# Multi/Single-stage structured zero-gradient-sum approach for prescribed-time
optimization

Shuaiyu Zhou, Yiheng Wei, Jinde Cao, and Yang Liu Shuaiyu Zhou, Yiheng Wei and
Jinde Cao are with the School of Mathematics, Southeast University, Nanjing
211189, China. J.D. Cao is also with the Yonsei Frontier Lab, Yonsei
University, Seoul 03722, South Korea (e-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>jdcao@seu.edu.cn). Yang Liu is with the Key Laboratory of
Intelligent Education Technology and Application of Zhejiang Province, School
of Mathematical Sciences, Zhejiang Normal University, Jinhua 321004, China
(e-mail<EMAIL_ADDRESS>
###### Abstract

Prescribed-time convergence mechanism has become a prominent research focus in
the current field of optimization and control due to its ability to precisely
control the target completion time. The recently arisen prescribed-time
algorithms for distributed optimization, currently necessitate multi-stage
structures to achieve global convergence. This paper introduces two modified
zero-gradient-sum algorithms, each based on a multi-stage and a single-stage
structural frameworks established in this work. These algorithms are designed
to achieve prescribed-time convergence and relax two common yet stringent
conditions. This work also bridges the gap in current research on single-stage
structured PTDO algorithm. The excellent convergence performance of the
proposed algorithms is validated through a case study.

###### Index Terms:

Consensus, Distributed convex optimization, Prescribed-time convergence,
Sliding mode control, Zero-gradient-sum.

##  I Introduction

Distributed optimization, focusing on achieving global objectives through
cooperation between multi-agent systems, has been extensively researched in
the past decade [1, 2]. Existing studies primarily tackled distributed
optimization for convex functions, particularly gradient-descent based methods
[3, 4, 5]. However, their reliance on local gradients necessitated diminishing
step-sizes for global exact convergence, leading to slower convergence rates.
Subsequently, new algorithms, such as proportional-integral based methods[6,
7], gradient tracking [8, 9, 10, 11], and zero-gradient-sum (ZGS) [12, 13,
14], were introduced to achieve global exact convergence with fixed step-
sizes.

However, most of these algorithms have unbounded theoretical convergence
times, limiting their practicality in those engineering problems with time
constraints. In recent years, a new category of distributed optimization
algorithms has emerged, guaranteeing bounded convergence times. Initial
efforts included finite-time convergence distributed algorithms [15, 16] based
on non-smooth formulations, but they still had unbounded bounds associated
with initial states of the system. To address this, fixed-time convergence
algorithms were developed [17, 18, 19], providing convergence time bounds
independent of initial states but reliant on system control parameters.
However, establishing a direct relationship between parameter selection and
desired convergence time still remains challenging [20]. Recently,
optimization research has centered on developing prescribed-time algorithms,
which enable the deliberate setting of convergence times without relying on
additional parameters. While initially applied in fields like system control
[21], they’ve only recently been integrated into distributed optimization
algorithms. Presently, prescribed-time algorithms for distributed optimization
(PTDO) take three primary forms: non-smooth formulation, time-base generator
functions [22], and time-varying scaling functions [23]. Of these, the latter
two, known for their innovation and efficiency, have gained more attention.

The goal of this paper is to construct high-performance and user-friendly ZGS
algorithms for PTDO in multi-stage/single-stage structures based on the
aforementioned time-varying scaling functions. The motivations behind this
research can be summarized into the following two main aspects,

  * 1)

current multi-stage prescribed-time ZGS methods involve separate stage
executions for tasks like ‘local minimization’ and ‘global convergence’, which
may introduce additional complexity and overhead. Hence, exploring multi-stage
ZGS algorithms without explicit stage separation is valuable.

  * 2)

a research gap currently exists regarding single-stage prescribed-time ZGS
algorithms. This is typically attributed to the initial value selection for
ZGS, frequently requiring an extra stage in algorithm design or analysis to
address this concern. Thus, developing a single-stage ZGS algorithm that
ensures both ‘initial condition satisfaction’ and ‘global prescribed-time
convergence’ in a single time interval addresses this gap.

The contributions of this paper are threefold.

  * 1)

Multi/Single-Stage Structure: This paper conducts a comprehensive study and
review of recent ZGS algorithms with different convergence rates, providing a
unified framework to incorporate these algorithms within the proposed
multi/single-stage structures.

  * 2)

Multi-Stage ZGS Algorithm: An implicit multi-stage ZGS algorithm is proposed
in this paper, allowing for prescribed-time convergence without requiring any
initial value constraints. The incorporation of the implicit multi-stage
structure enhances the algorithm’s applicability.

  * 3)

Single-Stage ZGS Algorithm: An novel single-stage ZGS algorithm is proposed in
this paper, enabling prescribed-time convergence without the need for initial
value constraint. Importantly, this algorithm doesn’t demand compliance with
another typical constraint found in ZGS algorithms. Unlike many related
studies, this algorithm is single-stage in both protocol design and
convergence analysis, bypassing the prerequisite of satisfying initial value
conditions before conducting convergence analysis.

The rest of this paper is organized as follows. Section II presents the
objective problem and introduces several fundamental concepts. In Section III,
the multi/single-stage structure and two novel algorithms are introduced, and
their prescribed-time convergence properties are analyzed. Section IV contains
the simulation results that illustrate the performance of the proposed
algorithms. Lastly, Section V provides the concluding remarks and reflections.

Notation: Throughout the paper, $\|\cdot\|$ denotes the 2-norm for vectors,
and the Frobenius norm for matrices. $\nabla^{2}f$ denotes the Hessian matrix
of the twice differentiable function $f$. $I_{N}$ is the identity matrices of
$N\times N$ and $\mathbf{1}_{N}$ is a vector of ones. $\otimes$ represents the
Kronecker product. $\mathcal{N}_{i}$ is the set of neighbor nodes of agent $i$
in a multi-agent system. $L^{\top}$ means the transpose of matrix $L$.
$\chi^{(i)},i=0,1,2$ represents the $i$th order derivative of $\chi$.

##  II Preliminaries

###  II-A Prescribed-time stability

Time-varying scaling function The following definition of time-varying scaling
function is first given, which is the core of the design of a distributed
optimizaiton with prescribed-time convergence performance.

######  Definition 1.

[23, 24] The time-varying scaling functions used here are defined as

$\varrho_{1}(t)\triangleq\varrho_{1}(t;t_{0},T_{1})=\left\\{\begin{array}[]{cl}\frac{T_{1}^{h_{1}}}{\left(T_{1}+t_{0}-t\right)^{h_{1}}},&\quad
t\in\left[t_{0},t_{0}+T_{1}\right),\\\ 1,&\quad otherwise,\end{array}\right.$
(1)
$\varrho_{2}(t)\triangleq\varrho_{1}(t;t_{1},T_{2})=\left\\{\begin{array}[]{cl}\frac{T_{2}^{h_{2}}}{\left(T_{2}+t_{1}-t\right)^{h_{2}}},&\quad
t\in\left[t_{1},t_{1}+T_{2}\right),\\\ 1,&\quad otherwise,\end{array}\right.$
(2)

where $h_{1},h_{2}>0$, $T_{1},T_{2}>0$ usually represent the desired
prescribed times and the duration of the time-scaling function, $t_{0}\geq 0$
and $t_{1}=t_{0}+T_{1}$ are the initial instant of each duration.

Prescribed-time stable The following definition and lemma regarding
prescribed-time stability are essential for the algorithm analysis in this
paper.

######  Definition 2.

[25] Considering a Lyapunov function $V(x,t)$, if the following equations
hold,

$\begin{array}[]{rl}1)&\lim\limits_{t\rightarrow\left(t_{0}+T\right)^{-}}V(x,t)=0,\\\
2)&V(x,t)\equiv 0,\forall t\in\left[t_{0}+T,\infty\right),\end{array}$ (3)

where $t_{0}$ is the initial instant, then $V(x,t)$ is globally prescribed-
time stable with prescribed time interval $T$.

######  Lemma 1.

[23, Lemma 1] Considering a Lyapunov function $V(x,t)$, if there exists
$\kappa>0$ such that

$\begin{array}[]{l}\dot{V}(x,t)=-\left[\kappa+2\frac{\dot{\varrho}\left(t;t_{0},T\right)}{\varrho\left(t;t_{0},T\right)}\right]V(x,t),t\in\left[t_{0},\infty\right),\end{array}$

then it can be derived that
$V(x,t)=\varrho\left(t;t_{0},T\right)^{-2}\mathrm{e}^{\left(-\kappa\left(t-t_{0}\right)\right)}V\left(x\left(t_{0}\right),t_{0}\right),t\in[t_{0},+\infty)$,
which satisfies the conditions (3) in Definition 2. Then $V(x,t)$ is globally
prescribed-time stable within $T$.

###  II-B Graph theory

A undirected graph $\mathcal{G}=(\mathcal{V},\mathcal{E},\mathcal{A})$ is
defined by a node set $\mathcal{V}=\left\\{v_{1},v_{2},...,v_{N}\right\\}$, an
undirected edge set $\mathcal{E}\subseteq\mathcal{V}\times\mathcal{V}$, and an
adjacency matrix $\mathcal{A}=\left[a_{ij}\right]_{N\times N}$ with non-
negative elements. The Laplacian matrix $L$ is defined as $D-A$ where $D={\rm
diag}\\{d(v_{1}),d(v_{2}),\ldots,d(v_{N})\\}$ is the degree matrix. The graph
$\overline{\mathcal{G}}$ is the complete graph of $\mathcal{G}$ if it has the
same number of agents and contains an edge between every pair of nodes.

The following lemmas provide the properties of undirected graphs.

######  Lemma 2.

[26] Given a connected and undirected graph $\mathcal{G}$ with $N$ nodes, it
follows that

  * 1)

$L$ is semi-positive definite and $\mathbf{1}_{N}^{\top}L=0$,
$L\mathbf{1}_{N}=0$ hold.

  * 2)

$L$ has one zero eigenvalue, while the remaining eigenvalues possess
nonnegative real parts which can be arranged in ascending order as
$\left[0,\lambda_{1}(L),\ldots,\lambda_{N}(L)\right]$.

  * 3)

a positive definite matrix $\Upsilon$ exists such that $L\Upsilon=\Upsilon
L=\Pi$ with $\Pi:=I_{N}-\frac{1}{N}\mathbf{1}_{N}\mathbf{1}_{N}^{\top}$.

######  Lemma 3.

[27] Given a connected and undirected graph $\mathcal{G}$, let its adjacency
matrix and Laplacian matrix be denoted as $A$ and $L$, respectively. Then for
$x_{1},x_{2},\ldots,x_{N}\in\mathbb{R}^{n}$ and
$\boldsymbol{x}=[x_{1}^{\top},x_{2}^{\top},\cdots,x_{N}^{\top}]^{\top}\in\mathbb{R}^{nN}$
the following equations hold $\boldsymbol{x}^{\top}(L\otimes
I_{n})\boldsymbol{x}=\frac{1}{2}\sum_{i=1}^{N}\sum_{j=1}^{N}a_{ij}(x_{i}-x_{j})^{\top}(x_{i}-x_{j})$.

######  Lemma 4.

[18] Given a connected and undirected graph $\mathcal{G}$ and its
corresponding complete graph $\overline{\mathcal{G}}$, with their Laplacian
matrices denoted as $L$ and $L_{\overline{\mathcal{G}}}$, the following
equation holds $\lambda_{2}(L)L_{\overline{\mathcal{G}}}\leq NL$.

###  II-C Convex analysis

######  Lemma 5.

[28] If function $f(x)$ is twice continuously differentiable and
$\gamma-$strongly convex, then the following equivalent conditions hold, i.e.,

$\begin{array}[]{l}f\left(x_{1}\right)-f\left(x_{2}\right)-\nabla^{\top}f\left(x_{2}\right)\left(x_{1}-x_{2}\right)\geqslant\frac{\gamma}{2}\left\|x_{1}-x_{2}\right\|^{2},\\\
\left[\nabla f\left(x_{1}\right)-\nabla
f\left(x_{2}\right)\right]^{\top}\left(x_{1}-x_{2}\right)\geqslant\gamma\left\|x_{1}-x_{2}\right\|^{2},\\\
\nabla^{2}f\left(x_{1}\right)\geqslant\gamma I_{n}.\end{array}$

Furthermore, if there exists $\Gamma$ such that
$\nabla^{2}f\left(x_{1}\right)\leq\Gamma I_{n}$, then the following equations
hold.

$\begin{array}[]{l}f\left(x_{1}\right)-f\left(x_{2}\right)-\nabla^{\top}f\left(x_{2}\right)\left(x_{1}-x_{2}\right)\leqslant\frac{\Gamma}{2}\left\|x_{1}-x_{2}\right\|^{2},\\\
\left[\nabla f\left(x_{1}\right)-\nabla
f\left(x_{2}\right)\right]^{\top}\left(x_{1}-x_{2}\right)\leqslant\Gamma\left\|x_{1}-x_{2}\right\|^{2}.\end{array}$

###  II-D Problem formulation

Consider a multi-agent network composed of $N$ agents, each possessing
computation and communication capabilities. The communication topology of the
network is abstracted as $\mathcal{G}$. Each agent independently possesses an
optimization variable for interaction with its neighbors and a private local
objective function. The objective is to minimize the sum of these local
objective functions, addressing the following optimization problem.

$\begin{array}[]{l}\min\limits_{x\in\mathbb{R}^{n}}\tilde{f}(x)=\sum\limits_{i=1}^{N}f_{i}\left(x\right),\end{array}$
(4)

which is equivalent to the following distributed form,

$\begin{array}[]{ll}\min\limits_{\boldsymbol{x}\in\mathbb{R}^{nN}}&f(\boldsymbol{x})=\sum\limits_{i=1}^{N}f_{i}\left(x_{i}\right),\\\
\text{ s.t }&\left(L\otimes I_{n}\right)\boldsymbol{x}=0,\end{array}$ (5)

where $f_{i}:\mathbb{R}^{n}\to\mathbb{R}$ is the local objective function of
agent $i$, $x_{i}\in\mathbb{R}^{n}$ is the local optimization variable of
agent $i$ and
$\boldsymbol{x}=[x_{1}^{\top},x_{2}^{\top},\cdots,x_{N}^{\top}]^{\top}\in\mathbb{R}^{nN}$.
Denote $\nabla f(\boldsymbol{x})=[\nabla f_{1}(x_{1})^{\top},\ldots,\nabla
f_{N}(x_{N})^{\top}]^{\top}$,
$\nabla^{2}f(\boldsymbol{x})=\left[\begin{smallmatrix}\nabla^{2}f_{1}\left(x_{1}\right)&&\\\
&\ddots&\\\ &&\nabla^{2}f_{N}\left(x_{N}\right)\end{smallmatrix}\right]$ for
further analysis.

######  Assumption 1.

Graph $\mathcal{G}$ is connected and undirected.

######  Assumption 2.

For each agent $i$, its local objective function $f_{i}(x_{i})$ is twice
continuously differentiable, $\gamma_{i}-$strongly convex, $\psi_{i}-$smooth
with $\gamma_{i},\psi_{i}>0$, and satisfies
$\gamma_{i}I_{n}\leqslant\nabla^{2}f_{i}\left(x_{i}(t)\right)\leqslant\Gamma_{i}I_{n}$
with a bounded $\Gamma_{i}>0$.

##  III Main Results

###  III-A Multi/Single-stage structure

Currently, most prescribed-time distributed optimization algorithms based on
time-varying scale functions require multi-stage structures to achieve global
convergence. In this context, multi-stage implies that each subsequent stage
relies solely on the results of the previous one, with each stage having
independent update processes and utilizing a single time-varying scale
function. The multi-stage and single-stage structures for prescribed-time
distributed optimization algorithms, as visualized in Fig. 1 and Fig. 2, can
be modeled as follows.

Multi-stage (MS):

$\begin{array}[]{lrl}&\text{Stage\,$1$:}\,\chi_{1}=&\Xi_{1}\big{(}\boldsymbol{x},f(\boldsymbol{x}),\nabla
f(\boldsymbol{x}),\nabla^{2}f(\boldsymbol{x}),\\\
&&\varrho_{1}(t;t_{0},T_{1})\big{)},\\\
&\text{Stage\,$2$:}\,\chi_{2}=&\Xi_{2}\big{(}\chi_{1}^{\ast},\boldsymbol{x},f(\boldsymbol{x}),\nabla
f(\boldsymbol{x}),\nabla^{2}f(\boldsymbol{x}),\\\
&&\varrho_{2}(t;t_{1},T_{2})\big{)},\\\ &&\cdots\\\
&\text{Stage\,$k$:}\,\chi_{k}=&\Xi_{k}\big{(}\chi_{1}^{\ast},\ldots,\chi_{k-1}^{\ast},\boldsymbol{x},f(\boldsymbol{x}),\nabla
f(\boldsymbol{x}),\\\
&&\nabla^{2}f(\boldsymbol{x}),\varrho_{k}(t;t_{k-1},T_{k})\big{)}.\end{array}$
(6)

For stage $k^{\prime}\,(1<k^{\prime}\leq k)$ in this model,
$\chi_{k^{\prime}}=[\chi_{k^{\prime}1}^{\top},\chi_{k^{\prime}2}^{\top},\ldots,\chi_{k^{\prime}m}^{\top}]^{\top}$
represents the $m$ different variables that need to be updated where the $i$th
variable
$\chi_{k^{\prime}i}^{\top}=[(\chi_{k^{\prime}i}^{(0)})^{\top},(\chi_{k^{\prime}i}^{(1)})^{\top},(\chi_{k^{\prime}i}^{(2)})^{\top}]^{\top}$
contains information of different orders of derivatives, and
$\chi_{1}^{\ast},\ldots,\chi_{k^{\prime}-1}^{\ast}$ are the optimal solutions
obtained from the updates in the previous stages.
$\varrho_{k^{\prime}}(t;t_{k^{\prime}-1},T_{k^{\prime}})$ is a time-varying
scaling function defined on the interval from $t_{k^{\prime}-1}$ to
$t_{k^{\prime}-1}+T_{k^{\prime}}$ and $\Xi_{k^{\prime}}$ represents the
dynamic input for updates.

######  Remark 1.

The aforementioned multi-stage model, due to its explicit stage divisions, can
also be referred to as explicit multi-stage (EMS) model. The multi-stage model
based distributed algorithms commonly involve stages such as ‘consensus
stage’, ‘global/local information estimation stage’, and ‘global/local
optimization stage’. Typically, the results from the previous stage are used
to facilitate the subsequent stages. For example, the two-stage ZGS algorithm
in [29] estimates the local minimizers in the first stage to satisfy the ZGS
initial conditions before applying the ZGS algorithm in the second stage.

Single-stage (SS):

$\begin{array}[]{l}\dot{\chi}=\Xi\big{(}\boldsymbol{x},f(\boldsymbol{x}),\nabla
f(\boldsymbol{x}),\nabla^{2}f(\boldsymbol{x}),\varrho_{1}(t;t_{0},T_{1})\big{)}.\end{array}$
(7)

Similarly,
$\chi=[\chi_{1}^{\top},\chi_{2}^{\top},\ldots,\chi_{m}^{\top}]^{\top}$ denotes
the $m$ different variables that need to be updated where the $i$th variable
$\chi_{i}^{\top}=[(\chi_{i}^{(0)})^{\top},(\chi_{i}^{(1)})^{\top},(\chi_{i}^{(2)})^{\top}]^{\top}$
contains information of different orders of derivatives. At this point, there
are no prior-stage results available for reference and use, as in (6).
$\varrho_{1}(t;t_{0},T_{1})$ is the only one time-varying scaling function
used in the algorithm and $\Xi$ represents the dynamic input for updates.

Figure 1: The multi/single-stage structure.

Comparing the two models (6) and (7) mentioned above, it can be observed that
single-stage model algorithms are not only structurally more concise but also
more convenient for practical use. However, multi-stage model algorithms have
the advantage of directly integrating various optimization algorithms
according to the objectives of each stage. In contrast, single-stage model
algorithms face challenges in both design and analysis because they need to
simultaneously achieve all the objectives of the multi-stage model within a
single predefined time interval. As a result, most current prescribed-time
distributed optimization algorithms based on time-varying scale functions are
multi-stage.

Figure 2: The explicit/implicit multi-stage structure.

######  Remark 2.

It is worth noting that even if some algorithms may appear to be structurally
similar to single-stage algorithms, they should be considered implicit multi-
stage (IMS) algorithms under the following circumstances, i.e.,

  * 1)

the simultaneous use of multiple time-varying scaling functions in one update.

  * 2)

the need for artificial temporal segmentation in the convergence analysis.

The above considerations also apply to finite-time convergence, fixed-time
convergence, and other prescribed-time convergence frameworks. For example,
the ZGS algorithm in [27], while not explicitly structured as multi-stage,
still defines two time intervals in its fixed-time convergence analysis: one
for satisfying the initial conditions of ZGS, and the other for ensuring
global convergence. Furthermore, the latter stage solely relies on the results
of the former stage, with non-intersecting updates, essentially making it a
two-stage algorithm.

###  III-B ZGS algorithm: research status

The above models, specific to the ZGS algorithm, usually takes the following
form.

$\begin{array}[]{rl}\dot{x}_{i}(t)=&-\left(\nabla^{2}f_{i}\left(x_{i}(t)\right)\right)^{-1}\Big{[}\zeta_{1}\big{(}\varrho_{\iota_{1}}(t),y_{i}(t)\big{)}\\\
&+\zeta_{2}\big{(}\varrho_{\iota_{2}}(t),\sum\limits_{j\in\mathcal{N}_{i}}a_{ij}\left(x_{i}(t)-x_{j}(t)\right)\big{)}\Big{]},\end{array}$
(8)

where $\zeta_{1}$ and $\zeta_{2}$ denote protocols related to convergence
control and consensus control, respectively. $\varrho_{\iota_{1}}(t)$ and
$\varrho_{\iota_{2}}(t)$ are two time-varying scaling functions that can be
either identical, signifying their equality in a single-stage algorithm, or
distinct, signifying their inequality in a multi-stage algorithm (in this
context, ‘equal’ or ‘not equal’ indicates whether they are defined over the
same time interval).

Recall from [12] that the traditional continuous-time ZGS has the following
form,

$\begin{array}[]{rl}\dot{x}_{i}(t)=&\varphi_{i}\left(x_{i}(t),\mathbf{x}_{\mathcal{N}_{i}}(t);f_{i},\mathbf{f}_{\mathcal{N}_{i}}\right),\forall
t\geq 0,\forall i\in\mathcal{V}\\\
x_{i}(t_{0})=&\chi_{i}\left(f_{i},\mathbf{f}_{\mathcal{N}_{i}}\right),\quad\forall
i\in\mathcal{V}\end{array}$ (9)

where $\mathbf{x}_{\mathcal{N}_{i}}(t)$ represents the optimization variable
information of node $i$’s neighbors, while $\mathbf{f}_{\mathcal{N}_{i}}$
represents the function value information of node $i$’s neighbors.
$\varphi_{i}$ is the local dynamic input to be designed which usually required
the following two stringent conditions,

$\displaystyle\sum\limits_{i\in\mathcal{V}}\nabla
f_{i}\left(x_{i}(t_{0})\right)=0,$ (10a)
$\displaystyle\sum_{i\in\mathcal{V}}\nabla^{2}f_{i}\left(x_{i}\right)\varphi_{i}\left(x_{i},\mathbf{x}_{\mathcal{N}_{i}};f_{i},\mathbf{f}_{\mathcal{N}_{i}}\right)=0.$
(10b)

The presence of these two conditions introduces certain inconveniences in both
the utilization and construction of ZGS algorithms. While feasible solutions
for (10a), such as introducing an additional stage for local objective
function minimization, are already available, there is still a lack of
research on algorithms addressing (10b).

###  III-C MS-PTZGS algorithm: multi-stage ZGS with prescribed-time
performance

To solve problem (4) without requiring (10a), the following multi-stage PTDO
algorithm using sliding mode control method is proposed,

$\displaystyle\dot{x}_{i}(t)=$
$\displaystyle\left(\nabla^{2}f_{i}\left(x_{i}(t)\right)\right)^{-1}\big{[}-(\kappa_{1}+\frac{\dot{\varrho}_{1}(t)}{\varrho_{1}(t)})s_{i}(t)$
$\displaystyle-c(\kappa_{2}\frac{\dot{\varrho}_{2}(t)}{\varrho_{2}(t)})\sum\nolimits_{j=1}^{N}a_{ij}\left(x_{i}(t)-x_{j}(t)\right)\big{]},$
(11a) $\displaystyle\dot{\phi}_{i}(t)=$
$\displaystyle\left(\kappa_{2}\frac{\dot{\varrho}_{2}(t)}{\varrho_{2}(t)}\right)\sum\nolimits_{j=1}^{N}a_{ij}\left(x_{i}(t)-x_{j}(t)\right),$
(11b) $\displaystyle s_{i}(t)=$ $\displaystyle\nabla
f_{i}\left(x_{i}(t)\right)+c\phi_{i}(t),$ (11c)

where $c,\kappa_{1},\kappa_{2}>0$ are tuning parameters,
$\varrho_{1}(t),\varrho_{2}(t)$ are the time-scaling functions defined in (1)
and (2), $s_{i}(t)$ is the distributed integral sliding manifold for agent $i$
to avoid additional local minimization to satisfy the requied initial
condition $\sum_{i=1}^{N}\nabla f_{i}(x_{i}(t_{0}))=0$ for ZGS.

######  Theorem 1.

Under Assumptions 1 and 2, for any initial value $x_{i}(t_{0})$, the proposed
algorithm (11) can solve the optimization problem (4) within a prescribed time
interval $T_{1}+T_{2}$, i.e.,
$\lim\nolimits_{t\rightarrow\left(t_{0}+T_{1}+T_{2}\right)^{-}}x_{i}=x^{\ast}$
where $x^{\ast}\in\mathbb{R}^{n}$ is the global minimizer.

###### Proof.

First, it is proven that the sliding manifold (11c) allows the satisfaction of
the required initial condition $\sum_{i=1}^{N}\nabla f_{i}(x_{i}(t_{0}))=0$
within the time interval $T_{1}$ without the need to calculate the local
minimizer $x^{\ast}_{i}$ of each $f_{i}(x)$. Define a Lyapunov function using
$s_{i}$

$V_{i}(t)=\frac{1}{2}s_{i}^{\top}s_{i}.$ (12)

To ensure formula simplicity, the $(t)$ in $x_{i},s_{i}$ will be omitted.
Differentiating $V_{i}(t)$ along (11c) yields

$\begin{array}[]{rl}\dot{V}_{i}(t)=&s_{i}^{\top}\dot{s}_{i}\\\
=&s_{i}^{\top}\big{[}\nabla^{2}f_{i}(x_{i})\dot{x_{i}}\\\
&+c\kappa_{1}\frac{\dot{\varrho}_{2}(t)}{\varrho_{2}(t)}\sum\nolimits_{j=1}^{N}a_{ij}\left(x_{i}-x_{j}\right)\big{]}\\\
=&s_{i}^{\top}\big{[}-(\kappa_{1}+\frac{\dot{\varrho}_{1}(t)}{\varrho_{1}(t)})s_{i}\\\
&-c\kappa_{2}\frac{\dot{\varrho}_{2}(t)}{\varrho_{2}(t)}\sum\nolimits_{j=1}^{N}a_{ij}\left(x_{i}-x_{j}\right)\\\
&+c\kappa_{2}\frac{\dot{\varrho}_{2}(t)}{\varrho_{2}(t)}\sum\nolimits_{j=1}^{N}a_{ij}\left(x_{i}-x_{j}\right)\big{]}\\\
=&-(\kappa_{1}+\frac{\dot{\varrho}_{1}(t)}{\varrho_{1}(t)})s_{i}^{\top}s_{i}\\\
=&-(2\kappa_{1}+2\frac{\dot{\varrho}_{1}(t)}{\varrho_{1}(t)})V_{i}(t).\end{array}$
(13)

Using Lemma 1, there is
$\lim\nolimits_{t\rightarrow\left(t_{0}+T_{1}\right)^{-}}V_{i}(t)=0$ and
$V(x,t)\equiv 0,\forall t\in\left[t_{0}+T_{1},\infty\right)$, i.e.,
$\lim\nolimits_{t\rightarrow\left(t_{0}+T_{1}\right)^{-}}s_{i}=0$ and
$s_{i}\equiv 0,\forall t\in\left[t_{0}+T_{1},\infty\right)$. Then when
$t\in[t_{0}+T_{1},+\infty)$ there is $s_{i}=0$.

Furthermore, since

$\begin{array}[]{l}\sum\nolimits_{i=1}^{N}c\phi_{i}(t)\\\
=\sum\nolimits_{i=1}^{N}c\int_{t_{0}}^{t}\left(\kappa_{2}\frac{\dot{\varrho}_{2}(\tau)}{\varrho_{2}(\tau)}\right)\sum\nolimits_{j=1}^{N}a_{ij}\left(x_{i}(\tau)-x_{j}(\tau)\right)\mathrm{d}\tau\\\
=0,\end{array}$ (14)

it is easy to obtain that $\sum_{i=1}^{N}s_{i}(t_{1})=\sum_{i=1}^{N}\nabla
f_{i}(x_{i}(t_{1}))$. Now it is proven that the condition
$\sum_{i=1}^{N}\nabla f_{i}(x_{i})=0$ can be achieved within time interval
$T_{1}$ using (11c).

Next proof that all local state variables $x_{i},i=1,\ldots,N$ will converge
to $x^{\ast}$ within the next time interval $[t_{1},t_{1}+T_{2})$. From the
analysis form above, since $s_{i}\equiv 0,\forall
t\in\left[t_{0}+T_{1},\infty\right)$, (11a) is equivalent to

$\begin{array}[]{rl}\dot{x}_{i}=&\left(\nabla^{2}f_{i}\left(x_{i}(t)\right)\right)^{-1}\big{[}-c(\kappa_{2}\frac{\dot{\varrho}_{2}(t)}{\varrho_{2}(t)})\\\
&\sum\nolimits_{j=1}^{N}a_{ij}\left(x_{i}-x_{j}\right)\big{]}.\end{array}$
(15)

Define another Lyapunov function as

$\begin{array}[]{l}V_{M}(t)=\sum\nolimits_{i=1}^{N}\big{[}f_{i}\left(x^{*}\right)-f_{i}\left(x_{i}\right)-\nabla^{\top}f_{i}\left(x_{i}\right)\left(x^{*}-x_{i}\right)\big{]}.\end{array}$
(16)

Differentiating $V_{M}(t)$ along (15) yields

$\begin{array}[]{rl}\dot{V}_{M}(t)=&\sum\nolimits_{i=1}^{N}\left[-\nabla^{2}f_{i}\left(x_{i}\right)\dot{x}_{i}\left(x^{*}-x_{i}\right)\right]\\\
=&c\kappa_{2}\frac{\dot{\varrho}_{2}(t)}{\varrho_{2}(t)}\sum\nolimits_{i=1}^{N}\sum\nolimits_{j=1}^{N}a_{ij}\left(x^{*}-x_{i}\right)^{\top}\left(x_{i}-x_{j}\right).\end{array}$
(17)

Using Lemma 3, there is

$\begin{array}[]{rl}\dot{V}_{M}(t)=&-\frac{1}{2}c\kappa_{2}\frac{\dot{\varrho}_{2}(t)}{\varrho_{2}(t)}\sum\nolimits_{i=1}^{N}\sum\nolimits_{j=1}^{N}a_{ij}\left(x_{i}-x_{j}\right)^{\top}\left(x_{i}-x_{j}\right)\\\
=&-c\kappa_{2}\frac{\dot{\varrho}_{2}(t)}{\varrho_{2}(t)}\boldsymbol{x}^{\top}(L\otimes
I_{n})\boldsymbol{x}.\end{array}$ (18)

Define $\overline{x}=\frac{1}{N}\sum_{i=1}^{N}x_{i}$. From
$\sum_{i=1}^{N}\nabla f_{i}(x_{i})=0$ and
$\tilde{f}(\overline{x})\geq\tilde{f}(x^{\ast})$, it can be obtained that

$\begin{array}[]{l}\sum\nolimits_{i=1}^{N}\left[f_{i}\left(\overline{x}\right)-f_{i}\left(x_{i}\right)-\nabla^{\top}f_{i}\left(x_{i}\right)\left(\overline{x}-x_{i}\right)\right]-V_{M}(t)\\\
=\sum\nolimits_{i=1}^{N}\left[f_{i}\left(\overline{x}\right)-f_{i}\left(x^{\ast}\right)-\nabla^{\top}f_{i}\left(x_{i}\right)(\overline{x}-x^{\ast})\right]\\\
=\tilde{f}(\overline{x})-\tilde{f}(x^{\ast})-(\overline{x}-x^{\ast})^{\top}\sum\nolimits_{i=1}^{N}\nabla
f_{i}\left(x_{i}\right)\\\ =\tilde{f}(\overline{x})-\tilde{f}(x^{\ast})\geq
0.\end{array}$ (19)

Then there is

$V_{M}(t)\leq\sum_{i=1}^{N}\left[f_{i}\left(\overline{x}\right)-f_{i}\left(x_{i}\right)-\nabla^{\top}f_{i}\left(x_{i}\right)\left(\overline{x}-x_{i}\right)\right].$
(20)

By Assumption 2 and Lemma 5, it can be derived that
$f_{i}\left(\overline{x}\right)-f_{i}\left(x_{i}\right)-\nabla^{\top}f_{i}\left(x_{i}\right)\left(\overline{x}-x_{i}\right)\leq\frac{\Gamma_{i}}{2}\left\|x_{i}-\overline{x}\right\|^{2}$.
Taking $\Gamma_{\max}$ as $\max_{i\in\mathcal{V}}\left\\{\Gamma_{i}\right\\}$,
there is
$V_{M}(t)\leq\frac{\Gamma_{\max}}{2}\sum_{i=1}^{N}\left\|x_{i}-\overline{x}\right\|^{2}$.
Using Lemma 3 and 4, it can be derived that

$\begin{array}[]{rl}V_{M}(t)\leq&\frac{\Gamma_{\max}}{2}\sum\nolimits_{i=1}^{N}\left\|x_{i}-\overline{x}\right\|^{2}\\\
=&\frac{\Gamma_{\max}}{2N^{2}}\sum\nolimits_{i=1}^{N}\left\|\sum\nolimits_{j=1}^{N}(x_{i}-x_{j})\right\|^{2}\\\
\leq&\frac{\Gamma_{\max}}{2N^{2}}N\sum\nolimits_{i=1}^{N}\sum\nolimits_{j=1}^{N}\left\|(x_{i}-x_{j})\right\|^{2}\\\
=&\frac{\Gamma_{\max}}{N}\boldsymbol{x}^{\top}(L_{\overline{\mathcal{G}}}\otimes
I_{n})\boldsymbol{x}\\\
\leq&\frac{\Gamma_{\max}}{\lambda_{2}(L)}\boldsymbol{x}^{\top}(L\otimes
I_{n})\boldsymbol{x}.\end{array}$ (21)

Combining (18) and (21), there is

$\dot{V}_{M}(t)\leq-\frac{\lambda_{2}(L)c\kappa_{2}}{\Gamma_{\max}}\frac{\dot{\varrho}_{2}(t)}{\varrho_{2}(t)}V_{M}(t).$
(22)

Letting $\alpha_{1}=\frac{\lambda_{2}(L)c\kappa_{2}}{\Gamma_{\max}}$ and
multiplying $\varrho_{2}^{\alpha_{1}}(t)$ on both sides of (22), it can be
obtained that
$\varrho_{2}^{\alpha_{1}}(t)\dot{V}_{M}(t)\leq-\alpha_{1}\dot{\varrho}_{2}(t)\varrho_{2}^{\alpha_{1}-1}(t)V_{M}(t)$.
Then there is $\frac{d(\varrho_{2}^{\alpha_{1}}(t)V_{M}(t))}{dt}\leq 0$ which
can be derived as

$V_{M}(t)\leq V_{M}(t_{0})\varrho_{2}^{-\alpha_{1}}(t),$ (23)

which satisfies the conditions (3) in Definition 1. Then there is
$\lim\nolimits_{t\rightarrow\left(t_{1}+T_{2}\right)^{-}}V_{M}(t)=0$ and
$V_{M}(t)\equiv 0,\forall t\in\left[t_{1}+T_{2},\infty\right)$, i.e.,
$\lim\nolimits_{t\rightarrow\left(t_{1}+T_{2}\right)^{-}}x_{i}(t)=x^{\ast}$
and $x_{i}(t)\equiv x^{\ast},\forall t\in\left[t_{1}+T_{2},\infty\right)$. The
proof is now complete. ∎

######  Remark 3.

In summary, the proposed MS-PTZGS demonstrates excellent prescribed-time
convergence performance. Compared to [29], this algorithm does not require
condition (10a), and compared to LMFZGS in [27], it achieves prescribed-time
convergence performance far exceeding fixed-time convergence performance.
However, this algorithm cannot guarantee the absence of condition (10b) in all
stages, only in the first stage. The upcoming single-stage algorithm, to be
introduced shortly, will address this issue.

######  Remark 4.

The MS-PTZGS algorithm proposed here belongs to the implicit multi-stage
algorithm mentioned earlier, which still maintains the simplicity of a similar
single-stage algorithm (i.e., it does not have to run each stage individually)
in terms of algorithmic operation as compared to the explicit multi-stage
algorithm.

###  III-D SS-PTZGS algorithm: single-stage ZGS with prescribed-time
performance

To solve problem (4) without requiring both (10a) and (10b), the following
single-stage PTDO algorithm using sliding mode control method is proposed,

$\displaystyle\dot{x}_{i}(t)=$
$\displaystyle\left(\nabla^{2}f_{i}\left(x_{i}(t)\right)\right)^{-1}\big{(}\kappa_{1}\frac{\dot{\varrho}_{1}(t)}{\varrho_{1}(t)}\big{)}\big{(}-\kappa_{2}s_{i}(t)$
$\displaystyle-c\sum\nolimits_{j=1}^{N}a_{ij}\left(x_{i}(t)-x_{j}(t)\right)\big{)},$
(24a) $\displaystyle s_{i}(t)=$ $\displaystyle
c\int_{0}^{t}\big{(}\kappa_{1}\frac{\dot{\varrho_{1}}(\tau)}{\varrho_{1}(\tau)}\big{)}\sum\nolimits_{j=1}^{N}a_{ij}\left(x_{i}(\tau)-x_{j}(\tau)\right)\mathrm{d}\tau$
$\displaystyle+\nabla f_{i}\left(x_{i}(t)\right),$ (24b)

where $c,\kappa_{1},\kappa_{2}>0$ are tuning parameters, $\varrho_{1}(t)$ is
the only used time-scaling function defined in (1), $s_{i}(t)$ is the
distributed integral sliding manifold for agent $i$ to avoid additional local
minimization to satisfy the requied initial condition (10a). Denote
$\boldsymbol{s}=[s_{1}^{\top},s_{2}^{\top},\cdots,s_{N}^{\top}]^{\top}\in\mathbb{R}^{nN}$
and $\boldsymbol{x}^{\ast}=\mathbf{1}_{N}\otimes x^{\ast}\in\mathbb{R}^{nN}$.

######  Remark 5.

There are two main motivations behind the construction of this single-stage
structured PTDO, i.e.,

  * 1)

currently, multi-stage non-prescribed-time convergence algorithms have the
drawback of significant disparity between theoretical convergence time and
practical convergence time (usually the former being larger). This inability
to accurately estimate convergence time is not conducive to practical
engineering applications.

  * 2)

in the existing literature related to ZGS algorithm, articles either exhibit a
multi-stage structure in algorithm design or convergence proofs. This is
primarily due to the inherent difficulty in simultaneously achieving the goals
of gradual gradient sum reduction to $0$ and achieving global convergence in a
single stage.

TABLE I: Comparison with existing ZGS algorithms Type | Algorithm | Protocals | Convergence rate | Is it not bound by
---|---|---|---|---
$\zeta_{1}$ | $\zeta_{2}$ | (10a) | (10b)
Multi-stage | Implicit | MS-PTZGS | $(\kappa_{1}+\frac{\dot{\varrho}_{1}}{\varrho_{1}})s_{i}$ | $c(\kappa_{2}\frac{\dot{\varrho}_{2}}{\varrho_{2}})\sum\limits_{j=1}^{N}a_{ij}\left(x_{i}-x_{j}\right)$ | prescribed-time | yes | no
Algorithm in [30] | $\varphi_{i}\left(z_{i}\right)+\frac{\partial\nabla f_{i}\left(x_{i},t\right)}{\partial t}$ | $\alpha\sum\limits_{j=1}^{N}a_{ij}\operatorname{sgn}\left(x_{i}-x_{j}\right)$ | finite-time | yes | yes
Algorithm in [27] | $\begin{array}[]{rl}&k_{1}\operatorname{sig}^{p_{1}}\left(s_{i}\right)\\\ +&k_{2}\operatorname{sig}^{q_{1}}\left(s_{i}\right)\end{array}$ | $\begin{array}[]{rl}&c_{1}\sum\limits_{j=1}^{N}a_{ij}\operatorname{sig}^{p_{2}}\left(x_{i}-x_{j}\right)\\\ +&c_{2}\sum\limits_{j=1}^{N}a_{ij}\operatorname{sig}^{q_{2}}\left(x_{i}-x_{j}\right)\end{array}$ | fixed-time | yes | no
Explicit | Algorithm in [29] | $\left(k_{1}\frac{\dot{\varrho}_{1}}{\varrho_{1}}\right)y_{i}(t_{\sigma}^{i})$ | $\begin{array}[]{rl}&(k_{2}\frac{\dot{\varrho}_{2}}{\varrho_{2}})\big{(}\sum\limits_{j=1}^{N}a_{ij}(x_{i}\left(t_{k}^{i}\right)\\\ &-x_{j}(t_{k^{\prime}}^{j}))\big{)}\end{array}$ | prescribed-time | yes | no
Algorithm in [31] | $\dot{\omega}_{1}\nabla f_{i}\left(x_{i}(t_{0})\right)$ | $\begin{array}[]{l}(\frac{\dot{\omega}_{2}}{1-\omega_{2}+\beta}+1)\times\\\ \sum\limits_{j=1}^{N}a_{ij}\left(\hat{x}_{j}^{i}-\hat{x}_{i}^{j}\right)\end{array}$ | prescribed-time | yes | no
Single-stage | SS-PTZGS | $\kappa_{1}\kappa_{2}\frac{\dot{\varrho}_{1}}{\varrho_{1}}s_{i}$ | $c(\kappa_{1}\frac{\dot{\varrho}_{1}}{\varrho_{1}})\sum\limits_{j=1}^{N}a_{ij}\left(x_{i}-x_{j}\right)$ | prescribed-time | yes | yes
Algorithm in [32] | not designed | $\begin{array}[]{rl}&\alpha_{1}\sum\limits_{j=1}^{N}a_{ij}\operatorname{sig}^{p}\left(x_{i}-x_{j}\right)\\\ +&\alpha_{2}\sum\limits_{j=1}^{N}a_{ij}\operatorname{sig}^{q}\left(x_{i}-x_{j}\right)\end{array}$ | finite-time | no | no
Algorithm in [33] | not designed | $\sum\limits_{j=1}^{N}a_{ij}\left(\hat{x}_{j}(t-\tau)-\hat{x}_{i}(t-\tau)\right)$ | not guaranteed | no | no
Algorithm in [28] | not designed | $\gamma\sum\limits_{j=1}^{N}a_{ij}\left(\hat{x}_{j}-\hat{x}_{i}\right)$ | not guaranteed | no | no
Algorithm in [14] | not designed | $\sum\limits_{j=1}^{N}a_{ij}\left(\theta_{i}+\theta_{j}\right)\left(x_{j}-x_{i}\right)^{-1}$ | fixed-time | no | yes
Algorithm in [26] | $c_{2}y_{i}$ | $c_{1}\sum\limits_{j=1}^{N}a_{ij}\left(x_{i}-x_{j}\right)$ | Exponential | yes | yes

######  Theorem 2.

Under Assumptions 1 and 2, for any initial value $x_{i}(t_{0})$, the proposed
algorithm (24) can solve the optimization problem (4) within a single
prescribed time interval $T_{1}$, i.e.,
$\lim\nolimits_{t\rightarrow\left(t_{0}+T_{1}\right)^{-}}x_{i}=x^{\ast}$ where
$x^{\ast}\in\mathbb{R}^{n}$ is the global minimizer.

###### Proof.

Similarly, to ensure formula simplicity, the $(t)$ in $x_{i},s_{i}$ will be
omitted. First, it can be obtained from (24) for $s_{i}$ that

$\begin{array}[]{rl}\dot{s}_{i}=&\nabla^{2}f_{i}(x_{i})\dot{x}_{i}+c\left(\kappa_{1}\frac{\dot{\varrho_{1}}(t)}{\varrho_{1}(t)}\right)\sum\nolimits_{j=1}^{N}a_{ij}\left(x_{i}-x_{j}\right)\\\
=&-\left(\kappa_{1}\frac{\dot{\varrho}_{1}(t)}{\varrho_{1}(t)}\right)\big{(}\kappa_{2}s_{i}+c\sum\nolimits_{j=1}^{N}a_{ij}\left(x_{i}-x_{j}\right)\big{)}\\\
&+c\left(\kappa_{1}\frac{\dot{\varrho_{1}}(t)}{\varrho_{1}(t)}\right)\sum\nolimits_{j=1}^{N}a_{ij}\left(x_{i}-x_{j}\right)\\\
=&-\left(\kappa_{1}\frac{\dot{\varrho}_{1}(t)}{\varrho_{1}(t)}\right)\kappa_{2}s_{i}.\end{array}$
(25)

Define another Lyapunov function as

$\begin{array}[]{rl}V_{S}(t)=&\frac{1}{2}p\boldsymbol{s}^{\top}\boldsymbol{s}+\sum\nolimits_{i=1}^{N}\big{[}f_{i}\left(x^{*}\right)-f_{i}\left(x_{i}\right)\\\
&-\nabla^{\top}f_{i}\left(x_{i}\right)\left(x^{*}-x_{i}\right)\big{]}\\\
\end{array}$ (26)

Taking $\Psi_{\max}$ as
$\max\limits_{i\in\mathcal{V}}\left\\{\psi_{i}\right\\}$, from Assumption 2,
there is

$\begin{array}[]{rl}V_{S}(t)\leq&\frac{\Psi_{\max}}{2}\left\|\boldsymbol{x}-\boldsymbol{x}^{\ast}\right\|^{2}+\frac{p}{2}\left\|\boldsymbol{s}\right\|^{2}.\end{array}$
(27)

Differentiating $V_{S}(t)$ along (24) yields

$\begin{array}[]{rl}\dot{V}_{S}(t)=&\sum\nolimits_{i=1}^{N}\left[-\nabla^{2}f_{i}\left(x_{i}\right)\dot{x}_{i}\left(x^{*}-x_{i}\right)\right]+p\boldsymbol{s}^{\top}\dot{\boldsymbol{s}}\\\
=&\kappa_{1}\frac{\dot{\varrho}_{1}(t)}{\varrho_{1}(t)}\sum\nolimits_{i=1}^{N}\left(x^{*}-x_{i}\right)^{\top}\big{[}c\sum\nolimits_{j=1}^{N}a_{ij}\left(x_{i}-x_{j}\right)\\\
&+\kappa_{2}s_{i}\big{]}-\kappa_{1}\frac{\dot{\varrho}_{1}(t)}{\varrho_{1}(t)}p\kappa_{2}\boldsymbol{s}^{\top}\boldsymbol{s}\\\
=&-\kappa_{1}\frac{\dot{\varrho}_{1}(t)}{\varrho_{1}(t)}\big{[}c(\boldsymbol{x}-\boldsymbol{x}^{\ast})^{\top}(L\otimes
I_{n})(\boldsymbol{x}-\boldsymbol{x}^{\ast})\\\
&+\kappa_{2}\boldsymbol{s}^{\top}(\boldsymbol{x}-\boldsymbol{x}^{\ast})\big{]}-\kappa_{1}\frac{\dot{\varrho}_{1}(t)}{\varrho_{1}(t)}p\kappa_{2}\boldsymbol{s}^{\top}\boldsymbol{s}.\end{array}$
(28)

Using Young’s inequality, there exists $\delta>0$ such that

$\begin{array}[]{rl}\dot{V}_{S}(t)\leq&\kappa_{1}\frac{\dot{\varrho}_{1}(t)}{\varrho_{1}(t)}\big{[}-c\lambda_{2}(L)\left\|\boldsymbol{x}-\boldsymbol{x}^{\ast}\right\|^{2}-p\kappa_{2}\boldsymbol{s}^{\top}\boldsymbol{s}\\\
&+\kappa_{2}\delta\boldsymbol{s}^{\top}\boldsymbol{s}+\frac{\kappa_{2}}{4\delta}\left\|\boldsymbol{x}-\boldsymbol{x}^{\ast}\right\|^{2}\big{]}\\\
=&-\kappa_{1}\frac{\dot{\varrho}_{1}(t)}{\varrho_{1}(t)}\big{[}(c\lambda_{2}(L)-\frac{\kappa_{2}}{4\delta})\left\|\boldsymbol{x}-\boldsymbol{x}^{\ast}\right\|^{2}\\\
&+\kappa_{2}(p-\delta)\boldsymbol{s}^{\top}\boldsymbol{s}\big{]}.\end{array}$
(29)

Assign values to $p$ and $\delta$ that satisfy the following condition,

$p>\delta>\frac{\kappa_{2}}{4c\lambda_{2}(L)}.$ (30)

Taking
$\sigma_{S}=\min\left\\{\frac{2c\lambda_{2}(L)}{\Psi_{\max}}-\frac{\kappa_{2}}{2\delta\Psi_{\max}},\frac{2\kappa_{2}(p-\delta)}{p}\right\\}$
and recalling (27), there is

$\dot{V}_{S}(t)\leq-\kappa_{1}\frac{\dot{\varrho}_{1}(t)}{\varrho_{1}(t)}\sigma_{S}V_{S}(t).$
(31)

Letting $\alpha_{2}=\kappa_{1}\sigma_{S}$ and multiplying
$\varrho_{1}^{\alpha_{2}}(t)$ on both sides of (31), it can be obtained that
$\varrho_{1}^{\alpha_{2}}(t)\dot{V}_{S}(t)\leq-\alpha_{2}\dot{\varrho}_{1}(t)\varrho_{1}^{\alpha_{2}-1}(t)V_{S}(t)$.
Then there is $\frac{d(\varrho_{1}^{\alpha_{2}}(t)V_{S}(t))}{dt}\leq 0$ which
can be derived as

$V_{S}(t)\leq V_{S}(t_{0})\varrho_{1}^{-\alpha_{2}}(t),$ (32)

which satisfies the conditions (3) in Definition 1. Then there is
$\lim\nolimits_{t\rightarrow\left(t_{0}+T_{1}\right)^{-}}V_{S}(t)=0$ and
$V_{S}(t)\equiv 0,\forall t\in\left[t_{0}+T_{1},\infty\right)$, i.e.,
$\lim\nolimits_{t\rightarrow\left(t_{0}+T_{1}\right)^{-}}x_{i}(t)=x^{\ast}$
and $x_{i}(t)\equiv x^{\ast},\forall t\in\left[t_{0}+T_{1},\infty\right)$. The
proof is now complete. ∎

######  Remark 6.

From the analysis above, it can be observed that the proposed SS-PTZGS
simultaneously avoids the requirements of conditions (10a) and (10b) while
achieving the objectives of making the gradient sum approach zero and
achieving global convergence within a single prescribed-time interval. The
approach to achieving this result is not overly complex. It involves adding
the quadratic term of a sliding mode surface to the traditional Lyapunov
function that guarantees global convergence. This ensures that each local
variable, driven by the algorithm, gradually approaches both the sliding mode
surface, ensuring the gradient sum becomes zero, and the global optimal
solution.

So far, the two algorithms proposed in this paper have both undergone a
comprehensive analysis. To further illustrate and compare their distinctions
and advantages compared to others’ work, Table I summarizes the recent
algorithmic developments related to the problems addressed, which also include
the two algorithms proposed later in this paper.

##  IV Simulation

This section will verify the excellent prescribed-time convergence performance
of the two proposed algorithms through a simulation. Consider an undirected
graph with $N=6$ agents as shown in Fig. 3 and the six local objective
functions for each agent are

$\displaystyle f_{1}(x)$ $\displaystyle=(x_{1}-1)^{2}+(x_{2}-2)^{2},\,$
$\displaystyle f_{4}(x)$ $\displaystyle=x_{1}^{2}+2x_{2}^{2},$ $\displaystyle
f_{2}(x)$ $\displaystyle=(x_{1}-3)^{2}+(x_{2}-4)^{2},\,$ $\displaystyle
f_{5}(x)$ $\displaystyle=2x_{1}^{2}+x_{2}^{2},$ $\displaystyle f_{3}(x)$
$\displaystyle=(x_{1}-5)^{2}+(x_{2}-6)^{2},\,$ $\displaystyle f_{6}(x)$
$\displaystyle=3x_{1}^{2}+2x_{2}^{2}.$

The selection of parameters for (11) and (24) is as follows.

Algorithm | $\kappa_{1}$ | $\kappa_{2}$ | $c$ | $T_{1}$ | $h_{1}$ | $T_{2}$ | $h_{2}$
---|---|---|---|---|---|---|---
MS-PTZGS (11) | 2 | 3 | 1 | 0.1s | 3 | 0.2s | 2.5
SS-PTZGS (24) | 2 | 3 | 1 | 0.3s | 2.3 |  |

Figure 3: Communication topology.

Fig. 4 illustrates the temporal variations of individual components $x_{ij}$
of the local optimization variables $x_{i}$ for the two proposed algorithms.
Fig. 5 further demonstrates the changes in residuals between each local
optimization variable and the global optimum, where the residual is denoted by
$\mathrm{Er}(x_{i}(t))=\frac{\left\|x_{i}(t)-x^{*}\right\|_{2}^{2}}{\left\|x_{i}(t_{0})-x^{*}\right\|_{2}^{2}}$.
Subsequently, Fig. 6 displays the temporal trends of the sliding mode surface
term $s_{i}$, confirming the effectiveness of eliminating the requirement of
(10a) using sliding mode control. Finally, Fig. 7 depicts the evolution of the
error between the global function value and the optimal value. From these four
figures, the following key points can be summarized as follows,

  * 1)

both of the proposed algorithms successfully achieve global convergence within
the prescribed time interval, and the small error between theoretical and
actual convergence times is advantageous for meeting the requirements of real-
world engineering applications that necessitate precise control over
convergence time.

  * 2)

MS-PTZGS, with its two-stage structure, exhibits a clear two-stage division in
the simulation results, while the single-stage SS-PTZGS shows only one-stage
convergence.

  * 3)

the applied sliding mode surface term accomplishes the desired convergence
towards zero within the prescribed time, fulfilling its expected role.

(a) MS-PTZGS

(b) SS-PTZGS

Figure 4: Motion of each agent’s state $x_{i}(t)$.

(a) MS-PTZGS

(b) SS-PTZGS

Figure 5: Motion of each agent’s normalized residual Er$(x_{i}(t))$.

(a) MS-PTZGS

(b) SS-PTZGS

Figure 6: Motion of each agent’s sliding mode surface $s_{i}(t)$.

(a) MS-PTZGS

(b) SS-PTZGS

Figure 7: Motion of the global function error
$|f(\boldsymbol{x}(t))-f(x^{\ast})|$.

##  V Conclusions

Starting from a unified perspective of multi-stage and single-stage
structures, this paper constructs two ZGS algorithms that guarantee
prescribed-time convergence performance. The proposed MS-PTZGS avoids the
requirement for the gradient sum to be zero and exhibits significant
improvements in overall applicability and convergence performance compared to
other ZGS algorithms. Furthermore, the introduced SS-PTZGS breaks through the
constraints in ZGS algorithm design, addressing the issue that previous
algorithms of this kind could not simultaneously achieve gradient sum
reduction to zero and global convergence within a single time interval.
Finally, the excellent performance of the proposed algorithms is validated
through simulation.

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