẋ∗ (t)
n
γ
= − [x∗ (t) − PX (x∗ (t))]T
∇fi (x∗ (t))
i=1
n

n
γ n
fi (x∗ (t)) −
fi (PX (x∗ (t))) (13)
≤ −
i=1
i=1
n
m
for t > T . Let Y = {s ∈ R | s −
Pn X (s) ≤ l1 } for some
constant l1 > 0 and ρ = mins∈∂¯Y
i=1 [fi (s) − fi (PX (s))],
¯ denotes the boundary of Y . Since PX (s) ∈ X,
where ∂Y
from the definition
of X, we have ρ > 0. Moreover, from

/ Y . Thus,
Lemma 3, ni=1 [fi (s) − fi (PX (s))] > ρ for any s ∈
V̇ (t) ≤ − nγ ρ for any x∗ (t) ∈
/ Y and t > T . This implies that

2243

there exists a constant T0 > T for any l1 > 0 such that x∗ (t) −
PX (x∗ (t)) ≤ l1 for all t > T0 . In view of the arbitrariness of l1 ,
letting l1 → 0, we have limt→+∞ x∗ (t) − PX (x∗ (t)) = 0. It
follows from the definition of X that the team objective function
(2) is minimized as t → +∞.

B. Distributed Adaptive Gradient Optimization Algorithm
In algorithm (4), it is required that the gains α and γ should be
known to all agents and it can only be used to deal with quadraticlike convex objective functions. In this subsection, we design a
distributed adaptive algorithm for (1) to solve the optimization
problem (2) for general convex local objective functions. The
algorithm is given by
ui (t) =

j ∈N i (t)

qij (t)sgn xj (t) − xi (t)

−∇fi (xi (t)),

sgn(maxs∈[t−c 0 ,t] xj (s) − xi (s)), if (i, j) ∈ G(t),
q̇ij (t) =
0, otherwise,
qij (0) = qj i (0) = 0,

(14)

where
c0 > 0 is an arbitrary constant. In (14), the role of the first
term, j ∈N i (t) qij (t)sgn xj (t) − xi (t) , is to drive all agents
to reach a consensus, while the second term, −∇fi (xi (t)), is
the negative gradient of fi (xi (t)) playing a role in minimizing
fi (xi (t)).
Theorem 2: Suppose that the graph G(t) is undirected and
connected for all t and Assumptions 1 and 2 hold. For system
(1) with algorithm (14), all agents reach a consensus in finite
time and minimize the team objective function (2) as t → +∞.
Proof: We first show that all xi (t) remain in a bounded
region. Under Assumption 1, it follows from Lemma 4 that all
Xi and X are nonempty closed bounded convex sets for all
i. Therefore, there is a closed bounded convex set Y such that
xi (0) ∈ Y, X ⊂ Y and Xi ⊂
Y for all i. Consider the Lyapunov
function candidate V (t) = ni=1 xi (t) − z2 for some z ∈ X.
Let Y be
sufficiently large for all zj ∈ Xj such that fi (xi (t)) −
/ Y.
fi (z) ≥ nj=1,j = i [fj (z) − fj (zj )] for all i and all xi (t) ∈
Calculating V̇ (t), we have
n

V̇ (t) =

i=1

×
−

xi (t) − z
j ∈N i (t)
n
i=1

T

qij (t)sgn xj (t) − xi (t)

xi (t) − z

T

∇fi (xi (t)).

(15)

Since z ∈ X, from the convexity of the function fi (xi (t)),
we have ∇fi (xi (t))T (z − xi (t)) ≤ fi (z) − fi (xi (t)). Moreover, since the graph G(t) is undirected, similar to the proof
of Proposition 1, we have
n
i=1

=

xi (t)T

j ∈N i (t)

n
i=1

j ∈N i (t)

qij (t)sgn xj (t) − xi (t)

qij (t)
T
xi (t) − xj (t)
2

× sgn xj (t) − xi (t) ≤ 0

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2244

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and
n
i=1

z

T
j ∈N i (t)

qij (t)sgn xj (t) − xi (t) = 0.

(17)


From (15), (16) and (17), we have V̇ (t) ≤ − ni=1 [fi (xi (t)) −
/ Y for some i0 , we have fi 0 (xi 0 (t)) −
fi (z)]. Ifxi 0 (t) ∈
fi 0 (z) ≥ nj=1,j = i 0 [fj (z) − fj (zj )] for all zj ∈ Xj and

hence V̇ (t) ≤ −[fi 0 (xi 0 (t)) − fi 0 (z)] + nj=1,j = i 0 [fj (z) −
fj (zj )] ≤ 0. This implies that all xi (t) remain in Y . Note that
each function fi (s) is differentiable and Y is bounded. Thus
each ∇fi (s) is bounded. That is, ∇fi (s) < ρf for some constant ρf > 0.
Now, we show that all agents reach a consensus in finite time.
Let 0 < tk 1 ≤ tk 2 < tk +1,1 ≤ tk +1,2 denote the contiguous
switching times for all k ∈ {1, 2, . . .} such that xi (t) = xj (t)
for some two integers i, j ∈ I and all t ∈ [tk 1 , tk 2 ) and xi (t) =
xj (t) for all i, j ∈ I and all t ∈ [tk 2 , tk +1,1
 ). Suppose that
consensus is not reached in finite time and +∞
k =1 (tk 2 − tk 1 ) <
+∞.
+∞. It is clear that tk 2 − tk 1 > 0 when k →

From the dynamics of qij (t), we have that limt→+∞ ni=1 nj=1 qij (t) =
+∞. Then by a similar approach to the following case of

+∞
k =1 (tk 2 − tk 1 ) = +∞, it can be shown that consensus can
be reached in finite
time, which is a contradiction.
+∞
Suppose that
k =1 (tk 2 − tk 1 ) = +∞. Then from the
dynamics of qij (t), there must exist a pair of agents, denoted by
i0 = j0 , such that limt→+∞ qi 0 j 0 (t) = +∞. In the following,
we prove that there exist a pair of agents, denoted by i1 = j1 ,
/ {(i0 , j0 ), (j0 , i0 )}, i1 ∈ {i0 , j0 } and
such that (i1 , j1 ) ∈
limt→+∞ qi 1 j 1 = +∞. If this is not true, we have qii 0 (t) < γq
and qij 0 (t) < γq for some constant γq > ρf , all t and all
i ∈ ∪s∈[0,+∞) [Ni 0 (s) ∪ Nj 0 (s)] with i = i0 and i = j0 . Since
limt→+∞ qi 0 j 0 (t) = +∞, there exists a sufficiently large
constant T0 > 0 for any γ0 > 6nmγq such that qi 0 j 0 (t) > γ0
for all t > T0 . By simple calculations based on (14), when
(i0 , j0 ) ∈ G(t) and xi 0 (t) − xj 0 (t) = 0 for t > T0 , we have
x i 0 (t)−x j 0 (t)
d
dt xi 0 (t) − xj 0 (t) ≤ x i (t)−x j (t) 2qi 0 j 0 (t)sgn xj 0 (t) −
0

0

xi 0 (t) + 2nmγq ≤ −2nmγq . When there exist at least an
agent i such that i ∈ Nĩ (t) and xĩ (t) − xi (t) = 0 for
/ G(t) or xi 0 (t) − xj 0 (t) = 0
ĩ ∈ {i0 , j0 } and either (i0 , j0 ) ∈
d
xi 0 (t) − xj 0 (t) ≤ 2nmγq for t > T0 . Since
holds, we have dt
all xi (t) remain in a bounded region, xi 0 (t) − xj 0 (t) < ρv for
some positive constant ρv . Let τa (T1 ) and τb (T1 ), respectively,
denote the total time in the interval (T0 , T1 ) for any T1 > T0
for the case when (i0 , j0 ) ∈ G(t) and xi 0 (t) − xj 0 (t) = 0
and the case when there exist at least an agent i such that
i ∈ Nĩ (t) and xĩ (t) − xi (t) = 0 for ĩ ∈ {i0 , j0 } and either
/ G(t) or xi 0 (t) − xj 0 (t) = 0 holds. Thus,
(i0 , j0 ) ∈
0 ≤ xi 0 (T1 ) − xj 0 (T1 )
≤ xi 0 (T0 ) − xj 0 (T0 ) + 2nmγq τb (T1 )
− 2nmγq τa (T1 )
≤ ρv + 2nmγq τb (T1 ) − 2nmγq τa (T1 ).

(18)

Since limt→+∞ qi 0 j 0 (t) = +∞, from the dynamics of qij (t),
we have limT 1 →+∞ τa (T1 ) = +∞ and hence from (18) we
have limT 1 →+∞ τb (T1 ) = +∞. That is, there exist a pair of

/ {(i0 , j0 ), (j0 , i0 )}, i1 ∈
agents i1 = j1 such that (i1 , j1 ) ∈
{i0 , j0 } and limt→+∞ qi 1 j 1 (t) = +∞. Similarly, it can be
proved that there exist a pair of agents i2 = j2 such that
/ {(i0 , j0 ), (j0 , i0 ), (i1 , j1 ), (j1 , i1 )}, i2 ∈ {i0 , j0 , i1 }
(i2 , j2 ) ∈
and limt→+∞ qi 2 j 2 (t) = +∞. By analogy, it can be proved that
limt→+∞ qij (t) = +∞ for all i, j. Since each ∇fi (xi (t))
is bounded for all t, there is a constant T2 > 0 such that
qij (t) > 2n maxk ∇fk (xk (t)) for all i, j and all t > T2 . Similar to the proof of Proposition 1, we have all agents reach a
consensus
in finite time. This contradicts with the precondition

that +∞
k =1 (tk 2 − tk 1 ) = +∞.
Summarizing the above analysis, all agents reach a consensus in finite time. Then there exists a number T > 0 such that
xi (t) = x∗ (t), where x∗ (t) is defined in (11), for all t > T
and all i ∈ I. Similar to the proof of Theorem 1, it can be
proved that the team objective function (2) is minimized as
t → +∞.

C. Distributed Optimization Algorithm With Nonuniform
Gradient Gains
In the existing works, the gradient gains are usually assumed
to be uniform and need to be known in advance, e.g., [1]. In this
subsection, we extend to consider the nonuniform gradient gains
based on the agents’ states for general convex local objective
functions. The algorithm is given by
q̇i (t) = arctan(ex i (t) ), qi (0) > 0,
ui (t) =

j ∈N i (t)

sgn xj (t) − xi (t) −

∇fi (xi (t))

qi (t)

(19)


for all i. Here, the gain 1/ qi (t) is used to ensure the weighted
gradient √df i (x i (t)) to tend to zero as time evolves. In practical
q i (t)dx i (t)

applications, it is hard for all agents to have a uniform system
clock and know its value accurately at any time. So, we do not
use the information of the system clock directly in the design of
the gradient gains.
Remark 2: Here, we use the inverse tangent functions and the
exponential functions to guarantee q̇i (t) to be upper and lower
bounded by two positive constants (here the two constants are
π
π
2 and 4 ). As a matter of fact, there are some other functions,
e.g., saturation function, that can be used to play the same role.
For easy readability, we do not give the general form of such
functions.
Theorem 3: Suppose that the graph G(t) is undirected and
connected for all t and Assumptions 1 and 2 hold. For system
(1) with algorithm (19), all agents reach a consensus in finite
time and minimize the team objective function (2) as t → +∞.
π/2 for
Proof: Note that π/4 ≤ arctan(ex i (s) ) ≤ √
all s and
all
i.
There
exists
a
constant
T
>
0
such
that
2
t
>
qi (t) >
√
t
for
all
i
and
all
t
>
T
.
Consider
the
Lyapunov
function
can2

didate V (t) = ni=1 xi (t) − z2 for z ∈ X and t > T . Under
Assumption 1, it follows from Lemma 4 that all Xi and X are
nonempty closed bounded convex sets for all i. Let Y be a closed
bounded convex set such
that xi (T ) ∈ Y, X ⊂ Y, Xi ⊂ Y
and fi (xi (t)) − fi (z) ≥ 4 nj=1,j = i [fj (z) − fj (zj )] for all i,

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LIN et al.: DISTRIBUTED CONTINUOUS-TIME OPTIMIZATION: NONUNIFORM GRADIENT GAINS, FINITE-TIME CONVERGENCE, AND CONVEX

all zj ∈ Xj and all xi (t) ∈
/ Y . It follows that √ 1 [fi (xi (t)) −
q i (t)
n
1
√
fi (z)] ≥ j =1,j = i
[fj (z) − fj (zj )] for all t > T , all i,
q j (t)

all zj ∈ Xj and all xi (t) ∈
/ Y . Then similar to the proof
of Theorem 2, it can be proved that all xi (t) and all
∇fi (xi (t)) are bounded for all t > T . That is, |fi (xi (t))| < μc
and ∇fi (xi (t)) < μc for some constant μc > 0, all i and all
t > T . Moreover, note that X is bounded and each fi (s) is diflarge such
ferentiable for all i and all s. Let μc be
 sufficiently

that μc > 2n ∇fi (xi (t)) and μc > fi (s) for all i, all t > T
√
and all s ∈ X. Let T0 > T be a constant such that 2T 0 > μc .
Similar to the proof of Proposition 1, it can be proved that all
agents reach a consensus in finite time. That is, there exists a
constant T1 > T0 such that xi (t) = x∗ (t) for all i and all t > T1 ,
where x∗ (t) is defined in (11).
Now, we prove that the team objective function (2) is minPX (s) ≤ l1 }
imized as t → +∞. Let E = {s ∈ Rm | s − 
for some constant l1 > 0 and ρ = mins∈∂¯E ni=1 [fi (s) −
¯
denotes the boundary of E.
fi (PX (s))], where ∂E
the definition of X, we have
Since PX (s) ∈ X, from 
n
ρ > 0. From Lemma 3,
i=1 [fi (s) − fi (PX (s))] > ρ for
any s ∈
/ E. Note that qi (t) − qi (T1 ) = qj (t) − qj (T1 ) =
t
x ∗ (s)
)ds  q ∗ (t) and qi (t) = q ∗ (t)/Δi (t) for all
T 1 arctan(e

1
≤ − 
n q ∗ (t)

n

1
≤ − 
n q ∗ (t)

n

i=1

i=1

φi (t) −
φi (t) + 

2245


1 − Δi (t)
φi (t) 
i=1
n q ∗ (t)
n

(21)

q ∗ (t)

√

√
for t > T2 . Since 2 t > q ∗ (t) > 2t for all t > T2 and
< ρ , then V̄˙ (t) ≤ − √ 1 [ ρ − ] ≤ − √1 [ ρ − 2 ] < 0 for
4n

q ∗ (t) n

t 2n

/ E and all t > T2 . Integrating both sides of this inany x∗ (t) ∈
√
√
equality from T2 to t, we have V̄˙ (t) ≤ −( t − T2 )[ nρ − 4 ].
This implies that there exists a constant T3 > T2 for any
l1 > 0 such that x∗ (t) − PX (x∗ (t)) ≤ l1 for all t > T3 .
In view of the arbitrariness of l1 , letting l1 → 0, we have
limt→+∞ x∗ (t) − PX (x∗ (t)) = 0. That is, the team objective
function (2) is minimized as t → +∞.

Remark 3: In the existing works, the gradient or subgradient
gains are usually assumed to be uniform for all agents at any time
instant, and moreover their values for all time instants need be
known in advance. For example, in [2], for discrete-time
multi
agentsystems, the gains should satisfy that ni=1 αk = +∞
and ni=1 αk2 = +∞, where αk denotes the uniform subgradient gain for all agents at time instant k. One example of the
1
q i (T 1 )
i, j and t > T1 , where Δi (t) = 1/(1 + q ∗ (t) ). Since the graph selection is αk = 1+k , k = 0, 1, . . .. To determine the gains,
the exact time clock (i.e., k in the discrete-time case) should be
G(t) is undirected and connected, it follows that for all t > T1 ,
known by all agents and the values of the gains for all agents
need be kept identical at any time instant. In contrast, in this
∗
paper, the gradient gains in algorithm (19) are state-dependent
ẋ (t)
and can be self-adjusted based on the agents’ current states. At
n
1
=
ẋi (t)
each time instant, the agents only need to know their current
i=1
n
 states xi (t) to determine their own gains and there is no need

n
to know the current time clock (i.e., t in the continuous-time
1
∇fi (xi (t))
=
sgn(xj (t) − xi (t)) − 
case). Note that while xi (t) is a function of time, the agents do
i=1
j
∈N
(t)
n
i
qi (t)
not
use the current time to calculate xi (t) but instead the states

n
Δi (t)∇fi (x∗ (t))
are obtained by measurements without the need for explicitly
1

=−
.
(20)
knowing the exact clock. The gradient gains can be different for
i=1
n
q ∗ (t)
different agents, which might distinctly relax the synchronization requirement on the system. In [5], a distributed algorithm
On the other hand, recall that π/4 ≤ arctan(ex i (s) ) ≤ π/2 for with nonuniform subgradient gains was also given to solve the
all s and all i. From the definition of q ∗ (t), there exists a constant
distributed optimization problem, but the algorithm can only be
√

√
ρ
such that 2 t > q ∗ (t) > 2t used in the stochastic sense. By taking the mathematical expecT2 > T1 for any 0 < < 4n

and 1 − Δi (t) < 2μ c for all i and all t > T2 . Let φi (t) = tation, it uses uniform subgradient gains for all agents in nature.
fi (x∗ (t)) − fi (PX (x∗ (t))) for all i. Since |fi (x∗ (t))|
 < μc and
D. Distributed Finite-Time Optimization Algorithm
|fi (PX (x∗ (t)))| < μc , it follows that |φi (t)[1 − Δi (t)]| <
for all i.
Most of the existing works on the distributed optimization
Consider the Lyapunov function candidate V̄ (t) = problem (e.g., [1]–[16]) as well as the algorithms introduced
1
˙
∗
∗
2
in Section III.A–III.C, studied only the asymptotic stability of
2 x (t) − PX (x (t)) for t > T2 . Calculating V̄ (t) along the
solutions
of
(20),
it
follows
from
Lemma
2
and
the
convexity
of
the algorithm, and are rarely concerned with the finite-time
n
convergence of the algorithms. To this end, in this subsection, we
i=1 fi (s) that
design one algorithm for (1) such that distributed optimization
can not only be achieved, but achieved in finite time.
V̄˙ (t) = [x∗ (t) − PX (x∗ (t))]T ẋ∗ (t)
The finite-time algorithm for system (1) is given by
1
∗
∗
T
[x (t) − PX (x (t))]
= − 
pij (t)sgn(θj (t) − θi (t)),
ψ̇i (t) =
n q ∗ (t)
×

n
i=1

∇fi (x∗ (t)) 1 − 1 +



j ∈N i (t)

Δi (t)

θi (t) = ψi (t) + ∇fi (xi (t)), ψi (0) = 0,

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IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 62, NO. 5, MAY 2017

ṗij (t) =


sgn(maxs∈[t−c 0 ,t] θj (s)−θi (s)), if (i, j) ∈ G(t),
0, otherwise,

pij (0) = pj i (0) = 0,
ui (t) =

j ∈N i (t)

−
q̇ij (t) =

qij (t)sgn(xj (t) − xi (t))



such that (s − s̄)T n1 ni=1 ∇fi (s) ≥ cx n1 ni=1 ∇fi (s) s −
s̄ for any s ∈ Z − X.
Proof: As all fi (s) are twice differentiable convex functions,

n
1
i=1 fi (s) is a twice differentiable convex function as well.
n
It follows that
1
n
1
≥
n

θi (t)
− gci (t),
θi (t)


sgn(maxs∈[t−c 0 ,t] xj (s)−xi (s)), if (i, j) ∈ G(t),
0, otherwise,

qij (0) = qj i (0) = 0,
⎧
⎨ 0, if θj (t) = θi (t)
and xi (t) = xj (t) for all j ∈ Ni (t),
gci (t) =
⎩
xi (t) − PX i (xi (t)), otherwise,
where c0 > 0 is an arbitrary constant, and θi (t) and ψi (t) are
the internal states of the dynamic averaging estimator for all
i. Here the dynamic averaging estimator is motivated by [21].
x
, we
Here, to eliminate the singular point of the function x
x
define x = 0 when x = 0.
In algorithm (22), the role of θi (t) is to estimate the average
derivative of all local objective functions fi (xi (t)) with respect
to xi (t), the role of the time-varying gains pij (t) is to ensure
the influence of ∇2 fi (x∗ (t))ẋ∗ (t) on the tracking of the average
derivative of all fi (xi (t)) to vanish to zero as time evolves, and
the role of the time-varying gains qij (t) is to force all agents to
reach a consensus and move along the negative direction of the
average derivative of all local objective functions fi (xi (t)).
Remark 4: There are three difficulties in the analysis of system (1) with algorithm (22): (a) this system is a time-varying
system with a strong nonlinearity since the interaction gains
pij (t) and qij (t) are time-varying and this system contains a
strongly nonlinear term ∇2 fi (x∗ (t))ẋ∗ (t) as shown later in (31);
(b) there exist four strong couplings: the first one is between the
variables θi (t) and xi (t) in each agent; the second one is among
the variables θi (t) for neighbor agents; the third one is among
the variables xi (t) for neighbor agents; and the last one is between the variables θi (t) and xj (t) for neighbor agents; (c) each
∇fi (xi (t)) and each ∇2 fi (x∗ (t)) are not bounded and hence
θi (t) might tend to infinity as time evolves, which might destroy
the system stability.
2
Assumption 4: Suppose that each [∇2 fi (s)]j k = ∂∂s jf∂(s)
s k is
continuous with respect to s, and either one of the following
conditions holds:
(a) There exists a scalar δ > 0 and a vector s̄ ∈ X such that
{ξ|ξ − s̄ ≤ δ} ⊂ X [1].
(b) There is a neighborhood of X, denoted by S, and a
T
uniform
s ≤ 1 such that (s − PX (s))
n constant 0 <1c
n
1
i=1 ∇fi (s) ≥ cs n
i=1 ∇fi (s) s − PX (s) for
n
all s ∈ S.
Below are some lemmas that will be used in the proof of the
main theorem.
Lemma 5: Let Z be a closed bounded convex set containing X, and s̄ be defined in Assumption 4(a). Under
Assumption 4(a), there exists a uniform constant 0 < cx ≤ 1

n
i=1
n
i=1

fi (s0 )
fi (s) +

1
n

n
i=1

∇fi (s)T (s0 − s),

i.e.,
(22)



1
n

1
≥
n

n
i=1

T
∇fi (s)
(s − s0 )

1
fi (s) −
i=1
n
n

(23)
n
i=1

0

fi (s ) > 0,

for all s0 ∈ X and all s ∈ Z − X. If this lemma
does not hold, there exists a sequence
of vectors

Z − X} such that limk →+∞ n1 ni=1 ∇fi (yk )T (yk −
{yk ∈ 
Since
each
s̄)/ n1 ni=1 ∇fi (yk ) /yk − s̄ = 0.
n
∂ 2 f (s)
1
2
[∇ fi (s)]j k = ∂ s j ∂ s k is continuous, n i=1 ∇fi (yk ) is contin
uous. Since Z is bounded, n1 ni=1 ∇fi (y
k ) and yk − s̄ are
both upper bounded. Thus, limk →+∞ n1 ni=1 ∇fi (yk )T (yk −
s̄) = 0. Let dk ∈ Rm be an arbitrary unit vector for any k.
Under Assumption 4(a), s̄ + 12 δdk ∈ X for all k, where δ
is defined in Assumption 4(a). In view of the arbitrariness
can adopt a proper dk such that
ofthe direction dk , we 
n
n
1
1
T
∇f
(y
)
d
=
) . As k → +∞,
i
k
k
i=1 ∇fi (yk
n i=1
n
n
n
1
1
1
T
∇f
(y
)
(y
−
s̄
−
δd
)/
i k
k
k
i=1
i=1 ∇fi (yk ) /yk −
n
2
n
1
1
1
s̄ − 2 δdk  = − 2 δ/yk − s̄ − 2 δdk . Since yk ∈ Z − X and
s̄ + 12 δdk ∈ X, we have that yk − s̄ − 12 δdk  is upper bounded
1
1
from the boundedness of X and Z and y
k − s̄ − 2 δdk  ≥ 2 δ
n
1
T
from the definition
 of s̄. Therefore, n i=1 ∇fi (yk ) (yk −
s̄ − 12 δdk )/ n1 ni=1 ∇fi (yk ) /yk − s̄ − 12 δdk  is upper
bounded by a negative constant as k → +∞, which contradicts
with (23).

Lemma
6:
Consider
the
system
given
by
ẏ(t)
=
n

.
Let
Z
be
a
closed
∇f
(y(t))
− ni=1 ∇fi (y(t))/
i
i=1
bounded convex set containing X. If y(t) ∈ Z for all t and
Assumption 4 holds, there exists a constant T > 0 such that
y(t) ∈ X for all t > T .
Proof: (a) Under Assumption 4(a), Lemma 5 holds and consider the Lyapunov function candidate Va (t) = y(t) − s̄ for
all t. Calculating V̇a (t), we have
V̇a (t) = −

(y (t)−s̄) T
y (t)−s̄

n

i = 1 ∇f i (y (t))

n

i = 1 ∇f i (y (t))

≤ −cx

for a constant cx > 0 and all y(t) ∈ Z − X. Integrating both
sides of this inequality, we have Va (t) − Va (0) ≤ −cx t for all
y(t) ∈ Z − X. It is clear that there exists a constant T > 0 such
that y(t) ∈ X for all t > T .

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LIN et al.: DISTRIBUTED CONTINUOUS-TIME OPTIMIZATION: NONUNIFORM GRADIENT GAINS, FINITE-TIME CONVERGENCE, AND CONVEX

(b) Under Assumption 4(b), there is a neighborhood of X,
denoted by S, and a uniform constant 0 < cs ≤ 1 such that
(y(t) − PX (y(t)))T
≥ cs

1
n

n
i=1

1
n

n
i=1

for any y ∈
/ Y and
maxi {y − PX (y), y − PX i (y)} ≤ 3dD

(25)

for all y ∈ Y . Then, from the triangle relationship, for any y ∈
/
Y , the angle between y − PX i (y) and y − z0 is no larger than
π
3 for all i. That is,

∇fi (y(t))

∇fi (y(t)) y(t) − PX (y(t))

(y − PX i (y))T (y − z0 ) ≥ 12 y − PX i (y)y − z0 

for all y(t) ∈ S. Similar to the proof of Lemma 4,

X (y )
¯
miny ∈∂¯S ni=1 ∇fi (y)T yy −P
−P X (y ) > 0, where ∂S denotes
the boundary of the set S. From Lemma 3, it follows

(t)−P X (y (t))
for some conthat  ≤ n1 ni=1 ∇fi (y(t))T yy (t)−P
X (y (t))
stant
n > 0 and any y(t) ∈ Z − S. Since Z is bounded,
1
i=1 ∇fi (y(t)) < cg for some constant cg > 0 and all
n
y(t) ∈ Z. Hence
n
1
(y(t) − PX (y(t)))
∇fi (y(t))
i=1
n
n
 1
∇fi (y(t)) y(t) − PX (y(t))
≥
i=1
cg n
T

for some constant  > 0 and any y(t) ∈ Z − S. Consider the
Lyapunov function candidate Vb (t) = y(t) − PX (y(t)) for
all t. In the same way as the proof of (a), it can be proved
that there exists a constant T > 0 such that y(t) ∈ X for all
t > T.

Remark 5: Under Assumption 4(a), X is a nonempty closed
convex set and contains at least one interior point while
Assumption 4(b) considers the case that X has no interior points
and excludes the singular situation where
(s − PX (s))T
s−P X (s)→0 s − PX (s)
 1
n
n
1
∇fi (s)
∇fi (s) = 0,
×
i=1
i=1
n
n

i.e., n1 ni=1 ∇fi (s) tends to be orthogonal to s − PX (s) as s
converges to X. In [25], some finite-time results are given for
nonconvex functions, but when convex functions are considered,
the results can only be used to a special case of Assumption 4(b)
because the convexity of the functions was not fully exploited.
Lemma 7: Suppose that the graph G(t) is undirected and
connected for all t and Assumptions 1, 2 and 4 hold. For system
(1) with algorithm (22), the following statements hold:
(a) xi (t) ∈ Z for all t and a closed bounded region Z and
there exists a constant T0 > 0 such that xi (t) = xj (t)
for all t > T0 ;
(b) Each θi (t) is bounded for all i and all t.
Proof: Under Assumption 1, it follows from Lemma 4
that all Xi and X are nonempty closed bounded convex sets
for
V (t) =
n all i. Consider2 the Lyapunov function candidate 
x
(t)
−
z

for
some
z
∈
X.
Let
d
=
max
y1 −
i
0
D
i=1
 0
n
and Y = {y | y − PX (y) ≤
y2  | y1 , y2 ∈ X i=1 Xi
L0 } ⊂ Rm for some constant L0 > 0 be a closed bounded convex set such that
lim

mini {y − PX (y), y − PX i (y)} ≥ dD

2247

(24)

(26)

for any y ∈
/ Y . Let Z = {y | y − PX (y) ≤ L1 } ⊂ Rm for
some constant L1 > L0 be a closed bounded convex set such
that Y ⊂ Z, xi (0) ∈ Z and
min{y − PX i (y), y − z0 }
i

> max{8n + 6ndD , 0.5 max z1 − z0 }
z 1 ∈Z

(27)

for all i and any y ∈
/ Z.
We first consider the case where θi (t) = θj (t) or xi (t) =
xj (t) for some i = j. Suppose that there exists an agent
/ Z. Then there must exist an agent i1
i0 such that xi 0 (t) ∈
such that xi 0 (t) = xi 1 (t) ∈
/ Z and gci 1 (t) = 0. If this is not
/ Z and θj (t) = θi 0 (t) for all
true, from (22), xj (t) = xi 0 (t) ∈
j ∈ Ni 0 (t). Since the graph G(t) is undirected and connected,
it follows that xi (t) = xj (t) ∈
/ Z and θi (t) = θj (t) for all i, j.
This yields a contradiction. Without loss of generality, suppose
/ Z, gci (t) =
that xi 1 (t) − z0  = max{xi (t) − z0  | xi (t) ∈
/ Z}.
0}. Clearly, xi 1 (t) − z0  = maxi {xi (t) − z0  | xi (t) ∈
Calculating V̇ (t), from (16) and (27), we have

n
θi (t)
+ gci (t)
(xi (t) − z0 )T
V̇ (t) = −
i=1
θi (t)

qij (t)sgn(xj (t) − xi (t))
−
j ∈N i (t)

n

≤

j ∈N i (t)

i=1

qij (t)
T
xi (t) − xj (t)
2

× sgn xj (t) − xi (t) +
−

n
i=1

n
i=1

xi (t) − z0 

(xi (t) − z0 )T gci (t)

≤ 2nxi 1 (t) − z0  −

n
i=1

(xi (t) − z0 )T gci (t).

From (22), (25) and (26), we have (xi (t) − z0 )T gci (t) ≤
3xi 1 (t) − z0 dD for any xi (t) ∈ Y and (xi (t) − z0 )T (xi (t) −
PX i (xi (t))) ≥ 12 xi (t) − z0 xi (t) − PX i (xi (t))
 for any
/ Y . It follows from (27) that − ni=1 (xi (t) −
xi (t) ∈
Thus,
z0 )T gci (t) ≤ −xi 1 (t) − z0 (4n + 3ndD − 3ndD ).
V̇ (t) ≤ −2nxi 1 (t) − z0 .
Now, we consider the case where θi (t) = θj (t) and xi (t) =
xj (t) for all i, j. Calculating V̇ (t), we have

n
θi (t)
V̇ (t) = −
(xi (t) − z0 )T
i=1
θi (t)

−
qij (t)sgn(xj (t) − xi (t))
j ∈N i (t)

≤ −

n
i=1

(xi (t) − z0 )T

θi (t)
.
θi (t)

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(28)

2248

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 62, NO. 5, MAY 2017

Since
and ψi (0) = 0 for all i, we have
n G(t) is connected 
n
i=1 ψ̇i (t) = 0 and thus
i=1 ψi (t) = 0, which implies that

Note that the number of all parameters pij is finite, denoted
by ne , and it takes at most ndμ time for each pij to increase
from 0 to ndμ at the rate of 1. Since θ̄˙k (t) ≤ dμ and especially
θ̄˙k (t) < 0 when α(t) > ndμ , we have that it takes at most
nne dμ time for α(t) to increase to ndμ when θ̄k (t) = θk (t).
Notethat when θ̄k (t) = θk (t), we have θ̄k (t) = θk (t)
 =
n
1
∗
∗
∗ (t)∈Z
∇f
∇f
(x
(t))
(x
(t))
.
≤
max
i
i
i,x
i=1
n
k
Hence, θ̄k (t) < θ̄k (T1 ) + nne d2μ for all t > T1 . Thus, θ̄k (t) is
upper bounded for all t and all k. In the same way, it can be
proved that θk (t) is lower bounded for all t and all k. Thus,

θi (t) is bounded for all i and all t.
Theorem 4: Suppose that the graph G(t) is undirected and
connected for all t and Assumptions 1, 2 and 4 hold. For system (1) with algorithm (22), all agents reach a consensus and
minimize the team objective function (2) in finite time.
Proof: Under Assumptions 1, 2 and 4, Lemma 7 holds.
Hence, xi (t) ∈ Z for all t and a closed bounded region Z and
there exists a constant T0 > 0 such that xi (t) = xj (t) = x∗ (t),
where x∗ (t) is defined in (11), for all i and all t > T0 . Moreover,
from Lemma 7, each θi (t) is bounded for all i and all t. Then,
similar to the proof of Theorem 2, it can be proved that all θi (t)
reach a consensus in finite time. That is, there exists a number
T1 > T0 such that θi (t) = θj (t)  θ∗ (t) for all t > T1 .
As a result, we have

n
n


1
1
∇fi (xi (t))
θi (t) =
n i=1
n i=1

for all t. Since θi (t) = θj (t) and xi (t) = xj (t) for all i, j, we
have
θi (t) =

n
n


1
1
∇fi (xi (t))
θi (t) =
n i=1
n i=1


n 
1
1
=
∇fi
n i=1
n

n
i=1


xi (t)

(29)

for all i. Moreover, since z0 ∈ X, from the convexity of the
functions fi (s), we have


n
n
n
1
1
1
0≤
fi
xi (t) −
fi (z0 )
i=1
i=1
i=1
n
n
n



n
n
n
1
1
1
∇fi
xi (t)
xi (t) − z0
≤
i=1
i=1
i=1
n
n
n
(30)
It follows from (28), (29) and (30) that V̇ (t) ≤ 0.
Summarizing both cases, we have V̇ (t) ≤ 0 if there exists an
agent i such that xi (t) ∈
/ Z. Since xi (0) ∈ Z, then xi (t) ∈ Z
for all i and all t. Then similar to the proof of Theorem 2,
it can be proved that there exists a constant T0 > 0 such that
xi (t) = xj (t) = x∗ (t), where x∗ (t) is defined in (11), for all i
and all t > T0 . It is clear that

n 
θi (t)
1
∗
+ gci (t)
ẋ (t) = −
n i=1 θi (t)
for all t > T0 . Define A1k (t)  {i | θik (t) = maxi {θik (t)}},
θ̄k (t)  |A 1 k1 (t)|
A2k (t)  {i | θik (t) = mini {θik (t)}},


1
i∈A 1 k (t) θik (t) and θ k (t)  |A 2 k (t)|
i∈A 2 k (t) θik (t), where
|A1k (t)| ≥ 1 and |A2k (t)| ≥ 1 denote, respectively, the cardinality of A1k (t) and A2k (t). It is clear that the kth component
of each θ̇i (t) can be written as
θ̇ik (t) =

j ∈N i (t)

pij (t)sgn(θj k (t) − θik (t))

+ [∇2 fi (x∗ (t))ẋ∗ (t)]k

ẋ∗ (t) = −θ∗ (t)/θ∗ (t)
for all t > T1 . Recalling (29), from Lemma 6, the team objective
function (2) will be minimized in finite time.

Remark 6: Due to the existence and the nonlinearity of the
objective functions, the existing approaches for the distributed
finite-time consensus problem (e.g., [17], [18]) cannot be extended directly to the distributed finite-time optimization problems, which need to consider the finite-time convergence of the
consensus of the agents and the finite-time convergence of the
objective functions simultaneously. Although some results have
been obtained in our previous works in [19], [20] for the distributed finite-time optimization problem, they are limited to a
special class of convex objective functions that have a quadraticlike form and the approaches cannot be applied to more general
convex objective functions.

(31)

for t > T0 .


Suppose
that
θ̄k (T1 ) ≥ maxi,x ∗ (t)∈Z ∇fi (x∗ (t))
for some T1 > T0 and θ̄k (t) = θk (t) for t > T1 . Let
/ A1k (t)}. Since
C1k (t)  {(i, j) ∈ E(G(t)) | i ∈ A1k (t), j ∈
the graph G(t) is connected and θ̄k (t) = θk (t), then C1k (t)
is nonempty. Moreover, since x∗ (t) ∈ Z and each entry
4, then
of ∇2 fi (x∗ (t)) is continuous from Assumption

is bounded. Let
dμ = maxi,x ∗ (t)∈Z ∇2 fi (x∗ (t))ẋ∗ (t)
α(t) = min(i,j )∈C 1 k (t) pij (t). Note that sgn(θhk (t) − θlk (t)) ≤
0 for any l ∈ A1k (t) and any h ∈ Nl (t). It is clear from (31)

that θ̄˙k (t) = |A 1 k1 (t)| i∈A 1 k (t) θ̇ik (t) ≤ dμ − αn(t) . From the
dynamics of pij (t), we have ṗij (t) = 1 for any (i, j) ∈ C1k (t).

IV. DISTRIBUTED CONTINUOUS-TIME OPTIMIZATION WITH A
COMMON CONVEX CONSTRAINT SET
In this section, we will extend the results in Sections III.C and
III.D and design algorithms for system (1) to solve a distributed
optimization problem with a common convex constraint set as
follows
minimize

n
i=1

fi (xi )

subject to xi = xj ∈ H ⊂ Rm ,
where H is a closed convex set.

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(32)

LIN et al.: DISTRIBUTED CONTINUOUS-TIME OPTIMIZATION: NONUNIFORM GRADIENT GAINS, FINITE-TIME CONVERGENCE, AND CONVEX

A. Distributed Optimization Algorithm With Nonuniform
Gradient Gains
In this subsection, we extend Theorem 3 to the problem (32)
for general convex local objective functions. Let X ⊂ Rm denote the optimal set of the problem (32).
When H is a closed bounded convex set, the algorithm is
given by

for all i. Calculating V̇i (t), we have for all t > T0 ,

V̇i (t) = −(xi (t) − PH (xi (t)))T gri (t) + gci (t)

−
sgn (xj (t) − xi (t))
j ∈N i (t)

j ∈N i (t)

∇fi (xi (t))

qi (t)

gci (t) =

γi (t)[xi (t) − PH (xi (t))]
xi (t) − PH (xi (t))

(33)

for all i, where γi (t) > |Ni (t)| + 1 and |Ni (t)| denotes the
cardinality of Ni (t).
Lemma 8: Under Assumption 1, X is a nonempty closed
bounded convex set.
Proof: When H is a bounded closed convex set, from the
property of continuous functions on closed bounded sets and
the convexity of the functions fi (s) and the set H, it is easy to
see that X is a nonempty closed bounded convex set.
Under Assumption 1, Lemma 
4 holds. Thus,
n
f
(y)
=
+∞
and
lim
limy →+∞
i
y →+∞
i=1 fi (y) = +∞.
n
Since i=1 fi (y) is lower bounded in H, its infimum exists.
Similar to the proof of Lemma 4, it can be proved that X is a
nonempty closed bounded convex set, when H is an unbounded
closed convex set.

Theorem 5: Suppose that the graph G(t) is undirected and
connected for all t and Assumptions 1 and 2 hold. For system (1)
with algorithm (33), all agents reach a consensus in finite time
and minimize the team objective function (32) as t → +∞.
π/2 for
Proof: Note that π/4 ≤ arctan(ex i (s) ) ≤ √
all s and
t
>
qi (t) >
all
i.
There
exist
a
constant
T
>
0
such
that
2
√
t
2 for all i and all t > T . Consider the Lyapunov function
candidate V (t) = 12 xi (t) − z2 for z ∈ H and all t. Under
Assumption 1, it follows from Lemma 4 that all Xi and X are
nonempty closed bounded convex sets for all i. Let Y be a closed
bounded convex set such
that xi (T ) ∈ Y, X ⊂ Y, Xi ⊂ Y
and fi (xi (t)) − fi (z) ≥ 4 nj=1,j = i [fj (z) − fj (zj )] for all i,
all zj ∈ Xj and all xi (t) ∈
/ Y . It follows that √ 1 [fi (xi (t)) −
q i (t)

fi (z)] ≥ nj=1,j = i √ 1 [fj (z) − fj (zj )] for all t > T , all i,
q j (t)

/ Y . Moreover, since z ∈ H, from
all zj ∈ Xj and all xi (t) ∈
(t)[x i (t)−P H (x i (t))]
Lemma 2, we have (xi (t) − z)T γ i x
≥ 0. Then
i (t)−P H (x i (t))
similar to the proof of Theorem 2, it can be proved that all
agents remain in a bounded region and each ∇fi (xi (t))
is bounded for all i and all t. Then √
there 
exist two √constants T0 > T and μc > 0 such that 2 t > qi (t) > 2t >
μc > 8n ∇fi (xi (t)) for all i and all t > T0 .
Consider the Lyapunov function candidate
Vi (t) =

1
xi (t) − PH (xi (t))2
2



7
≤ − xi (t) − PH (xi (t))
8
7
≤ −
2Vi (t)
8

sgn xj (t) − xi (t) − gri (t) − gci (t)

gri (t) =



1
≤ − xi (t) − PH (xi (t)) γi (t) − |Ni (t)| −
8n

q̇i (t) = arctan(ex i (t) ), qi (0) > 0,
ui (t) =

2249

where the second inequality holds since γi (t) > |Ni (t)| + 1
and n ≥ 1. It follows that √V̇ i (t) ≤ − 78 . Integrating both
2V i (t)

√
sides
of
this
inequality
from
T
to t, we have 2 Vi (t)/ 2 −
0

√
2 Vi (T0 )/ 2 ≤ − 78 (t − T0 ). Thus, Vi (t) vanishes to zero in
finite time. That is, thereexist a constant T1 > T0 such that
xi (t) ∈ H and ẋi (t) = j ∈N i (t) sgn xj (t) − xi (t) − gri (t)

for all i and all t > T1 . Since qi (t) > 8n ∇fi (xi (t)) for
all i and all t > T0 , similar to the proof of Proposition 1, it can
be proved that all agents reach a consensus in finite time. That is,
there exists a constant T2 > T1 such that xi (t) = x∗ (t), where
x∗ (t) is defined in (11), for all i and all t > T2 . For t > T2 , we
have
n

ẋ∗ (t) = − n1

i=1 gri (t).

Now, we prove that the team objective function (32) can be
minimized as t → +∞. Under Assumption 1, Lemma 8 holds
and hence X is a nonempty closed bounded convex set. Consider
the Lyapunov function candidate

V̄ (t) =

1 ∗
x (t) − PX (x∗ (t))2
2

for all t > T2 . After some calculations, we have
V̄˙ (t)

∗

∗

∇fi (x∗ (t))

i=1
qi (t)

1
n

n

T 1

n

= −[x∗ (t) − PX (x∗ (t))]T
= −[x (t) − PX (x (t))]

n


∇fi (x∗ (t)) q ∗ (t)


i=1
q ∗ (t)
qi (t)

where q ∗ (t) is defined in the proof of Theorem 3. Similar to the
proof of Theorem 3, we can
 let T2 be sufficiently
 large such that
√

√


2 t > q ∗ (t) > 2t and 1/ 1 + qqi ∗(T(t)0 ) − 1 < /μc for any
∗

(t)
= 1/(1 + qqi ∗(T(t)0 ) ),
small > 0, all i and all t > T2 . Since qq i (t)

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IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 62, NO. 5, MAY 2017

we have
V̄˙ (t)

it can be used to deal with the case of the unbounded closed
convex set.
Remark 9: In Theorems 1, 2, 3 and 5, we assume that each
local objective function fi (x) is differentiable for discussion
convenience. The results obtained in these theorems can be extended to more general nondifferentiable convex functions by
using a minimum norm subgradient, denoted by lsi (s). That
is, lsi (s) = arg minz ∈∂ f i (s) z, where ∂fi (s) denotes the subgradient set of fi (s) at s. However, it should be noted that from
the convexity of the convex function fi (s), it can be proved
that fi (s) is minimized if and only if lsi (s) = 0. It should
also be noted that when the minimum norm subgradients are
used, after all agents reach a consensus, the Lyapunov function
x∗ (t) − xe 2 for xe ∈ X or xe ∈ X should be used instead to
prove that all agents minimize the team objective function as
t → +∞.
Remark 10: Since the solution sets for linear inequalities
or equalities in the form of h(x) ≥ 0 or h(x) = 0 are usually
closed convex sets, Theorem 5 might be used to deal with the distributed optimization problem with linear inequality or equality
constraints if its optimal set is bounded.

≤ −[x∗ (t) − PX (x∗ (t))]T

1
n

∇fi (x∗ (t))

i=1
q ∗ (t)
n

+ x∗ (t) − PX (x∗ (t)) 
n q ∗ (t)
1
≤− 
n q ∗ (t)

n
i=1

[fi (x∗ (t)) − fi (PX (x∗ (t))

− x∗ (t) − PX (x∗ (t))],
for all t > T2 , where the second inequality has used the convexity of the functions fi (x∗ (t)), i.e.,
1
n

n
i=1

[fi (PX (x∗ (t)) − fi (x∗ (t))]

n
1
∇fi (x∗ (t))T [PX (x∗ (t)) − x∗ (t)].
i=1
n
Since all agents remain in a bounded region and X
is bounded, x∗ (t) − PX (x∗ (t)) is bounded. That is,
x∗ (t) − PX (x∗ (t)) < μp for some constant μp > 0. Let
constant 0 <
E = {s ∈ Rm | s − PX (s) ≤ l1 } for some 
l1 ≤ maxs∈H s − PX (s) and ρ = mins∈H∩∂¯E ni=1 [fi (s) −
¯ denotes the boundary of E. Since
fi (PX (s))], where ∂E
PX (s) ∈ X ,from the definition of X , we have ρ > 0. From
/ E and
Lemma 3, ni=1 [fi (s) − fi (PX (s))] > ρ for any s ∈
l1 > 0 such that
s ∈ H. Let T2 be further large for any given
√

√
< 4μρ p . Recall that 2 t > q ∗ (t) > 2t for all t > T2 . It
follows that for any t > T , V̄˙ (t) ≤ − 1√ ρ + 2√ μ < 0.

≥

2

2n t

n t p
T2 to t,

Integrating both sides of this inequality from
we
√
√
˙
have V̄ (t) ≤ (−ρ + 4μp )( t − T2 )/n. This implies that
there exists a constant T3 > T2 for any l1 > 0 such that
x∗ (t) − PX (x∗ (t)) < l1 and x∗ (t) ∈ E ∩ H for all t > T3 .
In view of the arbitrariness of l1 , letting l1 → 0, we have
limt→+∞ x∗ (t) − PX (x∗ (t)) = 0. That is, the team objective
function (32) is minimized as t → +∞.

Remark 7: On the boundary of H, there might rise a switch(t)[x i (t)−P H (x i (t))]
. But the term
ing surface due to the term γ i x
i (t)−P H (x i (t))
γ i (t)[x i (t)−P H (x i (t))]
does not decrease but increases the converx i (t)−P H (x i (t))
gence rate of the Lyapunov function V̄ (t) = 12 PX (x∗ (t)) −
x∗ (t)2 for all t > T2 . This is because at the switching
surface, the angle between the vectors PX (x∗ (t)) − x∗ (t)

(t)[x ∗ (t)−P H (x ∗ (t))]
is no larger than π2 , i.e.,
and − n1 ni=1 γ i x
∗ (t)−P (x ∗ (t))
H

(t)[x ∗ (t)−P H (x ∗ (t))]
≥ 0, from
−[PX (x∗ (t)) − x∗ (t)]T n1 ni=1 γ i x
∗ (t)−P (x ∗ (t))
H

Lemma 2.
Remark 8: The approach in [10] is to analyze the convergence of the largest distance from the agents to the constraint set
so as to yield a contradiction to prove the optimal convergence.
Due to the unboundedness of the local objective functions and
the nonuniformity of the gradient gains, the approach in [10]
cannot be applied in this paper. The approach in this paper is to
analyze the convergence rates of the consensus, the distance to
the constraint set, and the optimization by fully exploiting the
convexity of the objective functions and the constraint set, and

B. Distributed Finite-Time Optimization Algorithm
In this subsection, we extend Theorem 4 to the problem (32)
for general convex local objective functions. The algorithm is
given by
ψ̇i (t) =

j ∈N i (t)

pij (t)sgn(θj (t) − θi (t)),

θi (t) = ψi (t) + ∇fi (xi (t)), ψi (0) = 0,

sgn( max θj (s) − θi (s)), if (i, j) ∈ G(t),
s∈[t−c 0 ,t]
ṗij (t) =
0, otherwise,
pij (0) = pj i (0) = 0,
ui (t) =

j ∈N i (t)

−
q̇ij (t) =

qij (t)sgn(xj (t)−xi (t))−

θi (t)
θi (t)

γi (xi (t) − PH (xi (t)))
,
xi (t) − PH (xi (t))


sgn( max xj (s)−xi (s)), if (i, j) ∈ G(t),
s∈[t−c 0 ,t]

0, otherwise,

qij (0) = qj i (0) = 0,

(34)

where c0 > 0 is an arbitrary constant, γi > 1, and θi (t) and
ψi (t) are the internal states of the dynamic averaging estimator
for all i.
2
Assumption 5: Suppose that each [∇2 fi (s)]j k = ∂∂s jf∂(s)
s k is
continuous with respect to s, and either one of the following
conditions holds:
(a) There exists a scalar δ > 0 and a vector s̄ ∈ X such that
{ξ|ξ − s̄ ≤ δ} ⊂ X .
(b) There is a neighborhood of X , denoted by S,
≤ 1 such that
and a uniform
 constant 0 < cs 
(s − PX (s))T n1 ni=1 ∇fi (s) ≥ cs  n1 ni=1 ∇fi (s)
s − PX (s) for all s ∈ S.

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LIN et al.: DISTRIBUTED CONTINUOUS-TIME OPTIMIZATION: NONUNIFORM GRADIENT GAINS, FINITE-TIME CONVERGENCE, AND CONVEX

Fig. 1.

Fig. 2.

Fig. 3.

One undirected graph.

Fig. 4.

State trajectories of all agents using (22).

Fig. 5.

State trajectories of all agents using (33).

Fig. 6.

State trajectories of all agents using (34).

State trajectories of all agents using (14).

State trajectories of all agents using (19).

Theorem 6: Suppose that the graph G(t) is undirected and
connected for all t and Assumptions 1, 2 and 5 hold. For system (1) with algorithm (34), all agents reach a consensus and
minimize the team objective function (32) in finite time.
Proof: This theorem can also be proved based on the ideas
of the proofs of Theorems 4 and 5 and hence its proof is
omitted.

V. SIMULATIONS
Consider a multi-agent system with 8 continuous-time agents
in R2 . For the algorithms (14), (19), (22), (33) and (34), the
communication graph is randomly switched among connected
graphs, the union of which is shown in Fig. 1. The local objective
functions are adopted as
1 2
1
x11 + x212 ,
2
2
1
1
f2 (x2 ) = (x21 + 1)2 + x222 ,
2
2

f1 (x1 ) =

1 2
1
x + (x32 + 1)2 ,
2 31 2
1
1
f4 (x4 ) = (x41 + 1)2 + (x42 + 1)2 ,
2
2
1 4
1 4
f5 (x5 ) = x51 + x52 ,
4
4
1
1
f6 (x6 ) = (x61 + 1)4 + x462 ,
4
4
1
1
f7 (x7 ) = x471 + (x72 + 1)4
4
4

f3 (x3 ) =

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2251

2252

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 62, NO. 5, MAY 2017

and

the last two algorithms can be used to deal with a distributed
continuous-time optimization problem with a common convex
constraint set. Our future work will be directed towards the case
of directed graphs with nonuniform convex constraint sets.

1
1
f8 (x8 ) = (x81 + 1)4 + (x82 + 1)4 ,
4
4
where xi1 and xi2 denote the 1st and 2nd components of
xi . The constrained convex set is adopted as H = {s ∈ R2 |
 ≤ 1}. According to Lemma 1, by calculating the
s − [1, 1]T 
solution of ni=1 ∇fi (s) = 0, we have that the team objective
T
function (2) is minimized if and only
nif s = [−0.5, −0.5] .
From the convexity of the function i=1 fi (s), its local optimal point is also its global optimal point when there are
/ H, the team objective
no constraints. Since [−0.5, −0.5]T ∈
function (32) must have at least one optimal
 point at the
boundary of H. By calculating the values of ni=1 fi (s) along
.
the boundary of H, we have that s∗ = [0.2929, 0.2929]T is
one of the optimal
 points of the team objective function
(32). Note that ni=1 ∇fi (s) at s∗ is approximately equal to
of H at s∗ .
[7.5443, 7.5443]T and orthogonal to the
ntangent line
∗
) and y − s∗
Thus, the angle between the
nvectors ∗ Ti=1 ∇fi (s
∗
is smaller than π/2, i.e., i=1 ∇fi (s ) (y − s ) > 0, for any
Hence from the
fi ,
y ∈ H − {s∗ }.
nconvexity∗ ofT the functions

8
8
∗
∗
f
(y)
−
f
(s
)
≥
∇f
(s
)
(y
−
s
)
>
0
for
i
i
i
i=1
i=1
i=1
any y ∈ H − {s∗ }. That is, the team objective function (32)
is minimized if and only if s = s∗ . The simulation results are
shown in Figs. 2–6. We use dash-dot lines to denote the two
components of the optimal state. Specifically, for the algorithms (14) and (19), consensus is reached, respectively, at about
2.1 s and 2.2 s and the team objective function (2) is minimized
as t → +∞. For algorithm (22), consensus is reached at about
2.2 s and the team objective function (2) is minimized at about
2.5 s. For the algorithm (33), consensus is reached at about 1 s
and the team objective function (32) is minimized as t → +∞.
For algorithm (34), consensus is reached at about 1.7 s and the
team objective function (32) is minimized at about 2.6 s. Clearly,
all these simulation results are consistent with the obtained
theorems.
VI. CONCLUSION
In this paper, a distributed continuous-time optimization problem was studied with the consideration of nonuniform gradient
gains, finite-time convergence, and a common convex constraint
set. Six distributed algorithms were given. The first three and
the fifth dealt with a distributed gradient optimization problem
for general differentiable convex local objective functions. The
fourth and the sixth dealt with a distributed finite-time optimization problem using a combination of a distributed tracking
algorithm and a distributed dynamic averaging estimator. For
the first three and the fifth algorithms, it has been shown that
the agents reach a consensus in finite time while minimizing
the team objective function as time evolves. In particular, it
has been shown that the third and the fifth algorithms can be
used to deal with general differentiable convex local objective
functions with nonuniform gradient gains, and their gradient
gains are state-dependent and need not to be known in advance.
For the fourth and the sixth algorithms, it has been shown that
all agents reach a consensus while minimizing the team objective function in finite time. In addition, it has been shown that

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Peng Lin received the B.S. degree in automation from Xidian University, Xi’an, China, in 2004,
and the Ph.D. degree in control theory and applications from Beihang University, Beijing, China,
in 2010.
He is currently a Professor with the School
of Information Science and Engineering, Central South University, Changsha 410083, China.
His research interests include cooperative control, optimization theory, switching control and
robust control.

Wei Ren (F’16) received the Ph.D. degree in
electrical engineering from Brigham Young University, Provo, UT, in 2004.
He is currently a Professor with the Department of Electrical and Computer Engineering,
University of California, Riverside. From 2004
to 2005, he was a Postdoctoral Research Associate with the Department of Aerospace Engineering, University of Maryland, College Park.
He was an Assistant Professor (2005–2010) and
an Associate Professor (2010–2011) with the
Department of Electrical and Computer Engineering, Utah State University. He is an author of two books, Distributed Coordination of Multiagent Networks (Springer-Verlag, 2011) and Distributed Consensus in
Multi-vehicle Cooperative Control (Springer-Verlag, 2008). His research
focuses on distributed control of multi-agent systems and autonomous
control of unmanned vehicles.
Dr. Ren received the National Science Foundation CAREER Award in
2008. He is currently an Associate Editor for Automatica, Systems and
Control Letters, and IEEE Transactions on Control of Network Systems.

2253

Jay A. Farrell (F’08) received B.S. degrees in
physics and electrical engineering from Iowa
State University, and the M.S. and Ph.D. degrees
in electrical engineering from the University of
Notre Dame, Notre Dame, IN.
He is a Professor and two time Chair of the
Department of Electrical and Computer Engineering at the University of California, Riverside.
and author of over 200 technical publications.
He is author of the book “Aided Navigation: GPS
with High Rate Sensors” (McGraw-Hill 2008). He
is also co-author of the books “The Global Positioning System and Inertial Navigation” (McGraw-Hill, 1998) and “Adaptive Approximation Based
Control: Unifying Neural, Fuzzy and Traditional Adaptive Approximation
Approaches” (Wiley, 2006).
Dr. Farrell received the Engineering Vice President’s Best Technical
Publication Award in 1990, and Recognition Awards for Outstanding Performance and Achievement in 1991 and 1993. He was named a GNSS
Leader to Watch for 2009–2010 by GPS World Magazine in May 2009
and a winner of the Connected Vehicle Technology Challenge by the
U.S. Department of Transportation’s (DOT’s) Research and Innovative
Technology Administration in July 2011. He is a Fellow of AAAS, a Distinguished Member of IEEE CSS. He has served the IEEE Control Systems
Society (CSS) as Finance Chair for three IEEE CDC’s (1995, 2001, and
2003), on the Board of Governors for two terms (2003-2006, 2012-2014),
as Vice President Finance and Vice President of Technical Activities, as
CSS General Vice Chair of IEEE CDC-ECC 2011, as General Chair of
IEEE CDC 2012, and as President in 2014.

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