1500

IEEE TRANSACTIONS ON CONTROL OF NETWORK SYSTEMS, VOL. 12, NO. 2, JUNE 2025

Distributed Time-Varying Constrained Convex
Optimization: Finite-Time/Fixed-Time
Convergence
Ge Guo , Senior Member, IEEE, Zeng-Di Zhou , and Renyongkang Zhang

Abstract—This article investigates a distributed timevarying optimization problem with inequality constraints,
aiming to find finite-time and fixed-time convergent solutions free from initialization. A nonsmooth optimization
algorithm for state consensus achieving within a finite or
fixed time is presented by designing a projection-based
log-barrier penalty cost function to meet the constraints
and introducing integral sliding mode subsystems to guarantee zero-gradient-sum. With the use of the projection
idea, the penalized functions are always well defined (i.e.,
satisfying the logarithmic definition) for any system states,
which avoids initializing of certain parameters. An adaptive
gain scheme without any extra global information is presented. The time-varying zero-gradient-sum method here is
feasible for cost functions with nonidentical Hessian matrixes, and applicable to finite-time or fixed-time optimal
consensus tracking. The effectiveness and superiority of
our algorithms are verified with numerical simulations.
Index Terms—Distributed time-varying optimization
(TVO), finite-time/fixed-time convergence, inequality constraints, initialization free, zero-gradient-sum (ZGS).

I. INTRODUCTION
ISTRIBUTED optimization has received a great deal
of attention in recent decades due to its broad application in various systems, such as multirobot collaboration,
connected vehicles [1], [2], etc. In a distributed optimization
task, the goal is to minimize the team cost constructed from
local cost functions via local communications and computation. The major issues concerned in distributed optimization
include nonlinear dynamics [3], interference rejection [4], state

D

Received 28 June 2024; accepted 8 November 2024. Date of publication 6 January 2025; date of current version 20 June 2025. This work
was supported in part by the National Natural Science Foundation of
China under Grant 62173079 and Grant U1808205, in part by the 2024
Hebei Provincial Doctoral Candidate Innovation Ability Training Funding
Project under Grant CXZZBS2024184, and in part by the Fundamental
Research Funds for the Central Universities under Grant N2423049.
Recommended by Associate Editor Y. Wang. (Corresponding author:
Zeng-Di Zhou.)
Ge Guo is with the State Key Laboratory of Synthetical Automation for Process Industries, Northeastern University, Shenyang 110819,
China, and also with the School of Control Engineering, Northeastern University at Qinhuangdao, Qinhuangdao 066004, China (e-mail:
geguo@yeah.net).
Zeng-Di Zhou and Renyongkang Zhang are with the College of Information Science and Engineering, Northeastern University, Shenyang
110819, China (e-mail: zhouzd199912@163.com; zryk1998@163.
com).
Digital Object Identifier 10.1109/TCNS.2025.3526324

constraints [5], [6], etc. Most of the existing results are time
independent, while in many practical optimization tasks, the
objectives evolve over time [7], [8], and hence, there are timevarying optimization (TVO) algorithms, such as dynamic average tracking (DAT) and estimator-based average tracking (EAT)
methods [9].
The above TVO algorithms can achieve asymptotic or exponential convergence, in other words, the optimal trajectory
is obtained when time approaches infinity, which is clearly
not practical for many engineering applications. Therefore,
the DAT and EAT algorithms are extended or strengthened to
yield optimization methods that are finite-time/fixed-time convergent [10], [11], [12]. However, these DAT-based/EAT-based
algorithms still leave much room for improvement, as they are
not strictly finite-time or fixed-time optimal solutions, or need
extra information about the states or their estimators. To name
some, the improved DAT methods in [13] and [14] can achieve
finite-time and fixed-time consensus, but asymptotic optimum.
The extended EAT methods in [15] and [16] can drive the system
to the optimal state in a finite or fixed time but require sharing
of multiple virtual variables.
Recently, strict finite-time and fixed-time convergent distributed TVO methods were derived in [17] and [18] based on
zero-gradient-sum (ZGS) scheme [19]. The algorithm in [17]
achieves finite-time optimization via the use of a nonsmooth
consensus ZGS strategy and local auxiliary subsystems, assuming initial values of local gradients to be given. The algorithm
in [18] achieves fixed-time optimization by integrating the nonsmooth ZGS scheme with a sliding-mode manifold of local
gradients, supposing local Hessian matrixes to be identical. But
these methods cannot handle constraints, especially, inequality
constraints, which are common in practical situations [20],
[21], [22]. For distributed TVO problems with time-varying
inequality constraints, one scheme available is the log-barrier
penalty function-based method [23], which consists of a smooth
log-barrier penalty function designed to convert the constrained
problem into an unconstrained one, and a Hessian-based DAT
algorithm to ensure zero tracking errors. However, this method
requires known initial values of the constraints, which is not
expected in engineering practice. Removing such an initialization limitation is quite challenging for distributed time-varying
constrained optimization. More importantly, this algorithm can
just achieve asymptotic optimization, rather than finite-time or
fixed-time convergent solution.

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GUO et al.: DISTRIBUTED TIME-VARYING CONSTRAINED CONVEX OPTIMIZATION: FINITE-TIME/FIXED-TIME CONVERGENCE

This article aims to develop an initialization-free finitetime/fixed-time convergent algorithm for distributed optimization problems with time-varying cost functions and inequality
constraints. By regarding the TVO problem as a minimization
subproblem of the team cost function and a state consensus
seeking one, a time-varying ZGS framework with a projectionbased log-barrier penalty function is presented. The penalized
scheme here plays the role of converting the inequality constraints problem to an unconstrained one without additional
parameter presetting. A sliding manifold is involved to ensure
ZGS of the penalty cost functions. The nonsmooth ZGS protocol
serves the purpose of consensus seeking within a finite/fixed
time. Interestingly, an adaptive gain scheme is presented without
any extra interaction, which removes the need of knowledge of
certain conditions, such as upper bounds of the gradients’ partial
derivatives. Last but not the least, a continuous approximation
strategy is given to address input chattering due to the nonsmooth
sign functions.
The main contributions are given as follows.
1) Compared to the result in [23], the algorithms presented
here are free from initialization and can achieve zero
tracking error optimal consensus within a finite/fixed time.
2) Different from [17] and [18], the new algorithms can
address distributed TVO problems with time-varying inequality constraints but are free from initial values and
identical Hessians.
3) The gain adaption scheme here only requires exchanging
of the primal states in the network, refraining from further
global information, such as upper bounds of gradient
derivatives, which is superior to the DAT, EAT, and ZGS
methods in [13], [14], [15], [16], [17], and [18] and the
adaptive algorithm in [23].
II. PRELIMINARIES
A. Notations and Graph Theory
Let R, R+ , Rn , and Rn×m , respectively, represent the sets
of real numbers, nonnegative real numbers, n-dimensional real
vectors, and n × m-sized real matrices. Denote by 1n and 0n
the all-one and all-zero n-dimensional column vectors, respectively. I m is an m × m identity matrix. Denote P T (respectively, P −1 ) and P⊗ as the transpose (respectively, inverse)
of matrix P ∈ Rn×n and the Kronecker product of P and
I m . Besides, for matrixes A, B ∈ Rn×m , define A ∗ B as their
Hadamard product. Define a function sigp (z) = |z|p sgn(z),
where z ∈ R, p ∈ R+ , sgn(·) is the standard signum function,
sigp (xn )]T for a vector x ∈ Rn .
and sigp (x) = [sigp (x1 ), . . . ,
n
p 1/p
Denote by p-norm xp =
with p > 0, par√ ( i=1 |xi | )
ticularly, x2 ≤ x1 ≤ nx2 . Define diag(x) ∈ Rn×n as
the diagonal matrix with the main diagonal elements being
the elements of vector x. For a twice differentiable function
f (x, t) : Rn × R+ → R, its partial derivative with respect to
x, i.e., gradient (respectively, with respect to t, i.e., partial
derivative) is denoted as ∇x f (x, t) (respectively, ∇t f (x, t)).
Let ∇xx f (x, t) be the second-order partial derivative (or

1501

Hessian matrix), and ∇xt f (x, t) represent the gradient’s partial
derivative.
An undirected graph is denoted by G = (V, E, A) with node
set V = {1, 2, . . . , n}, edge set E ⊆ V × V, and weighted adjacency matrix A = [aij ] ∈ Rn×n , where aii = 0 and aij = 1 if
(j, i) ∈ E, and aij = 0, otherwise. Denote by Ni = {j|(j, i) ∈
n×n
the LaplaE} the neighbor set of node i, and
Ln = [lij ] ∈ R
cian matrix of G, where lii = j=1 aij and lij = −aij for
i = j. The undirected graph G is connected if for any pair of
nodes there is a path connecting them, where L is positive
semidefinite and its eigenvalues satisfy 0 = λ1 (L) < λ2 (L) ≤
· · · ≤ λn (L) in a nondecreasing order. Define the weighted
incidence matrix D = [dik ] ∈ Rn×|E| : dik = −djk = aij if i <
j for the kth edge (j, i) ∈ E, and dik = 0, otherwise, where
k ∈ {1, 2, . . . , |E|} and |E| is the number of edges. Note that for
an undirected graph, L1n = 0n and L = DDT .
B. Convex and Nonsmooth Analyses
In this section, we recall some properties of convex functions
and important definitions of nonsmooth systems, which will be
used and exploited in our main results.
For a θ-strongly convex (θ > 0) and twice continuously
differentiable function f (·) : Rn → R, the following equivalent
properties
satisfy:
(∇x f (xb ) − ∇x f (xa ))T (xb − xa ) ≥
2
f (xb ) − f (xa ) − ∇x f (xa )T (xb − xa ) ≥
θxb − xa 2 ;
2
θ/2xb − xa 2 ; and ∇xx f (xa ) ≥ θI n . Besides, for a ωLipschitz continuous (ω > 0) function f (·), the following property satisfies: f (xb ) − f (xa )2 ≤ ωxb − xa 2 . Furthermore,
if f (·) is a twice continuously differentiable and Θ-Lipschitz
continuous gradient function with Θ > 0, then the following
equivalent conditions hold: (∇x f (xb ) − ∇x f (xa ))T (xb − xa )
f (xb ) − f (xa ) − ∇x f (xa )T (xb − xa ) ≤
≤ Θxb − xa 22 ;
2
Θ/2xb − xa 2 ; and ∇xx f (xa ) ≤ ΘI n .
Definition 1 (Filippov Solution [24]): Consider the vector
differential equation ẋ = f (x, t), where f : Rn × R → Rn is
measurable and essentially locally bounded. A vector function
x(·) is called its solution on [t0 , t1 ] if x(·) is absolutely continuous on [t0 , t1 ] andẋ ∈ K[f
 ](x, t) for almost all t ∈ [t0 , t1 ],
where K[f ](x, t) := δ>0 μ(N )=0 cof (B(x, δ) − N, t), and

μ(N )=0 denotes the intersection over all sets N of Lebesgue
measure zero.
Definition 2 (Clarke’s Generalized Gradient [24]): For a
locally Lipschitz function V (x) : Rn → R, the generalized gradient of V at x is defined by ∂V (x) = co{lim ∇x V (xi )}|xi →
/ ΩV , where ΩV is the set of measure zero, where the
x, xi ∈
gradient of V is not defined.
Definition 3 (Chain Rule [24]): Let x(·) be a Filippov solution of ẋ = f (x, t) and V (x) : Rn → R be a locally Lipschitz
continuous function, then for almost all t
d
V (x(t)) ∈ Ṽ˙
dt
where Ṽ˙ is the set-valued Lie derivative defined as
Ṽ˙ := ξ∈∂V ξ T K[f ].

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Assumption 1: The topology graph G is undirected and

C. Problem Formulation
Consider an interactive network composed of n agents over
an undirected communication topology G. Suppose that each
agent satisfies the continuous-time dynamics as follows:
ẋi (t) = ui (t), i ∈ V

(1)

where xi ∈ Rm is the local state variable relative to agent i,
and ui ∈ Rm is the control input of agent i. Here, we focus on
distributed TVO problem with time-varying nonlinear inequality
constraints. The goal is to design controllers ui for all agents
using local information and interaction to cooperatively minimize the time-varying summation cost function with inequality
constraints, namely, to track the following optimal trajectory:
y ∗ (t) = arg min
y(t)

n


fi [y(t), t]

i=1

s.t. gi [y(t), t] ≤ 0ki , i ∈ V

where fi [y(t), t] : R × R+ → R are the local cost functions
and gi [y(t), t] : Rm × R+ → Rki are the local inequality constraint functions, which are only available to agent i. For dynamics (1),
problem (2) is equivalent to optimize the global cost
function ni=1 fi [xi (t), t] under inequality and state consensus
constraints, i.e., the following problem:
x∗ (t) = arg min
x(t)

where x(t) ∈ R

A. Distributed Protocol Design
Before deriving our distributed control algorithm, we define the following log-barrier penalty function (penalized cost
function):
L̃i (xi , t) = fi (xi , t) − ρi (t)

ki


log [σi (t) − gij (xi , t)] (4)

j=1

fi [xi (t), t]

i=1

s.t. gi [xi , t] ≤ 0ki , xi = xj
mn

Assumption 2: Both local cost functions fi (xi , t) and inequality constraint functions gi (xi , t) are twice continuously
differentiable with respect to xi , continuously differentiable with
respect to t. In addition, all fi (xi , t) and gi (xi , t) are strongly
convex and convex in xi , respectively, for ∀t ≥ 0.
Assumption 3: There exists at least one y satisfying
gi (y, t) < 0ki for all i ∈ V, t ≥ 0.
n 
T
Remark 1: Under Assumption 1,
i=1 j∈Ni xi (xi −
n
n
2
xj ) = 1/2 i,j=1 aij xi − xj 2 holds, and if i=1 xi = 0m ,


then 1/2 ni=1 j∈Ni xi − xj 22 = xT L⊗ x ≥ λ2 (L)xT x.
Assumption 2 implies that the optimal solution exists and is
unique. Assumption 3 indicates that the Slater’s condition holds
for any time. These assumptions are standard and reasonable in
distributed constrained convex optimization [23].

(2)

m

n


connected.

∀i, j ∈ V

(3)

is the stack of all state vectors xi (t).

D. Some Useful Lemmas
Lemma 1 (See [25]): Letχ1 , χ2 , . . . , 
χn ≥ 0. For all real
n
α
≥
(
values 0 < α ≤ 1,one has ni=1 χα
i 
i=1 χi ) . Also, if
n
n
α
(1−α)
α
( i=1 χi ) .
1 < α < ∞, then i=1 χi ≥ n
Lemma 2 (See [17]): For a Lyapunov stable differential
system ẋ(t) = f (x(t), t) and x(t0 ) = x0 , it is finite-time stable
at the origin if there exists a finite settling time T (x0 ) such that
limt→T (x0 ) x(t) = 0 and x(t) = 0∀t ≥ T (x0 ).
Lemma 3 (See [26]): For a system ẋ(t) = f (x(t), t) : Rn ×
R+ → Rn , if its semipositive-definite continuous Lyapunov
function V (x(t)) : Rn → R satisfies x(t) = 0n ⇔ V (x(t)) =
0, such that V̇ (x(t)) ≤ −κ1 V α (x(t)) − κ2 V β (x(t))∀x(t) ∈
Rn , where κ1 , κ2 > 0, 0 < α < 1, and β > 1, then the origin
is a globally fixed-time stable equilibrium and the settling-time
has an upper bound T = 1/(κ1 (1 − α)) + 1/(κ2 (β − 1)).
Lemma 4 (See [27]): Let Ω ⊂ Rk be a nonempty closed
convex set and PΩ (x) ∈ Ω be the projection of x onto Ω with
PΩ (x) = arg miny∈Ω x − y2 . Then, PΩ (x) − x2 is continuous on x and ∇x (PΩ (x) − x22 ) = −2(PΩ (x) − x).

III. FINITE-TIME METHOD
In this section, a distributed finite-time algorithm is proposed for the time-varying constrained optimization problem
(2) subject to system (1). Before description, some elementary
assumptions are listed as follows.

where gij (xi , t) is the jth component of constraint function gi (xi , t), and ρi (t) ∈ R (respectively, σi ∈ R) is the
time-varying barrier parameter (respectively, slack function)
satisfying

σ̇i = −a1 sigp1 σi − b1 sigq1 σi
(5)
ρ̇i = −a2 sigp2 ρi − b2 sigq2 ρi
with σi (0), ρi (0), a1−2 , b1−2 > 0, 0 < p1−2 < 1, and q1−2 >
1. Note that the domain of L̃i (xi , t) in (4) is Di (t) =
{xi |gi (xi , t) < σi (t)1ki }. In addition, some expressions about
∇x L̃i (xi , t), ∇xt L̃i (xi , t), and ∇xx L̃i (xi , t) can be found
in [23] (there are some differences due to the inconsistent
definition on ρi (t)). Next, a projection-based piecewise function
is designed as follows:

PΩi (t) (xi ), t ≤ tpi
Pi (xi , t) =
(6)
xi ,
t > tpi
where Ωi (t) = {xi |gi (xi , t) ≤ 0ki }, and tpi is the first time
node when gi (xi , t) < σi (t)1ki . The introduction of function
(6) will be explained in Remark 5. According to the piecewise
design, the projection-based penalized cost function is defined
as
Li (xi , t) = fi (xi , t)−ρi (t)

ki


log(σi (t)−gij [Pi (xi , t), t]) .

j=1

(7)
For simplicity, the index (xi , t) of penalized cost functions
Li (xi , t) and L̃i (xi , t) is removed in almost all remaining parts
and only kept in some necessary places.

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GUO et al.: DISTRIBUTED TIME-VARYING CONSTRAINED CONVEX OPTIMIZATION: FINITE-TIME/FIXED-TIME CONVERGENCE

Then, our initialization-free finite-time protocol is designed
as follows:

n


=

ui = φi +|φi |∗sgn[Pi (xi ,t)−xi ]+a3 sig [Pi (xi ,t)−xi ]
(8a)

φi = − [∇xx Li ]−1 [a4 sigp4 (si ) + b4 sigq4 (si )

sgn(xi − xj ) + ∇xt Li ]
+α
j∈Ni

si = ∇x Li + α

 t
0 j∈N
i

∇y fi [ỹ (t), t] +

i=1

p3

+ b3 sigq3 [Pi (xi , t) − xi ]

∗

i=1 j=1

(10)

holds only when y(t) = ỹ ∗ (t). Define the slack constrained
optimization problem as
ŷ ∗ (t) = arg min

i=1

∇y L̃i [ỹ ∗ (t), t]

n


=

n


fi [y(t), t] +

i=1

ki
n 


μij (t) {gij [y(t), t] − σi (t)} (12)

i=1 j=1

where µ is the stack vector of μij , and μij (t) ≥ 0 are the
Lagrangian multipliers. For problem (11), the corresponding
dual function is
h[µ(t)] = inf {Lag[y(t), µ(t)]} .
y(t)

i=1

(9)

(13)

Let µ̂∗ (t) be the optimal Lagrangian multiplier vector of (12).
Under Assumption 2, (11) is a convex optimization problem;
thus, there is the following property:
h[µ(t)] ≤ h[µ̂∗ (t)] = Lag[ŷ ∗ (t), µ̂∗ (t)] =

n


fi [ŷ ∗ (t), t]. (14)

i=1
ρi (t)
.
μ̃ij (t) = σi (t)−g
∗
ij [ỹ (t),t]

For μij (t) = μ̃ij (t), ỹ ∗ (t)
Define
minimizes the Lagrangian function Lag[y(t), µ(t)] due to that
if and only if y(t) = ỹ ∗ (t), and we have the following equation:
∇y Lag[y(t), µ̃(t)]
=

n


∇y fi [y(t), t] +

ki
n 


μ̃ij (t)∇y gij [y(t), t]

i=1 j=1

(15)

Therefore, the dual function h[µ(t)] at point µ̃(t) is
h[µ̃(t)]
=

n


fi [ỹ ∗ (t), t]+

i=1

=
L̃i [y(t), t]

(11)

Lag[y(t), µ(t)]

= 0m .

In this section, the finite-time convergence of system (1) to
the optimal trajectory (2) under the algorithm (8) is established,
and relevant lemmas and theorem are proposed.
Lemma 5: Let ỹ ∗ (t) be the optimal trajectory of uncon
strained optimization problem min ni=1 L̃i [y(t), t] defined by
(4). If Assumption 2 holds, then ỹ ∗ (t) will converge to y ∗ (t) in
(2) within a fixed time Tf1 , where Tf1 will be given in the proof.
Proof: According to the definition of ỹ ∗ (t), one has

n


fi [y(t), t]

i=1

and its Lagrangian function can be expressed as

B. Convergence Analysis

y(t)

n


s.t. gi [y(t), t] ≤ σi (t)1ki , i ∈ V

(8c)

i=1

ỹ ∗ (t) = arg min

σi (t) − gij [ỹ ∗ (t), t]

= 0m

(8b)

where a3−4 > 0, b3−4 > 0, 0 < p3−4 < 1, and q3−4 > 0, and
α ∈ R is a parameter to be designed. Before moving on, the
following assumption is needed for algorithm (8).
Assumption 4: For all the bounded states xi , there exists a
constant δ satisfying [∇xx L̃i ]−1 ∇xt L̃i ∞ ≤ δ.
Remark 2 [Roles of terms in (8)]: In our work, problem
(2) is transformed into an unconstrained distributed consensus
optimization problem by designing the log-barrier penalty cost
function (4). Protocol (8) is designed and executed in three
aspects for this unconstrained problem. Equation (8a) is a fixedtime projection method to avoid initialization, which drives
any initial state to the condition that L̃i (xi , t) is available. An
integral sliding-mode strategy (8c) can enable that the gradient
sum of penalized cost functions tends to zero as si → 0m . The
local-minimization-free ZGS algorithm is composed of (8b) and
(8c) to achieve optimal consensus, where (8b) is a time-varying
design of the ZGS method with a time
drift term ∇xt Li (xi , t)
and a nonsmooth consensus term −α j∈Ni sgn(xi − xj ).
Remark 3 (Assumption analysis): In Assumption 4, we
assume that [∇xx L̃i ]−1 ∇xt L̃i ∞ is bounded, which holds
only if both ∇xt fi (xi , t)∞ and ∇xt gi (xi , t)∞ are bounded.
There is an important class of situations in [23] such that
their bounds are existing. Therefore, the boundedness of
[∇xx L̃i ]−1 ∇xt L̃i ∞ holds for most common boundary constrained optimization problems, and similar conditions appear
in many studies aimed at TVO [9], [14], [17], [18], [28].

which implies that

ki
n 

ρi (t)∇y gij [ỹ ∗ (t), t]

y(t)

sgn(xi − xj )dτ

1503

n


ki
n 

i=1 j=1

fi [ỹ ∗ (t), t] −

i=1

≤ (a)

μ̃ij (t){gij [ỹ ∗ (t), t]−σi (t)}

n


ki
n 


ρi (t)

i=1 j=1

fi [ŷ ∗ (t), t]

(16)

i=1

where
(a) holds according
to (14). There is a fact that
n
n
∗
∗
f
[ŷ
(t),
t]
≤
f
[ỹ
(t), t], and then it follows from
i
i
i=1
i=1

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approach Di (t) until that xi ∈ Di (t). According to the definition
of tpi , Pi (xi , t) will switch to xi once that the local state xi
belongs
to the constraint set Di (t). Consequently, if initial states
(17)
do not satisfy the constraint, then under the protocol (8) and
i=1
i=1
i=1 j=1
design (6), all xi will be within the set Di (t) in time tpi , and
A similar result can be obtained by using the perturbation and
2(1−p3 )/2
2(1−q3 )/2 m(q3 −1)/2
. Similar to case 1), it
b3 (q3 −1)
sensitivity analysis in between problems (2) and (11) (see [23, tpi ≤ T3 = a3 (1−p3 ) +
is concluded that xi ∈ Di (t) for tpi ≤ t ≤ Tfinite , where Tfinite
Thm. 1] for more details), given as follows:
is an enough large but finite time. In summary, all xi from any


ki
n
n
n 

 

initial values will belong to Di (t) in t ≥ tpi and Li → L̃i within


fi [ŷ ∗ (t), t]−
fi [y ∗ (t), t] ≤
μ∗ij (t)σi (t).

a fixed time T3 . The proof is completed.


i=1
i=1
i=1 j=1
Theorem 1: If Assumptions 1–4 hold, system (1) under
(18)
δ+
( > 0) will track the
algorithm
(8) with α ≥ λ̄ [(∇
2
−1 ]
Define a Lyapunov function V1,i = 1/2σi , and then,
min
xx L̃)
∗
(1+p1 )/2
(1−q1 )/2
optimal
trajectory
y
(t)
in
(2)
in
a
finite
time Tc1 without any
−m
from (5), one has V̇1,i = −a1 (2V1,i )
(1+q1 )/2
(·)
represents
the smallest positive
initialization,
where
λ̄
min
b1 (2V1,i )
. Hence, limt→T1 σi (t) = 0, where T1 =
(1−p1 )/2
(1−q1 )/2
eigenvalue
of
(·).
2
2
a1 (1−p1 ) + b1 (q2 −1) . Similarly, limt→T2 ρi (t) = 0 with T2 =
Proof: Here, we first
(1−p2 )/2
 prove that the gradient sum of all pe2
2(1−q2 )/2
nalized cost functions ni=1 ∇x L̃i will converge to zero within
a2 (1−p2 ) + b2 (q2 −1) . Therefore, we have


fixed time under our sliding-mode design (8c), and next give a
n
n





∗
∗
(19) stability analysis that all local states xi will attain consensus in
lim 
fi [ŷ (t), t] −
fi [y (t), t] = 0
t→Tf1 

a finite time.
i=1
i=1
For t ≥ T3 , one has (20) according to Lemma 6. From
where Tf1 = max{T1 , T2 }. This ends the proof.
(8c), we can obtain that ṡi = −a4 sigp4 (si ) − b4 sigq4 (si ).
Lemma 6: Suppose Assumptions 1–3 hold. For system (1)
Next, through a Lyapunov analysis similar to the proof
under protocol (8), from any initial condition, each local state xi
of Lemma 5, it is inferred that si → 0m as t → T3 +
(1−p4 )/2
(1−q )/2
(q4 −1)/2
belongs to the constraint set Di (t) = {xi |gi (xi , t) < σi (t)1ki }
+ 2 4b4 (qm
. For
T4 for ∀i ∈ V, where T4 = a2 4 (1−p
4 −1)
for t ≥ tpi , and limt→T3 Li = L̃i .
n4 )
n
i=1 ∇x L̃i =
i=1 si for t ≥ T3 due to
Proof: The proof is considered in the following two Assumption
n 1,
sgn(x
−
x
)
=
0m , which implies that
that
i
j
i=1
situations.
i
j∈N
n
1) For xi (0) ∈ Di (0), i.e., gi [xi (0), 0] < σi (0)1ki . In this limt→T3 +T4 i=1 ∇x L̃i = 0m .

case, ẋi = φi and Li = L̃i for ∀t ≥ tpi = 0. Next, we
For t ≥ T3 + T4 , ẋi = −[∇xx L̃i ]−1 [α j∈Ni sgn(xi −
prove that L̃i is always well defined.
xj ) + ∇xt L̃i ]. Define Γ = [∇xt L̃T1 , ∇xt L̃T2 , . . . , ∇xt L̃Tn ]T and
From (8), the time derivative of L̃i satisfies
[∇xx L̃]−1 = diag{[∇xx L̃1 ]−1 , [∇xx L̃2 ]−1 , . . . , [∇xx L̃n ]−1 },

˙ x L̃i = −a4 sigp4 (si )−b4 sigq4 (si )−α
∇
sgn(xi − xj ). then a compact form for ẋi is as follows:

−1
j∈Ni
T
αD⊗ sgn(D⊗
ẋ = − ∇xx L̃
x) + Γ .
(22)
(20)
L̃
Since
∇
[x
(0),
0]
is
well
defined
and
x i i

T
α j∈Ni sgn(xi − xj ) is bounded, then according to Choose a Lyapunov candidate V3 [x(t)] = D⊗ x1 ; then, from
˙ L̃ always exists. Therefore, Definition 2, its generalized gradient is given by
the definition of s (8c), ∇
(16) that:


ki
n
n
n 

 



∗
∗
fi [ŷ (t), t] −
fi [ỹ (t), t] ≤
ρi (t).




i

x

i

∇x L̃i is bounded in a finite time. Note that ∇x L̃i is
continuous for t ≥ 0 from (20) but unbounded at the
boundary of Di (t), thus xi ∈ Di (t) in a finite time.
/ Di (0), i.e., gi [xi (0), 0] ≥ σi (0)1ki . In this
2) For xi (0) ∈
case, we will demonstrate that xi will converge to the set
Di (t) within fixed time.
For 0 ≤ t ≤ tpi , Pi (xi , t) = PΩi (t) (xi ). Define a proper Lyapunov function V2,i = 1/2PΩi (t) (xi ) − xi 22 and derive it
along (8a) according to Lemmas 1 and 4, one has
V̇2,i = −[PΩi (xi ) − xi ]T ẋi
= −φTi [PΩi (xi ) − xi ] − |φi | ∗ |PΩi (xi ) − xi |

1+p3
2

1−q3

− m 2 b3 (2V2,i )

1+q3
2

where SGN is the multivalued function defined as
⎧
if z > 0
⎨1,
SGN(z) = [ − 1, 1], if z = 0
⎩
1,
if z < 0.

(23)

(24)

According to Definition 3, we can obtain the set-valued Lie
derivative of V3 as follows:

T
Ṽ˙ 3 [x(t)] =
ξ T D⊗
K[f ]
(25)
T x)
ξ∈SGN(D⊗

1+q3
3
−a3 PΩi (xi )−xi 1+p
1+p3 −b3 PΩi (xi )−xi 1+q3

≤ −a3 (2V2,i )

T
∂V3 = D⊗ [SGN(D⊗
x)]

(21)

from which we can know that within tpi , xi will move toward
the domain Ωi (t). Note that Ωi (t) ⊆ Di (t), namely, xi will

T
x) + Γ] is
where K[f ] = −[∇xx L̃]−1 [αD⊗ SGN(D⊗
set-valued Filippov map of (22).
If Ṽ˙ 3 [x(t)] = ∅, let c̃ ∈ Ṽ˙ 3 [x(t)], which follows that:

−1
T
∇xx L̃
c̃ ≤ −ξ T D⊗
[αD⊗ ξ + Γ]

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the

GUO et al.: DISTRIBUTED TIME-VARYING CONSTRAINED CONVEX OPTIMIZATION: FINITE-TIME/FIXED-TIME CONVERGENCE


≤ −αλ̄min (∇xx L̃)−1 D⊗ ξ22 + δ D⊗ ξ1



≤ − αλ̄min (∇xx L̃)−1 − δ D⊗ ξ1 .

T
x). From (25) and (26), for (29),
SGN[(DB)T⊗ x] = SGN(D⊗
we have the following inequality:

(26)

If V3 [x(t)] = 0, then D⊗ ξ1 ≥ 2 due to the inevitable existence
of a pair states satisfying xi2 − xj2 = 0m and xj2 − xi2 = 0m .
δ+
, thus it follows that c̃ ≤ −D⊗ ξ1 ≤
Since α ≥ λ̄ [(∇
−1 ]
min
xx L̃)
−2, namely, Ṽ˙ [x(t)] ≤ −2. Therefore, we have
3

V3 [x(t)] − V3 [x(t0 )] =

 t
t0

Ṽ˙ 3 [x(τ )]dτ ≤ −2(t − t0 ) (27)

where t0 = T3 + T4 . It should be pointed out that only if xi (t) =
xj (t) for all i, j ∈ V, we have V3 [x(t)] = 0. Then, from (27),
for t ≥ t0 , all local states xi (t) will achieve consensus within
a finite time that is smaller than or equal to V3 [x(t0 )]/(2).
Combining our previous proof that the gradient sum of penalized
cost functions converges to zero within a fixed time, system
(1) will track ỹ ∗ (t) within a finite time Tf2 , where Tf2 =
T
x(t0 )1 /(2). Ultimately, each state xi (t) reaches the
t0 + D⊗
optimal trajectory y ∗ (t) in (2) with zero tracking errors in a
finite time Tc1 = max{Tf1 , Tf2 } owing to that the evolution of
σi (t) and ρi (t) in our log-barrier penalty function design (4) is
independent of xi (t) in algorithm (8). The proof ends.
In Remark 3, we mentioned that Assumption 4 commonly
appears in distributed TVO, but in reality this upper bound
is difficult to obtain accurately. To remove Assumption 4, we
propose the following proposition by introducing a gain adaption
scheme.
Proposition 1: If Assumptions 1–3 hold, all states of system
(1) will asymptotically converge to the optimal trajectory y ∗ (t)
in (2) under the following controllers (i ∈ V):
ui = φi + |φi | ∗ sgn[Pi (xi , t) − xi ] + Pi (xi , t) − xi (28a)
⎡
⎤

φi = −[∇xx Li ]−1 ⎣si + βij sgn(xi −xj )+∇xt Li ⎦ (28b)
si = ∇x Li +

 t
0 j∈N
i

j∈Ni

βij sgn(xi − xj )dτ

β̇ij = sgn(xi − xj 1 ), βij (0) = βji (0) ≥ 1

1505


c̃ ≤ −λ̄min (∇xx L̃)−1 (DB)⊗ ξ22 + δ (DB)⊗ ξ1 (30)
where c̃ ∈ Ṽ˙ 3 [x(t)] for Ṽ˙ 3 [x(t)] = ∅. Since βij ≥ 1, then
(DB) ξ2
(DB)⊗ ξ22 ≥ (DB)⊗ ξ1 . Let χ(t) = (DB)⊗⊗ ξ21 . If there exists a pair states xi2 = xj2 , χ(t) will increase as βi2 j2 according
to (28d). Actually, χ(t) is nondecreasing for any xi (i ∈ V).
There is a finite time t1 such that χ(t1 ) > δ/λ̄min [(∇xx L̃)−1 ]
if all states do not achieve consensus in t ≤ t2 (t2 > t1 ). Then,
for t > t1 , Ṽ˙ 3 [x(t)] < 0; therefore, V3 [x(t)] will converge to
zero eventually, i.e., the system’s states reach consensus. From
an analysis similar to the proof of Lemma 5, it is inferred that
this proposition holds.
Remark 4 (Choice of methods for constrained optimization): The existing distributed methods for handling inequality

constraints include the following:
1) -exactly penalty function method [29];
2) DAT-based penalty function method [5];
3) epigraph transformation [30];
4) our log-barrier penalized cost function design from [23].
Among them, both 1) and 3) can only solve an approximate solution of the optimization problem, not an exact one.
For example, the accuracy of the result obtained from 1) is
positively correlated with the value of . Method 2) estimates
the global penalty dual gradient utilizing distributed consensus
tracking and converges to the optimal solution with zero error.
However, it requires extra variables sharing, resulting in higher
communication cost. Moreover, they are only applied in distributed TIO problems and may not necessarily be feasible to
our topic “TVO.” Fortunately, 4) is a continuous and error-free
scheme, and has been applied in the time-varying system [23].
Consequently, the log-barrier penalty function design (4) is our
best choice to address the inequality constraint.
Remark 5 (Roles of the piecewise function (6) design):

(28c)
(28d)

where Li (xi , t) is defined in (7), and σi (t) and ρi (t)
are revised as σi (t) = ai1 e−bi1 t and ρi (t) = ai2 e−bi2 t with
ai1 , bi1 , ai2 , bi2 > 0, respectively.
Proof: For (28), the proofs regarding constraint satisfaction
and ZGS implementation are similar to those in (8), so it is
omitted here. Next, we show that all states will reach a consensus.
Define a diagonal matrix B = diag[βij ] ∈ R|E|×|E| for (i, j) ∈
E. Note that βij (t) is nondecreasing, thus B is positive definite.
Similar to (22), it follows from (28) that a compact form is given
as:

−1
T
(DB)⊗ sgn(D⊗
x) + Γ .
(29)
ẋ = − ∇xx L̃
Choose a Lyapunov candidate V3 [x(t)] = (DB)T⊗ x1 , its genT
x)] due to that
eralized gradient is ∂V3 = (DB⊗ [SGN(D⊗

Li (xi ) constructed by (4) and (6) is well defined for any states
xi (t). Due to the availability of penalized function (4) being
based on the premise that gij (xi , t) < σi (t), we adopt the projection method to map xi onto a proper domain. Note that for
(4), it is not feasible for xi to be located at the boundary of
Di (t), yet projection cannot be avoided, which prevents us from
directly projecting xi into Di (t). We choose a compromise range
Ωi (t) ⊆ Di (t), to let all xi ∈ Ωi (t) in a period of time, i.e.,
Pi (xi , t) = PΩi (t) (xi ), t ≤ tpi . From Lemma 6, we know that
once xi belongs to Di (t), then gij < σi (t) will always hold
under our protocol (8). Therefore, we set Pi (xi , t) = xi for
t > tpi , which implies that after this moment, Li is equivalent
to L̃i .
Remark 6 (Piecewise protocol): Algorithm (8) seems somewhat complex. To facilitate the implementation of engineering,
we propose a piecewise protocol as follows:
ui = a3 sigp3[PΩi(t) (xi )−xi ]+b3 sigq3[PΩi(t) (xi )−xi ]
for 0 ≤ t ≤ T3

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(31a)

1506

IEEE TRANSACTIONS ON CONTROL OF NETWORK SYSTEMS, VOL. 12, NO. 2, JUNE 2025


ui = −[∇xx L̃i ]

+α

−1

a4 sigp4 (si ) + b4 sigq4 (si )




sgn(xi − xj ) + ∇xt Li , for t > T3

j∈Ni

si = ∇x L̃i + α

 t
0 j∈N
i

sgn(xi − xj )dτ

(31b)

B. Convergence Analysis
(31c)

where these parameters are same as (8). Algorithm (31) is a
simplified form of (8), which directly adopts the projection
method and removes the design (6) and (7). Its proof is omitted.
Note that (31) is a simplification of (8) for the ease of engineering practice, but it does not mean that it has better performance.
They theoretically have the same upper bound on convergence
time. However, due to the segmented design of (31) and the
conservatism of the fixed-time proof, the system under (31) will
require longer convergence time than (8), mainly because for
(31), the controller needs to switch from (31a) to (31b) at time
T3 . Their relationship is similar to the methods in [12] and [31].
Remark 7 (Some comparisons): Both protocols (8) and
(31) can provide finite-time results, and compared to [17], they
can handle time-varying inequality constraints. Our projectionbased initialization-free scheme designed for distributed timevarying constraint optimization can seamlessly integrate into the
method in [23]. In addition, the adaptive gain protocol (28) has
the superiority of avoiding some global information, and does
not require the interaction of extra variables compared to the
algorithm (39) in [23].
IV. FIXED-TIME METHOD
This section gives a distributed fixed-time algorithm for TVO
problem (2) and lists some comparisons with other distributed
TVO methods.
A. Distributed Protocol Design
The fixed-time convergent controller designed for agent i is
as follows:
ui = Φi + |Φi | ∗ sgn[Pi (xi , t) − xi ]
+ a3 sigp3 [Pi (xi , t)−xi ]+b3 sigq3 [Pi (xi , t)−xi ] (32a)
Φi = − [∇xx Li ]

−1

p4

q4

T
T
ẋ = − [∇xx L̃]−1 a5 D⊗ sigp5 (D⊗
x)+b5 D⊗ sigq5 (D⊗
x)
T
x) +Γ .
+ γD⊗ sgn(D⊗

(33)

T
x1 as its Lyapunov candidate; then,
We choose V3 [x(t)] = D⊗
˙
if Ṽ [x(t)] = ∅, it satisfies
3




Ṽ˙ 3 ≤ − γ λ̄min (∇xx L̃)−1 − δ D⊗ ξ1

T
T
a5 D⊗ sigp5 (D⊗
− λ̄min (∇xx L̃)−1 ξ T D⊗
x)
T
x)
+ b5 D⊗ sigq5 (D⊗


 T q5
T  p5
x p 5 + b5  D ⊗
xq5 )
≤(a) −λ̃(a5 D⊗

(34)

where γ ≥ λ̄ [(∇δ L̃)−1 ] and λ̃ = λ̄min [(∇xx L̃)−1 ]λ̄min (DT D).
min
xx
Next, we will discuss why the inequality (a) holds.
Under Assumptions 1 and 5, |E| = n − 1 or n. Based on the
undirected connectivity of the graph G, rank(D) = rank(L) =
n − 1. For |E| = n − 1, i.e., rank(D) = |E|, then λ1 (DT D) >
0. It is obtained that P T (DT D)P = Λ from matrix diagonalization transformation, where P = [p1 , p2 , . . . , p|E| ] and Λ =
diag{λ1 (DT D), λ2 (DT D), . . . , λ|E| (DT D)}. There exists the
following result (η ∈ R|E| ):
η T (DT D)η = η T P ΛP T η =

|E|


λi (DT D)η T pi pTi η

i=1

≥ λ1 (D D)η P P T η = λ1 (DT D) η22 . (35)
(32b)

j∈Ni

+ γsgn(xi − xj )]
 t
si = ∇x Li +
i dτ

In this section, two fixed-time convergence results are presented for system (1) tending to the optimal trajectory (2).
Theorem 2: If Assumptions 1–5 hold, for system (1) under
controllers (32), all states xi (t) will converge to the optimal
trajectory y ∗ (t) in (2) within a fixed time Tc2 .
Proof: According to the proof of Theorem 1, one has a
compact form of (32a) for t ≥ T3 + T4

T

[a4 sig (si ) + b4 sig (si )

+ i + ∇xt Li ]

[a5 sigp5 (xi − xj ) + b5 sigq5 (xi − xj )
i =

Remark 8 (Assumption analysis): For an undirected graph,
at least n − 1 edges are needed to make it connected. Under
Assumption 5, if a graph G is undirected and connected, then
|E| = n − 1 or n. Thus, Assumption 5 does not conflict with
Assumption 1.

(32c)
(32d)

0

where a3−5 > 0, b3−5 > 0, 0 < p3−5 < 1, and q3−5 > 0, and
γ ∈ R is a parameter to be designed. For achieving fixed-time
convergence, an additional assumption is needed.
Assumption 5: The number of edges E for graph G is less
than or equal to the number of nodes V, i.e., |E| ≤ n.

T

Similarly, for a > 0, one has SGNT (η)(DT D)siga (η) ≥
λ1 (DT D)SGN(η)T P P T siga (η) = λ1 (DT D)ηaa . Therefore,
the inequality (a) holds for system (1) with n − 1 edges.
For |E| = n, rank(D) < |E|, then λ2 (DT D) > λ1 (DT D) =
0. Define nodes with only one edge adjacent as alone nodes, and
call nodes connected from the beginning to end as a hoop. For
example, if the communication topology of a graph is 1 ↔ 2 ↔
3 ↔ 4 ↔ 2, then node 1 is an alone node and nodes 2–4 form a
hoop. Next, we consider the following two scenarios.
1) If there are no alone nodes, then all nodes constitute a
hoop. Without losing generality, D = [di,j ] ∈ Rn×n with di,i =
1, dk+1,k = d1,n = −1 (k ∈ {1, . . . , n − 1}) and di,j = 0, otherwise. The unit eigenvector corresponding to its zero eigenvalue

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GUO et al.: DISTRIBUTED TIME-VARYING CONSTRAINED CONVEX OPTIMIZATION: FINITE-TIME/FIXED-TIME CONVERGENCE

√
is p1 = 1/ n1n . If 1Tn η = 0, namely, pT1 η = 0, then the following inequality holds:
η T (DT D)η =

n

i=2

≥ λ2 (DT D)

n

i=1

η T pi pTi η = λ2 (DT D) η22 .
(36)

Similarly, SGNT (η)(DT D)siga (η) ≥ λ2 (DT D)ηaa . Note
that Dp1 = 0n always holds in this case, which means that
T
x = 0. Consequently, here (a) holds.
(p1 )T⊗ D⊗
2) If there exists r alone nodes (r ∈ {1, 2, . . . , n − 3}), then
D has r row vectors that contain only one nonzero element (1 or
−1). Their corresponding r elements in unit eigenvector p1 of
DT D are zero. Note that other n − r nodes constitute a hoop, let
D ∈ R(n−r)×(n−r) as its incidence matrix without considering
the connection of alone nodes. Similar
to the analysis of 1), the
1/ (n − r)1n−r . Without
unit eigenvector of (D )T D is p1 = 
losing generality, choosing p1 = 1/ (n − r)[0Tr , 1Tn−r ]T , one
T
x = 0. Refer to the result of case 1), we are
still has (p1 )T⊗ D⊗
informed that (a) also holds for case 2).
Above all, for Assumptions 1 and 5, if V3 [x(t)] = 0, we can
obtain
 T  p5
 T q5
Ṽ˙ 3 ≤ −λ̃(a5 D⊗
xp5 + b5 D⊗
xq5 )
≤ −λ̃(a5 V3p5 + b5 (mn)(1−q5 ) V3q5 ).

(37)

Since V3 is locally continuous differentiable and only nondifferentiable at some points, (35) similarly applies to Lemma
3. Consequently, all states xi (t) will converge to the optimal
solution ỹ ∗ (t) of the unconstrained optimization problem (9)
(q5 −1)
1
+ (mn)
.
in a fixed time Tf3 , where Tf3 = t0 + λ̃a (1−p
λ̃b5 (q5 −1)
5
5)
Eventually, according to Lemma 5, the optimal trajectory y ∗ (t)
in (2) will be tracked within fixed time Tc2 = max{Tf1 , Tf3 }.
This completes the proof.
However, Theorem 2 may not be available for |E| > n. Next,
we give a result without the limitation of edges E, which requires
the following assumption.
Assumption 6: There exists a positive constant ω such
that ∇t L̃i (x, t) − ∇t L̃i (y, t) ≤ ωx − y2 for bounded x, y ∈
Rm , i.e., ∇t L̃i is ω-Lipschitz continuous.
Proposition 2: With Assumptions 1–3 and 6, all states for
system (1) will track the optimal trajectory y ∗ (t) of problem
(2) without the chattering of input in a fixed time, under the
following controllers (i ∈ V):
ui = Φi + |Φi | ∗ ĥ[Pi (xi , t) − xi ]
+ a3 sigp3 [Pi (xi , t)−xi ]+b3 sigq3 [Pi (xi , t)−xi ] (38a)
Φi = − [∇xx Li ]−1 [a4 sigp4 (si ) + b4 sigq4 (si )
+ i + ∇xt Li ]

[a5 sigp5 (xi − xj ) + b5 sigq5 (xi − xj )
i =
j∈Ni

+ γ  ĥ(xi − xj )
si = ∇x Li +

λi (DT D)η T pi pTi η

(38b)

1507

 t
0

(38c)

i dτ

(38d)

z1
z2
m
where ĥ(z) = [ |z1 |+e(t)
, |z2 |+e(t)
, . . . , |zmz|+e(t)
]T , z ∈ Rm is
the continuous approximation of sgn(·), ė = −a6 sigp6 (e) −
b6 sigq6 with e(0), a3−6 , b3−6 > 0, 0 < p3−6 < 1, and q3−6 >
0, and γ  ∈ R is a parameter to be designed.
Proof: According to the design of ĥ, it is known that e(t) → 0
(1−p6 )/2
(1−q )/2
as t → T5 with T5 = a2 6 (1−p
+ 2b6 (q66−1) , which indicates that
6)

ĥ will be approximately equivalent to the sign function. Next,
we prove that for t ≥ T3 + T4 + T5 , all xi → ỹ ∗ (t) in a fixed
time.

Choose a Lyapunov candidate V4 = ni=1 [L̃∗i − L̃i −
∗
∇x L̃i (ỹ ∗ − xi )], where L̃∗i represents
n L̃i (ỹ ,∗t). Taking its time
derivative, one can obtain V̇4 = i=1 [∇t L̃i − ∇t L̃i + (xi −
ỹ ∗ )T ∇xx L̃i ẋi ], where ẋi evolves along the following trajectory:


−1
ẋi = − [∇xx L̃i ]
a5 sigp5 (xi − xj )
j∈Ni
q5




+ b5 sig (xi − xj ) + γ ĥ(xi − xj )

.

(39)



Let V4a = ni=1 [∇t L̃∗i − ∇t L̃i ] and V4b = ni=1 [(xi −

ỹ√∗ )T ∇xx L̃i ẋi ], then we have V4a ≤ ω ni=1 xi − ỹ ∗ 2 ≤
nωx̂2 , where x̂ = [x̂T1 , x̂T2 , . . . , x̂Tn ]T with x̂i = xi − ỹ ∗ .
Substituting (37) into V4b yields
n 

a5
b5
5
5
xi −xj 1+p
xi −xj 1+q
V4b = −
1+p5 +
1+q5
2
2
i=1 j∈Ni

γ
+ xi − xj 1 .
(40)
2

, x̄T2 , . . . , x̄Tn ]T . AcDenote x̄i = xi − n1 ni=1 xi and x̄ = [x̄T1
T
cording to [25], one has x̄ L⊗ x̄ = 1/2 ni=1 x̄i − x̄j 22 ≥
λ2 x̄T x̄ for Assumption 1, thus V4b follows from Lemma 1 that:
1+p5

V4b ≤ −

a5 λ2 2

1−p5
2

5
x̄1+p
−
2

2
√
γ  λ2
− √ x̄2 .
2

q5 −1

1+q5

2 2 b5 λ 2 2
[n (n − 1) m]

q5 −1
2

1+q5

x̄2 2

(41)

n
Notice that for ∀t ≥ T3 + T4 + T5 , since
i=1 ∇x L̃i =
n
− L̃i (xi , t) − ∇x L̃i (xi , t)(ȳ − xi )] −
0m , then i=1 [L̃i (ȳ, t)
V4 ≥ 0, where ȳ = 1/n ni=1 xi . Hence
V4 ≤

1

x̄22
−1
2λ̄min (∇xx L̃)

(42)

where 1/λ̄min [(∇xx L̃)−1 ] is equivalent to the Lipschitz gradient constant of L̃i (xi , t) with respect to xi . Similarly,
V4 ≥ λ̄min (∇xx L̃)/2x̂22 . For simplicity, next we replace
λ̄min [(∇xx L̃)−1 ] with λ̄ and λ̄min (∇xx L̃) with λ. From (39)

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and (40), one has

following variant:

V4b ≤ −2p5 (λ̄λ2 )
−



1+p5
2

1+p5

a5 V4 2 −

2q5 (λ̄λ2 )

1+q5
2

[n(n−1)m]

b5

q5 −1
2

1+q5

V4 2

1

λ̄λ2 γ  V42

√

+ a3 sigp3 [Pi (xi , t)−xi ]+b3 sigq3 [Pi (xi , t)−xi ] (46a)
(43)

√

1

2nω
V42 . Thus, by choosing γ  ≥ √2nω , we have
and V4a ≤ √
λ

λ̄λλ2

V̇4 ≤ −2p5 (λ̄λ2 )

1+p5
2

1+p5

a5 V4 2 −

1+q5

2q5 (λ̄λ2 ) 2 b5
[n(n−1)m] 2

T6 =

(λ̄λ2 )

1+p5
2

a5 (1 − p5 )

+

(λ̄λ2 )

1+q5
2

b5 (q5 − 1)

+ νi + ∇xt Li ]

[a5 sigp5 (xi − xj ) + b5 sigq5 (xi − xj )
νi =

(46b)

+ γsgn(xi − xj )]
p4

(46c)
q4

ṡi = −a4 sig (si )−b4 sig (si ), si (0) = ∇x Li (xi (0),0).
(46d)
All parameters are same as (32).

q5 −1

[n(n − 1)m] 2 21−q5

ϕi = − [∇xx Li ]−1 [a4 sigp4 (si ) + b4 sigq4 (si )

j∈Ni

1+q5

2
. (44)
q5 −1 V4

According to Lemma 3, limt→Tf  xi (t) = ỹ ∗ (t) with Tf 3 =
3
T3 + T4 + T5 + T6 , where
21−p5

ui = ϕi + |ϕi | ∗ sgn[Pi (xi , t) − xi ]

. (45)

Consequently, the time-varying constrained optimization problem (2) is solved within fixed time Tc2 = max{Tf1 , Tf 3 }. This
proof is completed.
Remark 9 (Effectiveness of our continuous approximation strategy ĥ(z)): Let the ith element of ĥ(z) be [ĥ(z)]i .

There are the following results that we are informed:
1) [ĥ(z)]i → 1 (or −1) for zi  e(t) > 0 (or zi  −e(t) <
0);
2) [ĥ(z)]i → 1/2 (or −1/2) for zi → e(t) (or zi → −e(t));
3) [ĥ(z)]i → 0 for −e(t)  zi  e(t).
Suppose that the initial states of system (1) satisfy |xi (0) −
xj (0)|  e(0)I m , then all xi (t) will be close to each other under
our method (38) until |xi (t) − xj (t)| approaches e(t). Although
each state has not been equal, xi − xj → 0m as e(t) → 0.
Namely, all xi (t) will be consensus over time. In the process
of system states approaching each other, [ĥ(z)]i is continuously
close to 0 from 1 or -1, instead of jumping like sgn(·). Therefore,
ĥ(z) can effectively avoid input chattering.
Remark 10 [Difference of our fixed-time methods (32) and
(38)]: Algorithms (32) and (38) are roughly the same in form,

and their difference is mainly reflected in the assumptions,
which affects the values of compensation parameters γ and γ  .
Controllers (32) are applicable for the case that the number
of edges in the communication graph is less than or equal to
the number of nodes and [∇xx L̃i ]−1 ∇xt L̃i is bounded. The
difference is that (38) is still feasible for |E| > n, but there is
a special assumption on penalized cost functions that all ∇t L̃i
are Lipschitz continuous and the constant ω is known, which is
difficult to obtain in practice. In addition, (38) is a continuous
algorithm, whose approximation strategy avoids the chattering
of control inputs.
Remark 11 (Variant from the work [17]): Compared with
our sliding-mode scheme, the design of local auxiliary subsystems in [17] requires preallocation of initial gradients si (0) =
∇x Li (xi (0), 0). In engineering, such initialization is unreasonable or at least unsatisfactory. However, the characteristic of such
subsystem is that the gradient no longer needs to be calculated
after the initial allocation. Focusing on this feature, (32) has the

Remark 12 (Discussion on extending our algorithms to
time-synchronized optimization): It is worth noting that

finite-time/fixed-time stability cannot guarantee that all agents
reach equilibrium at the same time. However, some practical
missions require all agents to arrive synchronously, namely,
achieve time-synchronized optimization [32], [33], [34]. To
avoid staying on a fake equilibrium state that partial agents are
consensus, one potential solution is the switching terminal sliding mode based on the norm-normalized sign function in [32],
where the switching condition is determined by the defined
trigger sliding-mode variable. In this article, finite-time and
fixed-time convergences are established under the classical sign
function. This poses a challenge for us to overcome the situation
where norm-normalized sign functions are not available in our
method. Furthermore, the different control objectives require
us to improve the switching law and reselect appropriate trigger
sliding-mode variable. It is meaningful to expand our algorithms
to time-synchronized optimization, which is our next work.
Remark 13 (Some discussions of our algorithms and (6)
in [23]): There are results in this article that can achieve

finite-time and fixed-time convergence without initialization.
The faster performance owes to functions sig(·) and sliding
mode variables si (t). In which, sigq (xi − xj ) can quickly reduce the difference between states xi and xj to around 1, while
sigp (xi − xj ) can quickly drive their difference from 1 to 0.
It is worth mentioning that both sig(xi − xj ) and si do not
require extra information exchange and only share the primal
state among agents, as the same as [23, (6)]. Therefore, proposed protocols do not need more information. However, there
are some methods achieving faster convergence through more
information sharing, such as the distributed estimators-based
method in [16], resulting in high communication cost and low
privacy. This is undesired.
As for computational complexity, our algorithms add some
basic operations, such as projection, and “if” statements, on the
basis of [23, (6)], but does not add or remove any loop. That
is to say, their time complexity is of the same level. However,
algorithms in this article define more intermediate variables than
[23, (6)], so they have a greater spatial complexity. The way
of sacrificing space for better performance is common [35].
Increased complexity may impose a burden on computers and

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GUO et al.: DISTRIBUTED TIME-VARYING CONSTRAINED CONVEX OPTIMIZATION: FINITE-TIME/FIXED-TIME CONVERGENCE

1509

TABLE I
COMPARISON OF SEVERAL DISTRIBUTED TVO METHODS

make it unsuitable for practical applications, which deserves
further research and improvement.
Moreover, compared to most existing TVO methods [9],
[17], [18], all algorithms in this article can handle inequality
constraints. In Table I , we give a clear comparison from the perspectives of whether it can be applied to constrained problems,
convergent form, the need of initialization, the dimension of
required sharing variables, and whether it contains boundedness
assumptions.

Fig. 1. Evolutions of states xi (t) under finite-time algorithm (8). The
blue dashed line is the optimal solution and the other solid lines are the
state trajectories of all agents. (a) States x1i ; (b) States x2i .

V. NUMERICAL SIMULATIONS
In this section, we consider a network composed of
five agents with m = 2. Their communication topology
is the undirected graph as 1 ↔ 2 ↔ 3 ↔ 4 ↔ 5 ↔ 1. Let
xi (t) = [x1i (t), x2i (t)]T ∈ R2 represent the state of agent i,
i ∈ {1, 2, . . . , 5}. Each agent i is equipped with a timedependent local cost function fi = 12 [x1i (t) + i cos(t)]2 +
3 2
2
2 [xi (t) − i sin(t)] . The time-varying inequality constraint
functions are denoted by gi = x2i (t) − x1i (t) − 5i e−t . Next, we
will verify the effectiveness of our proposed algorithms and their
superiorities in convergence speed.

Fig. 2. Evolutions of states xi (t) under adaptive gain algorithm (28).
The blue dashed line is the optimal solution and the other solid lines are
the state trajectories of all agents. (a) States x1i ; (b) States x2i

A. Effectiveness Verification
In this section, we aim to validate that our methods can
solve the constrained optimization problem (2) from any initial states and without any initialization, which is different
from [23]. All parameters and some variables’ initial values in our protocols are set as a1−6 = b1−6 = 1, p1,3−5 =
0.5, q1,3−5 = 1.5, p2,6 = 0.9, q2,6 = 1.1, α = γ = γ  = 6, and
σi (0) = 5, ρi (0) = 0.1. According to Theorem 2 and Proposition 2, the upper bounds of the settling time with protocols
(32) and (38) are, respectively, calculated as Tc2 = 26.8279 and

= 37.1598 s. Choose initial states arbitrarily for agents, here
Tc2
are xi (0) = [9 − 3i, 3i − 9]T .
/ Di (0) from the above, thus the
We are informed that x1 (0) ∈
method in [23] is infeasible under our settings. However, it is
applicable for our projection-based initialization-free algorithms
(8), (28), (31), (32), and (38) from the simulation results. As
shown in Figs. 1–4, we can clearly see that all the agents track
the optimal trajectory y ∗ (t). The constraint result is indicated
in Fig. 5, taking (8) as an example, which shows that in our

Fig. 3. Evolutions of states xi (t) under piecewise algorithm (31). The
blue dashed line is the optimal solution and the other solid lines are the
state trajectories of all agents. (a) States x1i ; (b) States x2i

algorithms, once the local state satisfies the slack constraint gi <
σi (t), it will always satisfy. From the results of Figs. 1 and 3, it
can be clearly seen that under the piecewise protocol (31), the
system converges slower, as analyzed in Remark 6. Moreover,
we give the plots of inputs under discontinuous and continuous
protocols, respectively, in Fig. 6, which demonstrates that there

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IEEE TRANSACTIONS ON CONTROL OF NETWORK SYSTEMS, VOL. 12, NO. 2, JUNE 2025

Fig. 7. Evolutions of tracking errors
gorithms in [23], (8), and (32).

Fig. 4. Evolutions of states xi (t) under fixed-time algorithms (32) and
(38). The blue dashed line is the optimal solution and the other solid
lines are the state trajectories of all agents. (a) and (b) States under
protocol (32). (c) and (d) States under protocol (38).

5
i=1

xi (t) − y ∗ (t)1 under al-

of consistency, modify σi (0) = 10 in (8) and (32). Set xi (0) =
[3i − 9, 9 − 3i]T . Other parameters remain consistent with the
above. Note that under this setting, Li = L̃i for any t ≥ 0, then
(8) and (32) can be simplified. Fig. 7 shows the evolutions
of tracking errors under [23, (6)], the reduced (8) and (32)
with same settings. The result indicates that our finite-time and
fixed-time convergent methods solve the problem (2) in t = 3.5
s, but the asymptotic method (6) of [23] using more time, about
t = 4.6 s. Although there is almost the same tracking time under
(8) and (32), which is likely due to our similar design for si (t),
the fixed-time method can achieve state consensus faster from
Figs. 5 and 3. Consequently, there is better performance for our
proposed algorithms compared to [23, (6)].

Fig. 5. Curves of the inequality constraints with system (1) under
algorithm (8).

VI. CONCLUSION

Fig. 6. Illustration of control inputs ui (t) under algorithms (32) and
(38). (a) and (b) Inputs under protocol (32). (c) and (d) Inputs under
protocol (38).

is no chattering of ui (t) for system (1) under the continuous
approximation algorithm (38).
B. Performance Comparison
In this section, we conduct comparative simulations between
the finite-time/fixed-time methods (8), (32), and the asymptotic
method (6) in [23]. Keeping the initial states of the system,
set σi (t) = 10e−t for [23, (6)] to meet the initial constraint
condition gi (xi (t), t) − σi (t) < 0. According to the principle

This article proposes projection-based initialization-free
distributed time-varying constrained optimization algorithms,
which provide finite-time and fixed-time solutions. They are
composed of the penalized cost function design that can convert constrained TVO into unconstrained one for solution, an
integral sliding mode for driving the system’s gradient to zero
sum, and the nonsmooth time-varying ZGS algorithm that seeks
consensus. To relax the upper bound requirement, we design
an adaptive gain method without extra variables. Furthermore,
an approximation strategy for the nonsmooth sign function is
proposed to enable the control input continuous.
Future research content comprises the generalization of this
work to high-order system, nonconvex optimization, etc. It is
hard to obtain Hessian of the cost function in practical scenarios,
and there are no results that are free from Hessian in TVO
methods, which is undoubtedly an interesting challenge.
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Ge Guo (Senior Member, IEEE) received the
B.S. degree in automation instrumentation and
devices and the Ph.D. in control theory and control engineering from Northeastern University,
Shenyang, China, in 1994 and 1998, respectively.
From 2000 to 2005, he was with the Lanzhou
University of Technology, China, as a Professor,
and the Director with the Institute of Intelligent
Control and Robots. He then joined Dalian Maritime University, Dalian, China, as a Professor
with the Department of Automation. Since 2018, he has been a Professor with Northeastern University and the Dean with the School of
Control Engineering, Qinhuangdao Campus. He has authored or coauthored more than 190 international journal papers within the areas of
his research interests, which include intelligent transportation systems,
cyber-physical systems, etc.
Dr. Guo is an Associate Editor for IEEE TRANSACTIONS ON INTELLIGENT TRANSPORTATION SYSTEMS, IEEE TRANSACTIONS ON VEHICULAR
TECHNOLOGY, IEEE TRANSACTIONS ON INTELLIGENT VEHICLES, Information Sciences, IEEE Intelligent Transportation Systems Magazine, ACTA
Automatica Sinica, China Journal of Highway and Transport, and Journal of Control and Decision. He was the recipient of the New Century
Excellent Talents in University, Ministry of Education, China, in 2004,
and a nominee for Gansu Top Ten Excellent Youths by the Gansu
Provincial Government in 2005, the CAA Young Scientist Award in 2017,
and the First Prize of Natural Science Award of Hebei Province in 2020.

Zeng-Di Zhou received the B.S. degree in measurement and control technology and instrument from Northeastern University, Qinhuangdao, China, in 2021. He is currently working
toward the Ph.D. degree in control science and
engineering with the College of Information Science and Engineering, Northeastern University,
Shenyang, China.
His current research focuses on control and
optimization of multiagent systems.

Renyongkang Zhang received the B.S. degree in automation from Northeastern University, Qinhuangdao, China, in 2020. He is currently working toward the Ph.D. degree in control science and engineering with the College
of Information Science and Engineering, Northeastern University, Shenyang, China.
His current research focuses on control and
optimization of multiagent systems, with application to intelligent transportation.
Dr. Zhang was the recipient of the Best Paper
Award at the 5th International Conference on Industrial Artificial Intelligence (IAI 2023).

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