\section{Introduction}\label{SectionI}
Due to the wide applications in the swarm robotics \cite{raffard2004distributed}, sensor networks \cite{rabbat2004distributed} and machine learning \cite{boyd2011distributed},
the distributed multi-agent optimization problems have attracted much attention in academia in the past decade.
For the multi-agent optimization problem $\min \sum\nolimits_{i = 1}^n {{Q_i}(x)}$, each agent $i$ has a local convex cost function $Q_i(x)$. For example,
$Q_i(x)$ can be used to represent the cost of vehicle $i$ moving from the current location to location $x$. In this case,
the objective of the optimization problem $\min \sum\nolimits_{i = 1}^n {{Q_i}(x)}$ is to find out the location $x^*$ where all agents gather,
such that the total cost $\sum\nolimits_{i = 1}^n {{Q_i}(x^*)}$ is the minimum.

For the above mentioned multi-agent optimization problem, there have been lots of relevant research. Among these research, one common approach is to utilize the distributed gradient optimization method.
\cite{nedic2009distributed} firstly proposed the classical gradient-descent optimization method to analyze the convergence based on the consensus problem. In details, they designed a distributed iterative algorithm,
which combines the consensus algorithm with the gradient-descent method. Under this algorithm, each agent's local estimate for the global decision variable will reach consensus and the global cost is the minimum.
Furthermore, they analyze the convergence of the gradient-descent method in the condition of constrained states \cite{nedic2010constrained},
asynchronous communication \cite{nedic2010asynchronous} and time-varying topology \cite{nedic2014distributed}.
Afterwards, other researchers began to utilize the gradient-descent method to deal with various multi-agent optimization problems.
\cite{zhu2011distributed} dealt with the multi-agent optimization problem with both equality and inequality constraints.
\cite{agarwal2011distributed} utilized time-delay stochastic gradient information to design the distributed optimization algorithm.
In \cite{kia2015distributed}, a continuous-time optimization algorithm was designed with discrete-time data.
\cite{lin2016distributed} gave the accelerated optimization algorithm based on the quadratic gradient information.
More relevant research can be referred to \cite{cao2012overview} and references therein.

In the gradient-descent method, the choice of step-size is of great concern. There are two main approaches on the choice of step-size:
fixed stepsizes and diminishing stepsizes.
The approach of diminishing stepsizes usually requires that the sum of stepsizes tends to infinity, but the sum of squared stepsizes is finite.
Under this requirement, \cite{nedic2009distributed,nedic2010constrained} proved that it can guarantee that the local estimates asymptotically converge to the optimal solution. Furthermore, in order to improve the convergence speed, \cite{wang2017distributed} proposed an optimization algorithm,
which does not require the sum of the squared stepsizes to be limited.
However, the convergence speed of the optimization algorithm based on diminishing stepsizes is still slow.
Compared with the diminishing stepsizes, the method of fixed stepsizes can guarantee faster convergence.
Nevertheless, it only makes the local estimate converge to a certain neighborhood of the optimal solution.
By requiring the local cost functions to be strongly convex and Lipschitz continuous,
literature \cite{nedic2017achieving,xi2018linear,saadatniaki2020decentralized}
proposed a scheme with fixed stepsizes,
where the optimal solution can be linearly converged to. Here, the fixed stepsizes should be smaller than a certain value determined by the cost functions.

The previously mentioned research on optimization all require all agents in the system to be able to collaboratively calculate the global minimum point. However, if some individuals in the system become uncontrolled or spread wrong information, the above obtained results are no longer valid,
and the original optimization problem $\min \sum\nolimits_{i = 1}^n {{Q_i}(x)}$
also becomes meaningless.
More reasonably, we should make those agents working normally gather and minimize the cost of these normal agents.
That is to say, the optimization problem becomes $\min\sum\nolimits_{i\in {{\mathcal{V}}_{\mathcal{N}}}}{{{Q}_{i}}(x)}$,
where $\mathcal V_\mathcal N$ represents the set of normal agents.










The rest of the paper is organized as follows.
In Section \ref{SectionII}, some necessary preliminaries on the graph theory are provided and the resilient consensus problem is formulated.
The convergence analysis of the TSR-based resilient consensus algorithm is presented in Section \ref{SectionIII}. 
Finally, conclusions are summarized in Section \ref{SectionVI}.

\emph{Notation:}
{Symbols $\mathbb R$, $\mathbb N$ and $\mathbb Z_+$ denote the sets of real numbers, natural numbers and positive integers, respectively.
The sets of $n \times m$ real matrices and $n$-dimensional real column vectors are respectively represented by $\mathbb R^{n \times m}$ and $\mathbb R^n$.
Let $I_n$ represent the identity matrix of dimension $n$, which means that each diagonal entry of the diagonal matrix is equal to $1$.
$|\mathcal V|$ denotes the number of elements in set $\mathcal V$.
Symbols $\max \{\cdot\}$ and $\min \{\cdot\}$ are used to denote, respectively, the maximum and minimum values of all entries of the vectors.}

\section{Preliminaries and Problem Formulation}\label{SectionII}

\subsection{Graph Theory}

Let $\mathcal{G}(\mathcal{V}, \mathcal{E})$ be a directed graph (digraph) with the node set $\mathcal{V}=\{1,\ldots,n\}$ and edge set $\mathcal{E}\subseteq\mathcal{V}\times\mathcal{V}$. The edge $\left( {j,i} \right)\in \mathcal{E}$ indicates that node $i$ can receive information from node $j$.
The neighbor set of $i$ is denoted by ${{\mathcal N}_i} = \{ j\left| {(j,i) \in {\mathcal E}} \right.\}$. The adjacency matrix $A=[a_{ij}]\in\mathbb{R}^{n\times n}$ associated with the digraph $\mathcal{G}$ is defined by ${a_{ij}} > 0$ if $(j,i)\in\mathcal{E}$, otherwise $a_{ij}=0$. Furthermore, self-loops are not allowed, i.e., $a_{ii}=0$ for $i=1,2,\ldots,n$. A directed path from node $v_1$ to node $v_p$ is an edge sequence $\{({v_1},{v_2}),({v_2},{v_3}),\ldots,({v_{p - 1}},{v_p})\}$ in which $(v_{i},v_{i+1})\in\mathcal{E}$ for $i=1,\ldots,p-1$. A directed tree is a digraph in which there exists a node, called the root, such that any other node can be reached by one and only one directed path starting at the root.
In a digraph, if there exists one node which has a directed path to all other nodes, then this digraph is said to contain a directed spanning tree.
A digraph, denoted by $\mathcal{G}_\mathcal{S}(\mathcal{V}_\mathcal{S},\mathcal{E}_\mathcal{S})$, is called a subgraph of $\mathcal{G}(\mathcal{V},\mathcal{E})$ if $\mathcal{V}_\mathcal{S} \subseteq \mathcal{V}$ and $\mathcal{E}_\mathcal{S} \subseteq \mathcal{E}$.
Furthermore, if ${{\mathcal E}_{\mathcal S}} = \{ \left( {j,i} \right)\left|\left( {j,i} \right) \in {\mathcal E}, i,j \in {{\mathcal V}_\mathcal S}\right.\}$, $\mathcal{G}_\mathcal{S}(\mathcal{V}_\mathcal{S},\mathcal{E}_\mathcal{S})$
is called {the subgraph induced by $\mathcal V_\mathcal S$}.

\subsection{Problem Formulation}
Considering a MAS consisting of normal agents and Byzantine agents, which cooperates on digraph $\mathcal G(\mathcal V,\mathcal E)$.
Assume that each agent $i \in \mathcal V$ has a local cost function $Q_i(x):\mathbb R^d\rightarrow \mathbb R$,
$x\in \mathbb R^d$ is the global decision variable.
The objective of this paper is to minimize the following multi-agent optimization problem
under $f$-total Byzantine attacks:
\begin{equation}\label{OptimizationModel}
\text{minimize}\ \ \ \ \sum\limits_{i\in {{\mathcal{V}}_{\mathcal{N}}}}{{{Q}_{i}}(x)},
\end{equation}
where $\mathcal V_{\mathcal N}$ is the set of normal agents.
The local cost function of each normal agent is convex and differentiable.
The set of Byzantine agents is denoted by $\mathcal V_{\mathcal B}$.

In general, the similar optimization problem is solved by the distributed manner.
$x_{i}[k]$ is used to represent agent $i$'s local estimate of the optimal solution
$x^* \in \arg \min \sum\nolimits_{i \in {\mathcal V_\mathcal N}} {{Q_i}(x)}$ at time step $k$.
Based on the communication graph $\mathcal G$, each agent transmits the local estimate $x_i[k]$ and the gradient $\nabla{{Q}_{i}}({{x}_{i}}[k])$
to its out-neighbors at time step $k$.
Nevertheless, the received data may be false and deceptive due to the presence of Byzantine agents.
In detail, respectively denote the local estimate and gradient transmitted from agent $j$ to agent $i$ by $m_{ij}[k]$ and $g_{ij}[k]$.
If agent $j$ is a normal agent, i.e., $j \in \mathcal N_i \cap \mathcal V_\mathcal N$,
then $m_{ij}[k]=x_j[k]$ and $g_{ij}[k]=\nabla{{Q}_{j}}({{x}_{j}}[k])$.
Instead, if agent $j$ is a Byzantine agent, i.e., $j \in \mathcal N_i \cap \mathcal V_\mathcal B$,
then $m_{ij}[k]$ and $g_{ij}[k]$ may be the arbitrary $d$-dimensional vectors.
Unfortunately, each agent does not know that the received data is from normal agents or Byzantine agents.
\begin{definition}
Under $f$-total Byzantine attacks, if for any initial local estimates ${x_1}[0],{x_2}[0],\ldots,{x_n}[0]$ and
\textbf{any possible set of Byzantine agents $\mathcal V_\mathcal B \subseteq \mathcal V$},
there exists some distributed algorithm such that all normal agents' local estimates reach consensus at the minimum point.
For details, the following two conditions are satisfied:

1) $\mathop {\lim }\nolimits_{k \to \infty } \left\| {{x_i}[k] - {x_j}[k]} \right\|_2= 0, \;\;\;\;\; \forall i,j \in \mathcal V_\mathcal N$;

2) $\mathop {\lim }\nolimits_{k \to \infty } {x_i}[k] \in \arg \min \sum\nolimits_{j \in {\mathcal V_\mathcal N}} {{Q_j}(x)}, \;\;\;\; \forall i \in \mathcal V_\mathcal N$.\\
Then, this algorithm is said to be resilient to the $f$-total Byzantine attacks.
\end{definition}
\begin{remark}
The most literature on resilient optimization only gives an approximate optimal solution
\cite{su2020byzantine, zhao2019resilient, kuwaranancharoen2020byzantine, sundaram2018distributed, yan2020resilient},
i.e., the estimates of the optimal solution finally converge to a bounded region containing the optimal solution $x^*$.
Furthermore, this paper will present an approach to make the local estimates converge to the exact optimal solution $x^*$.
\end{remark}

The literature \cite{su2016fault} shows the fact that the optimal solution of the problem (\ref{OptimizationModel}) cannot be obtained in the general cases.
To give an approach to obtain the optimal solution, the literature \cite{gupta2020fault,gupta2021byzantine} constrain the cost functions
to satisfy the newly proposed $2f$-redundancy condition. Furthermore, they prove that the proposed $2f$-redundancy condition is necessary and sufficient
to obtain the exact optimal solution. Next, we will give the definition of $2f$-redundancy.
\begin{definition}\label{Redundancy}
\cite{gupta2020fault,gupta2021byzantine}
($2f$-redundancy)
For the given set of normal agents $\mathcal V_\mathcal N$,
if for any subset $S\subseteq \mathcal V_\mathcal N$ with $\left| S \right| \ge n - 2f$, it holds that
\[\mathop {\arg \min }\limits_{x \in {\mathbb R^d}} {\mkern 1mu} \sum\limits_{i \in S} {{Q_i}(x)}= \mathop {\arg \min }\limits_{x \in {\mathbb R^d}} \sum\limits_{i \in {{\cal V}_{\cal N}}} {{Q_i}(x)},\]
then the normal agents' cost functions are said to satisfy the $2f$-redundancy condition.
\end{definition}
\begin{remark}
The $2f$-redundancy condition shows that the minimum point of the sum of any $n-2f$ normal agents' cost functions
is the same as the minimum point of all normal agents' cost functions. From this point of view,
this redundancy condition seems to be strict. However, the redundancy in cost functions often happens in the actual scenarios,
such as the distributed sensing \cite{gupta2019byzantine,chong2015observability}
and the homogeneous distributed learning \cite{blanchard2017machine,gupta2020byzantine}.
\end{remark}

In general, we also need to make the following necessary assumptions on the cost functions $Q_i(x), i \in \mathcal V$.
\begin{assumption}\label{BoundedSolutionAssumption}
(Existence)
$\arg \min \sum\nolimits_{i\in {{\mathcal{V}}_{\mathcal{N}}}}{{{Q}_{i}}(x)}$ is nonempty and bounded,
i.e., there exists $a\in \arg \min \sum\nolimits_{i\in {{\mathcal{V}}_{\mathcal{N}}}}{{{Q}_{i}}(x)}$ and $\xi \in (0,\infty)$
such that $a\in \mathcal X=[-\xi ,\xi ]^{d}\subseteq \mathbb R^{d}$,
where
${[-\xi ,\xi ]^d} = \{ ({x^1},\ldots,{x^d}) \in {\mathbb R^d}\left| {{x^1} \in [ - \xi ,\xi ],\ldots,{x^d} \in [ - \xi ,\xi ]\} } \right.$.
\end{assumption}
\begin{remark}
$\xi \in \mathbb R$ can be chosen large enough such that the normal agent's initial local estimate $x_i[0]\in \mathcal X, i \in \mathcal V_\mathcal N$.
\end{remark}
\begin{assumption}\label{GradientContinuousAssumption}
(Lipschitz Continuous Gradients)
For each normal agent $i \in \mathcal V_\mathcal N$, the local cost function $Q_i(x)$ is differential
and the gradient $\nabla {{Q}_{i}}(x)$ is Lipschitz continuous in set $\mathcal X$. For details,
for any $i\in \mathcal V_\mathcal N$, there exists $\mu<\infty$ such that
\[\left\| \nabla {{Q}_{i}}(x)-\nabla {{Q}_{i}}(y) \right\|_2\le \mu \left\| x-y \right\|_2, \;\;\;\;\;\;\forall x,y\in \mathcal X.\]
\end{assumption}
\begin{assumption}\label{StrongConvexityAssumption}
(Strongly Convex Global Cost Functions)
The average normal agents' cost function
\[Q(x) \buildrel \Delta \over = \frac{1}{{\left| {{{\cal V}_{\cal N}}} \right|}}\sum\nolimits_{i \in {{\cal V}_{\cal N}}} {{Q_i}(x)}\]
is strongly convex in set $\mathcal X$. For details, there exists a real number $0<\lambda <\infty$ such that
\[\left\langle x-y,\nabla Q(x)-\nabla Q(y) \right\rangle \ge \lambda \left\| x-y \right\|_{2}^{2}, \;\;\;\;\;\;\forall x,y\in \mathcal X.\]
\end{assumption}
\begin{remark}
Based on the literature \cite{gupta2020fault, gupta2021byzantine}, the parameters $\mu$ and $\lambda$
in Assumption \ref{GradientContinuousAssumption} and Assumption \ref{StrongConvexityAssumption} have the following relation
under the $2f$-redundancy condition:
$\mu \geq \lambda$.
\end{remark}

In the literature \cite{gupta2020fault,gupta2021byzantine} on resilient optimization with redundancy in cost functions,
they either consider the server-based system architecture (centralized control) or peer-to-peer system architecture (distributed control)
with the complete communication topology.
Compared with these relevant researches, our work in this paper not only is independent on the centralized control,
but also does not require the communication topology relied on by the distributed control to be complete.


Therefore, we take the distributed control manner to solve the Byzantine-resilient optimization problem (\ref{OptimizationModel}) in this paper.

For details, we have the following assumption on the communication topology (in-degree of each normal agent):
\begin{assumption}\label{CommunicationTopologyAssumption}
For any $i \in \mathcal V_\mathcal N$, the number of its in-neighbors satisfies the condition that
\[\left| {{{\mathcal N}_i}} \right| > 3f + 1 + (1 - \frac{\lambda }{{\sqrt d \mu }})n.\]
\end{assumption}

\subsection{MSR-based Distributed Comparative Gradient Elimination (MSR-DCGE) Resilient Optimization Algorithm}
For the server-based system architecture under the $f$-total Byzantine attack,
the literature \cite{gupta2020fault} proposes the Comparative Gradient Clipping (CGC) method
to correct the $f$ gradients with the largest $2$-norm. In the literature \cite{Arxivgupta2021byzantine} considering
the peer-to-peer system architecture under the $f$-total Byzantine attack,
each normal agent utilizes the classical MSR method to filter out the suspicious local estimates and
the Comparative Gradient Elimination (CGE) method to filter out the $f$ gradients with the largest $2$-norm.
However, the communication topology is required to be complete.
In this paper, we try to utilize the distributed control manner to deal with the Byzantine optimization problem.
Moreover, we do not require the communication topology to be complete, i.e

For the resilient optimization problem (\ref{OptimizationModel}) under the $f$-total Byzantine attacks,
we will design an algorithm only using the information from in-neighbors to iteratively update the local estimate
$x_i[k], i \in \mathcal V_\mathcal N$.

In details,
the MSR-based Distributed Comparative Gradient Elimination (MSR-DCGE) resilient optimization algorithm has the following seven steps
in the $(k+1)$-th ($k=0,1,\ldots$) iteration:

1) All agents transmit their local estimates and the corresponding gradients to their out-neighbors;

2) Each normal agent $i \in \mathcal V_\mathcal N$ sorts each component $m_{ij}^l[k]$
of the received local estimates $m_{ij}[k],j \in \mathcal N_i$ in an increasing order:
\[m_{i{{j}_{1}}}^{l}[k]\le ...\le m_{i{{j}_{\left| {{\mathcal N}_{i}} \right|}}}^{l}[k],\;\;\;\;\;\;\;l\in \{1,2,...,d\},\]
where $l$ represents the $l$-th component;

3)Denote
\[{\mathcal I}_i^l[k] \buildrel \Delta \over = \{ j \in {{\mathcal N}_i}\left| {m_{ij}^l[k] > x_i^l[k]} \right.\},\]
\[{\mathcal J}_i^l[k] \buildrel \Delta \over = \{ j \in {{\mathcal N}_i}\left| {m_{ij}^l[k] < x_i^l[k]} \right.\},\]
where $l\in \{1,2,...,d\}$, $i \in \mathcal V_\mathcal N$.
If $\left| {{\mathcal I}_i^l[k]} \right| < f$, then let $w_{ij}^{l}[k]=0$ for all $j \in \mathcal I_i^l[k]$.
Otherwise, let $w_{ij}^{l}[k]=0$ for $j \in \{{j_{\left| {{{\mathcal N}_i}} \right| - f + 1}},\ldots,{j_{\left| {{{\mathcal N}_i}} \right|}}\}$.
Similarly, if $\left| {{\mathcal J}_i^l[k]} \right| < f$, let $w_{ij}^{l}[k]=0$ for all $j \in \mathcal J_i^l[k]$.
Otherwise, let $w_{ij}^{l}[k]=0$ for $j \in \{j_1,\ldots,j_f\}$.
Denote the set of remaining in-neighbors as $\mathcal R_i^l[k]$,
let
$w_{ij}^l[k] = {1 \mathord{\left/{\vphantom {1 {\left| {{\mathcal R}_i^l[k]} \right|}}} \right.\kern-\nulldelimiterspace} {(\left| {{\mathcal R}_i^l[k]}\right|+1)}}$ for $j \in \mathcal R_i^l[k]\cup\{i\}$;

4) For $l\in \{1,\ldots, d\}$, $i \in \mathcal V_\mathcal N$, let
\[u_{i}^{l}[k]=\sum\limits_{j\in {{\mathcal{N}}_{i}\cup\{i\}}}{w_{ij}^{l}[k]m_{ij}^{l}[k]},\]
where $m_{ii}^{l}[k]=x_i^l[k]$;

5) Each normal agent $i \in \mathcal V_\mathcal N$ calculates the $2$-norm of the received gradients and sorts them in an increasing order:
\[{{\left\|{{g}_{i{{s}_{1}}}}[k] \right\|}_{2}}\le \ldots \le {{\left\|{{g}_{i{{s}_{\left| {{\mathcal N}_{i}} \right|}}}}[k] \right\|}_{2}}.\]
Denote $\mathcal M_i[k]=\{s_1,\ldots,s_{\left| {{{\mathcal N}_i}} \right| - f}\}$. Particularly, $\mathcal M_i[k]=\emptyset$
if $\left| {{{\mathcal N}_i}} \right|\leq f$.

6) Let
\begin{equation}\label{GradientAggregation}
{{c}_{i}}[k]=\sum\limits_{j\in {{\mathcal{M}}_{i}}[k]\cup \{i\}}{{{g}_{ij}}[k]},
\end{equation}
where $g_{ii}[k]=\nabla Q_i(x_i[k]), i\in \mathcal V_\mathcal N$.

7) Each normal agent $i\in \mathcal V_\mathcal N$ updates its local estimate as
\begin{equation}\label{OptimalAlgorithm}
{{x}_{i}}[k+1]={{P}_{\mathcal X}}({{u}_{i}}[k]-{{\alpha}_{k}}c_i[k]),
\end{equation}
where
\[{{P}_{\mathcal X}}(x)=\arg \underset{y\in \mathcal X}{\mathop{\min }}\,\left\| x-y \right\|_2,\]
$\alpha_k>0$ is the step-size at the $(k+1)$-th iteration.

To guarantee the effectiveness of the MSR-DCGE resilient optimization algorithm,
we have certain requirements for the step-size $\alpha_k$.
\begin{assumption}\label{StepSizeAssumption}
The sequence of step-size $\{\alpha_k\}_{k=0}^{\infty}$ satisfies the conditions:
\[{{\alpha }_{k+1}}\le {{\alpha }_{k}},\;\;
\sum\limits_{k=0}^{\infty }{{{\alpha }_{k}}}=\infty,\;\;
\sum\limits_{k=0}^{\infty }{\alpha _{k}^{2}}<\infty.\]
\end{assumption}

\section{Convergence Analysis of MSR-DCGE Resilient Optimization Algorithm}\label{SectionIII}
This section will specifically analyze how the local estimates of decision variable $x$ converge to the optimal solution to problem (\ref{OptimizationModel})
under the proposed MSR-DCGE resilient optimization algorithm.

First, we give several necessary lemmas.

For $v \in \mathbb R$, we define the function
\[{(v)_ + } \buildrel \Delta \over =  \left\{{\begin{array}{*{20}{c}}
{v,\;\;\;v > 0},\\
{0,\;\;\;v \le 0}.
\end{array}} \right.\]
\begin{lemma}\label{FiniteLimitLemma}
\cite{bottou1998online}
For a non-negative sequence of real numbers $\left\{ {{\delta _k}} \right\}_{k = 0}^\infty$, if it holds that $\sum\nolimits_{k = 0}^\infty {{{({\delta _{k + 1}} - {\delta _k})}_ + }}  < \infty$,
then there exists some non-negative real number $\delta < \infty$ such that $\mathop {\lim }\nolimits_{k \to \infty } {\delta _k} = \delta$.
\end{lemma}
\begin{lemma}\label{SequenceLimitZero}
For the non-negative sequences of real numbers $\{{\tau_k}\}_{k = 0}^\infty$ and $\{{\delta_k}\}_{k = 0}^\infty$,
if there holds that $\sum\nolimits_{k = 0}^\infty  {{\tau_k}}  = \infty$,
$\mathop {\lim }\nolimits_{k \to \infty } {\delta_k}=\delta < \infty$ and $\sum\nolimits_{k = 0}^\infty  {{\tau_k}{\delta_k}} < \infty$,
then $\delta = 0$.
\end{lemma}
\begin{proof}
It is obvious to know that $\delta \geq 0$.

We prove it by contradiction. Suppose that $\mathop {\lim }\nolimits_{k \to \infty } {\delta_k} = \delta> 0$.
Therefore, there exists $K \in \mathbb N$ such that
\[{\delta_k} > \frac{\delta }{2}\]
for $k> K$.
It is subsequently derived that
\[\sum\limits_{k = K + 1}^\infty  {{\tau _k}{\delta _k}}  > \frac{\delta }{2}\sum\limits_{k = K + 1}^\infty {{\tau _k}}
= \frac{\delta }{2}\left(\sum\limits_{k = 0}^\infty {{\tau _k}} - \sum\limits_{k = 0}^K {{\tau_k}} \right).\]
Furthermore, we have
\[\sum\limits_{k = K + 1}^\infty  {{\tau_k}{\delta _k}}= \infty\]
due to $\sum\nolimits_{k = 0}^\infty  {{\tau _k}}  = \infty$.
Since $\{ {\tau_k}\}_{k = 0}^\infty$ and $\{{\delta_k}\}_{k = 0}^\infty$ are both the sequences of non-negative real numbers,
it is obtained that
\[\sum\limits_{k = 0}^\infty {{\tau _k}{\delta _k}}\ge \sum\limits_{k = K + 1}^\infty{{\tau_k}{\delta_k}}= \infty,\]
which is a contradiction. Therefore, we have $\mathop {\lim }\nolimits_{k \to \infty} {\delta_k}=\delta=0$.
\end{proof}
\begin{lemma}\label{NedicLemma}
\cite{nedic2010constrained}
Let $\mathcal X \subseteq \mathbb R^d$ be a non-empty closed convex set, then for any $x \in \mathbb R^d$ and $y \in \mathcal X$, the following two results are established:

(i)
${({P_{\mathcal X}}(x) - x)^\mathrm T}(x - y) \le  - \left\| {{P_{\mathcal X}}(x) - x} \right\|_2^2$;

(ii)
$\left\| {{P_{\mathcal X}}(x) - y} \right\|_2^2 \le \left\| {x - y} \right\|_2^2 - \left\| {{P_{\mathcal X}}(x) - x} \right\|_2^2$.

\end{lemma}






Next, we will present several important propositions, which play crucial roles in the proof of Theorem \ref{MainResult}.
\begin{proposition}\label{FilteringGradientBoundedness}
Under (\ref{GradientAggregation}), there exists a positive real number $L<\infty$ such that
\[{{\left\|{{c}_{i}}[k] \right\|}_{2}}\le L\]
for any $i \in \mathcal V_\mathcal N$ and $k\geq 0$ if Assumptions \ref{BoundedSolutionAssumption}, \ref{GradientContinuousAssumption} and \ref{CommunicationTopologyAssumption} hold.
\end{proposition}
\begin{proof}
Under the MSR-DCGE resilient optimization algorithm, we have
\begin{equation}\label{BoundedGradientInequality}
\begin{array}{l}
{\left\| {{c_i}[k]} \right\|_2} = {\left\| {\sum\nolimits_{j = 1}^{\left| {{{\cal N}_i}} \right| - f} {{g_{i{s_j}}}[k] + {g_{ii}}[k]} } \right\|_2}\\
\;\;\;\;\;\;\;\;\;\;\;\;\; \le (\left| {{{\cal N}_i}} \right| - f){\left\| {{g_{i{s_l}}}[k]} \right\|_2} + {\left\| {{g_{ii}}[k]} \right\|_2}
\end{array}
\end{equation}
for any $l \in \{|\mathcal N_i|-f,|\mathcal N_i|-f+1,\ldots, |\mathcal N_i|\}$.
Combine (\ref{BoundedGradientInequality}) with the fact that there are at most $f$ Byzantine agents under the $f$-total Byzantine attacks,
there exists some normal agent $j_i \in \{s_{|\mathcal N_i|-f},\ldots,s_{|\mathcal N_i|}\}\cap \mathcal V_\mathcal N$ such that
\begin{equation}\label{BiggerInequality}
{{\left\| {{c}_{i}}[k] \right\|}_{2}}\le (\left| {{\mathcal{N}}_{i}} \right|-f){{\left\| \nabla {{Q}_{{{j}_{i}}}}({{x}_{{{j}_{i}}}}[k]) \right\|}_{2}}+{{\left\| \nabla {{Q}_{i}}({{x}_{i}}[k]) \right\|}_{2}}.
\end{equation}
Based on the definition of projection $P_\mathcal X(x)$, it is known that $x_j[k]\in \mathcal X$ for any $j \in \mathcal V_\mathcal N$ and $k\geq 0$.
Also, the cost functions of normal agents are differential in $\mathcal X$, it can be obtained that
\begin{equation}\label{SetMaxValue}
{{\left\| \nabla {{Q}_{j}}({{x}_{j}}[k]) \right\|}_{2}}\le \underset{x\in \mathcal X}{\mathop{\max }}\,{{\left\| \nabla {{Q}_{j}}(x) \right\|}_{2}}<\infty
\end{equation}
for any $j \in \mathcal V_\mathcal N$ and $k \geq 0$.
Combining (\ref{BiggerInequality}) with (\ref{SetMaxValue}) yields that
\[\begin{array}{l}
{\left\| {{c_i}[k]} \right\|_2} \le (\left| {{{\cal N}_i}} \right| - f)\mathop {\max }\limits_{x \in {\cal X}} {\mkern 1mu} {\left\| {\nabla {Q_{{j_i}}}(x)} \right\|_2} + \mathop {\max }\limits_{x \in {\cal X}} {\mkern 1mu} {\left\| {\nabla {Q_i}(x)} \right\|_2}\\
\;\;\;\;\;\;\;\;\;\;\;\;\; \le (\left| {{{\cal N}_i}} \right| - f + 1)\mathop {\max }\limits_{j \in {{\cal V}_{\cal N}}} {\mkern 1mu} \mathop {\max }\limits_{x \in {\cal X}} {\mkern 1mu} {\left\| {\nabla {Q_j}(x)} \right\|_2}\\
\;\;\;\;\;\;\;\;\;\;\;\;\; \le (n - f)\mathop {\max }\limits_{j \in {{\cal V}_{\cal N}}} {\mkern 1mu} \mathop {\max }\limits_{x \in {\cal X}} {\mkern 1mu} {\left\| {\nabla {Q_j}(x)} \right\|_2}.
\end{array}\]
Denote
$L\triangleq(n-f)\underset{j\in {{\mathcal{V}}_{\mathcal{N}}}}{\mathop{\max }}\,\underset{x\in \mathcal{X}}{\mathop{\max }}\,{{\left\| \nabla {{Q}_{j}}(x) \right\|}_{2}}<\infty$,
the result can be subsequently proved.
\end{proof}
\begin{proposition}\label{EstimationErrorBound}
Under the MSR-DCGE resilient optimization algorithm, if Assumptions \ref{BoundedSolutionAssumption}, \ref{GradientContinuousAssumption}, \ref{CommunicationTopologyAssumption} and \ref{StepSizeAssumption} hold,
there exist $0<\theta<1$ and $\eta <\infty$ such that
\[\left| x_{i}^{l}[k]-x_{j}^{l}[k] \right|\le \left| {{\mathcal V}_{\mathcal N}} \right|({{\theta }^{k}}\underset{p\in {{\mathcal V}_{\mathcal N}}}{\mathop{\max }}\,\left| x_{p}^{l}[0] \right|+\eta \sum\limits_{s=0}^{k}{{{\alpha }_{s}}{{\theta}^{k-s}}})\]
for any $i,j \in \mathcal V_\mathcal N$, $l\in\{1,2,\ldots,d\}$ and $k\geq 0$.
\end{proposition}
\begin{proof}
Denote the $l$-th component of $P_{\mathcal X}(x)$ as $P_{\mathcal X}^{l}(x)$. Based on the definition of $P_{\mathcal X}(x)$, we have
\begin{equation}\label{ProjectionTranslation}
P_{\mathcal X}^{l}(x)={{P}_{[-\xi ,\xi ]}}({{x}^{l}})\triangleq \underset{y\in [-\xi ,\xi ]}{\mathop{\arg \min }}\,\left| {{x}^{l}}-y \right|.
\end{equation}
Therefore, for any $l \in \{1,2,\ldots, d\}$, it can be derived from (\ref{OptimalAlgorithm}) that
\[\begin{array}{l}
x_i^l[k + 1] = {P_{[ - \xi ,\xi ]}}(u_i^l[k] - {\alpha _k}c_i^l[k])\\
\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; = u_i^l[k] - {\alpha _k}c_i^l[k] + {P_{[ - \xi ,\xi ]}}(u_i^l[k] - {\alpha _k}c_i^l[k])\\
\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;- (u_i^l[k] - {\alpha _k}c_i^l[k]).
\end{array}\]
Let
${{e}_{i}}[k]\triangleq{{P}_{\mathcal X}}({{u}_{i}}[k]-{{\alpha }_{k}}{{c}_{i}}[k])-({{u}_{i}}[k]-{{\alpha }_{k}}{{c}_{i}}[k])$,
it is easy to obtain that
\[e_{i}^{l}[k]={{P}_{[-\xi ,\xi ]}}(u_{i}^{l}[k]-{{\alpha}_{k}}c_{i}^{l}[k])-(u_{i}^{l}[k]-{{\alpha }_{k}}c_{i}^{l}[k])\]
for any $l \in \{1,2,\ldots, d\}$.
Furthermore, we have
\begin{equation}\label{ErrorQuantity}
x_{i}^{l}[k+1]=u_{i}^{l}[k]-{{\alpha }_{k}}c_{i}^{l}[k]+e_{i}^{l}[k].
\end{equation}
Next, we prove that
$\left| e_{i}^{l}[k] \right|\le {{\alpha }_{k}}L$.

Based on (\ref{ErrorQuantity}), it can be derived that
\[e_{i}^{l}[k]=x_{i}^{l}[k+1]-(u_{i}^{l}[k]-{{\alpha }_{k}}c_{i}^{l}[k]).\]
Because $u_i^l[k]$ is obtained by the MSR method, it is known that $u_i^l[k]$ can be represented as a convex combination of all normal agents' local estimates, i.e.,
\begin{equation}\label{ConvexCombination}
u_{i}^{l}[k]=\sum\nolimits_{j\in {{\mathcal V}_{\mathcal N}}}{\beta_{ij}^{l}[k]x_{j}^{l}[k]},
\end{equation}
where $\beta_{ij}^l[k]\geq 0$, $\sum\nolimits_{j\in {{\mathcal V}_{\mathcal N}}}{\beta _{ij}^{l}[k]=1}$.
Moreover, it is established that $x_j^l[k] \in [-\xi ,\xi ]$ for any $j \in \mathcal V_\mathcal N$ and $k \in \mathbb N$.
Therefore, we have $u_i^l[k] \in [-\xi,\xi]$.
According to the result (ii) in Lemma \ref{NedicLemma}, we have
\begin{equation}\label{SingleDimensionProjection}
{{\left\| {{P}_{\mathcal X}}({{\phi}_{1}})-{{\phi}_{2}} \right\|}_2^{2}}\le {{\left\| {{\phi}_{1}}-{{\phi}_{2}} \right\|}_2^{2}}-{{\left\| {{P}_{\mathcal X}}({{\phi}_{1}})-{{\phi}_{1}} \right\|}_2^{2}}
\end{equation}
for any $\phi_1 \in \mathbb R^d$ and $\phi_2 \in \mathcal X$.
For the case of $d=1$, let
\[{\phi_1} = u_i^l[k] - {\alpha _k}c_i^l[k],\]
\[{\phi_2} = u_i^l[k].\]
It is derived from (\ref{SingleDimensionProjection}) that
\[\begin{array}{l}
{\left( {x_i^l[k + 1] - u_i^l[k]} \right)^2}
 \le {\left( {u_i^l[k] - {\alpha _k}c_i^l[k] - u_i^l[k]} \right)^2}\\
\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;- {\left( {x_i^l[k + 1] - (u_i^l[k] - {\alpha _k}c_i^l[k])} \right)^2}\\
\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;= {({\alpha _k}c_i^l[k])^2} - {(e_i^l[k])^2}.
\end{array}\]
Furthermore, we have
\[{(e_i^l[k])^2} \le {({\alpha _k}c_i^l[k])^2} - {\left( {x_i^l[k + 1] - u_i^l[k]} \right)^2} \le {({\alpha _k}c_i^l[k])^2}.\]
Therefore, it is obtained from Proposition \ref{FilteringGradientBoundedness} that
\begin{equation}\label{ErrorBound}
\left| {e_i^l[k]} \right| \le {\alpha _k}\left| {c_i^l[k]} \right| \le {\alpha _k}{\left\| {{c_i}[k]} \right\|_2} \le {\alpha _k}L
\end{equation}
for any $i\in \mathcal V_\mathcal N$.

Based on [ref. \cite{su2020byzantine}, Lemma 2 and its proof], combining (\ref{ErrorQuantity}) with (\ref{ErrorBound}) yields the result:
there exist $0<\theta<1$ and $\eta <\infty$ such that
\[\left| x_{i}^{l}[k]-x_{j}^{l}[k] \right|\le \left| {{\mathcal{V}}_{\mathcal{N}}} \right|{{\theta }^{k}}\underset{p\in {{\mathcal{V}}_{\mathcal{N}}}}{\mathop{\max }}\,\left| x_{p}^{l}[0] \right|+\eta \left| {{\mathcal{V}}_{\mathcal{N}}} \right|\sum\limits_{s=0}^{k}{{{\alpha }_{s}}{{\theta }^{k-s}}}.\]
\end{proof}
\begin{proposition}\label{StepEstimationErrorFinite}
Under the MSR-DCGE resilient optimization algorithm, if Assumptions \ref{BoundedSolutionAssumption}, \ref{GradientContinuousAssumption}, \ref{CommunicationTopologyAssumption} and \ref{StepSizeAssumption} hold,
then it is established that
\[\sum\limits_{k=0}^{\infty }{{{\alpha }_{k}}}\underset{i,j\in {{\mathcal V}_{\mathcal N}}}{\mathop{\max }}\,\left| x_{i}^{l}[k]-x_{j}^{l}[k] \right|<\infty\]
for any $l \in \{1,2,\ldots,d\}$.
\end{proposition}
\begin{proof}
Based on Proposition \ref{EstimationErrorBound}, it is known that there exist $0<\theta<1$ and $\eta <\infty$ such that
\begin{equation}\label{MaxEstimationErrorBound}
\begin{array}{l}
\mathop {\max }\limits_{i,j \in {{\cal V}_{\cal N}}} {\mkern 1mu} \left| {x_i^l[k] - x_j^l[k]} \right| \le \\
\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left| {{{\cal V}_{\cal N}}} \right|({\theta ^k}\mathop {\max }\limits_{p \in {{\cal V}_{\cal N}}} {\mkern 1mu} \left| {x_p^l[0]} \right| + \eta \sum\limits_{s = 0}^k {{\alpha _s}{\theta ^{k - s}}} )
\end{array}
\end{equation}
for any $l \in \{1,2,\ldots,d\}$ and $k\geq 0$.
Multiply $\alpha_k$ on both sides of equation (\ref{MaxEstimationErrorBound}) and sum it on $k=0,1,\ldots$, we have
\[\begin{array}{l}
\sum\limits_{k = 0}^\infty  {{\alpha _k}} \mathop {\max }\limits_{i,j \in {{\cal V}_{\cal N}}} {\mkern 1mu} \left| {x_i^l[k] - x_j^l[k]} \right| \le \\
\left| {{{\cal V}_{\cal N}}} \right|\mathop {\max }\limits_{p \in {{\cal V}_{\cal N}}} {\mkern 1mu} \left| {x_p^l[0]} \right|\sum\limits_{k = 0}^\infty  {{\alpha _k}{\theta ^k}}  + \eta \left| {{{\cal V}_{\cal N}}} \right|\sum\limits_{k = 0}^\infty  {({\alpha _k}\sum\limits_{s = 0}^k {{\alpha _s}{\theta ^{k - s}}} )},
\end{array}\]
where
\[\begin{array}{l}
\left| {{{\cal V}_{\cal N}}} \right|\mathop {\max }\limits_{p \in {{\cal V}_{\cal N}}} {\mkern 1mu} \left| {x_p^l[0]} \right|\sum\limits_{k = 0}^\infty  {{\alpha _k}{\theta ^k} \le {\alpha _0}\left| {{{\cal V}_{\cal N}}} \right|\mathop {\max }\limits_{p \in {{\cal V}_{\cal N}}} {\mkern 1mu} \left| {x_p^l[0]} \right|} \sum\limits_{k = 0}^\infty  {{\theta ^k}} \\
\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; = \dfrac{{{\alpha _0}\left| {{{\cal V}_{\cal N}}} \right|\mathop {\max }\limits_{p \in {{\cal V}_{\cal N}}} {\mkern 1mu} \left| {x_p^l[0]} \right|}}{{1 - \theta }} < \infty .
\end{array}\]
As a result, we next only need to prove that
\[\sum\nolimits_{k=0}^{\infty }{({{\alpha}_{k}}\sum\nolimits_{s=0}^{k}{{{\alpha }_{s}}{{\theta }^{k-s}}})}<\infty,\]
which can conclude the proof.

$\sum\nolimits_{s=0}^{k}{{{\alpha }_{s}}{{\theta }^{k-s}}}$ can be divided into the sum of the following two parts:
\[\sum\limits_{s=0}^{k}{{{\alpha }_{s}}{{\theta }^{k-s}}}=\sum\limits_{s=0}^{\left\lfloor {k}/{2} \right\rfloor }{{{\alpha }_{s}}{{\theta }^{k-s}}}+
\sum\limits_{s=\left\lfloor {k}/{2} \right\rfloor +1}^{k}{{{\alpha }_{s}}{{\theta }^{k-s}}},\]
where
\[\sum\limits_{s=0}^{\left\lfloor {k}/{2} \right\rfloor }{{{\alpha }_{s}}{{\theta }^{k-s}}}\le {{\alpha }_{0}}\sum\limits_{s=0}^{\left\lfloor {k}/{2} \right\rfloor }{{{\theta }^{k-s}}}\le {{\alpha }_{0}}{{\theta }^{{k}/{2}\;}}\sum\limits_{s=0}^{\infty }{{{\theta }^{s}}}=\frac{{{\alpha }_{0}}{{\theta }^{{k}/{2}\;}}}{1-\theta },\]
\[\sum\limits_{s=\left\lfloor {k}/{2} \right\rfloor +1}^{k}{{{\alpha }_{s}}{{\theta }^{k-s}}}\le {{\alpha }_{\left\lfloor {k}/{2} \right\rfloor +1}}\sum\limits_{s=0}^{\infty }{{{\theta }^{s}}}=\frac{{{\alpha }_{\left\lfloor {k}/{2} \right\rfloor +1}}}{1-\theta }.\]
Therefore, we have
\[\sum\limits_{s=0}^{k}{{{\alpha }_{s}}{{\theta }^{k-s}}}\le \frac{{{\alpha }_{0}}{{\theta }^{{k}/{2}\;}}}{1-\theta }+\frac{{{\alpha }_{\left\lfloor {k}/{2} \right\rfloor +1}}}{1-\theta }=\frac{{{\alpha }_{0}}{{\theta }^{{k}/{2}}}+{{\alpha }_{\left\lfloor {k}/{2} \right\rfloor +1}}}{1-\theta }.\]
Furthermore, it follows that
\begin{equation}\label{FiniteJointItem}
\begin{array}{l}
\sum\limits_{k = 0}^\infty  {({\alpha _k}\sum\limits_{s = 0}^k {{\alpha _s}{\theta ^{k - s}}} )} \\
 \;\;\;\;\;\;\;\;\le \dfrac{1}{{1 - \theta }}({\alpha _0}\sum\limits_{k = 0}^\infty  {{\alpha _k}{\theta ^{k/2\;}} + \sum\limits_{k = 0}^\infty  {{\alpha _k}{\alpha _{\left\lfloor {k/2} \right\rfloor  + 1}}} } ).
\end{array}
\end{equation}
For the first term of the right side of (\ref{FiniteJointItem}), it is obtained from Assumption \ref{StepSizeAssumption} that
\[\sum\limits_{k=0}^{\infty }{{{\alpha }_{k}}{{\theta }^{{k}/{2}\;}}\le {{\alpha }_{0}}\sum\limits_{k=0}^{\infty }{{{(\sqrt{\theta })}^{k}}}}=\frac{{{\alpha }_{0}}}{1-\sqrt{\theta }}<\infty.\]
For the second term of the right side of (\ref{FiniteJointItem}), we have
\[\sum\limits_{k=0}^{\infty }{{{\alpha }_{k}}{{\alpha }_{\left\lfloor {k}/{2} \right\rfloor +1}}}\le \sum\limits_{k=0}^{\infty }{\alpha _{\left\lfloor {k}/{2} \right\rfloor }^{2}}=2\sum\limits_{k=0}^{\infty }{\alpha _{k}^{2}}<\infty.\]
In summary, it is obtained from (\ref{FiniteJointItem}) that $\sum\nolimits_{k=0}^{\infty }{({{\alpha }_{k}}\sum\nolimits_{s=0}^{k}{{{\alpha }_{s}}{{\theta }^{k-s}}})}<\infty$, which concludes the proof.
\end{proof}
\begin{proposition}\label{WAbsoluteBoundedness}
Under the MSR-DCGE resilient optimization algorithm, if Assumptions \ref{BoundedSolutionAssumption}, \ref{GradientContinuousAssumption} and \ref{CommunicationTopologyAssumption} hold,
then for any $i,j \in \mathcal V_\mathcal N$ and $l \in \{1,2,\ldots,d\}$, it is established that
\[\left| (u_{j}^{l}[k]-x_{i}^{l}[k])c_{j}^{l}[k] \right|\le L\underset{s\in {{\mathcal V}_{\mathcal N}}}{\mathop{\max }}\,\left| x_{s}^{l}[k]-x_{i}^{l}[k] \right|,\]
where
$L=(n-f)\underset{j\in {{\mathcal{V}}_{\mathcal{N}}}}{\mathop{\max }}\,\underset{x\in \mathcal{X}}{\mathop{\max }}\,{{\left\| \nabla {{Q}_{j}}(x) \right\|}_{2}}$.
\end{proposition}
\begin{proof}
Based on (\ref{ConvexCombination}), it is known that
\begin{equation}\label{ConvexCombinationII}
u_{j}^{l}[k]=\sum\limits_{s\in {{\mathcal V}_{\mathcal N}}}{\beta_{js}^{l}[k]x_{s}^{l}[k]}
\end{equation}
for any $l \in \{1,2,\ldots, d\}$, where
$\beta_{js}^l[k]\geq 0$, $\sum\nolimits_{s\in {{\mathcal V}_{\mathcal N}}}{\beta_{js}^{l}[k]=1}$.
Therefore, it can be inferred that
\[u_{j}^{l}[k]-x_{i}^{l}[k]=\sum\limits_{s\in {{\mathcal V}_{\mathcal N}}}{\beta_{js}^{l}[k](x_{s}^{l}[k]-x_{i}^{l}[k])}.\]
Furthermore, we have
\begin{equation}\label{PlusOne}
\begin{array}{l}
\left| {u_j^l[k] - x_i^l[k]} \right| \le \sum\limits_{s \in {{\cal V}_{\cal N}}} {\beta _{js}^l[k]\left| {x_s^l[k] - x_i^l[k]} \right|} \\
\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \le \mathop {\max }\limits_{s \in {{\cal V}_{\cal N}}} {\mkern 1mu} \left| {x_s^l[k] - x_i^l[k]} \right|.
\end{array}
\end{equation}
Under Assumptions \ref{BoundedSolutionAssumption}, \ref{GradientContinuousAssumption} and \ref{CommunicationTopologyAssumption}, it is known from Proposition \ref{FilteringGradientBoundedness} and its proof that
\[\left\| {{c}_{j}}[k] \right\|_2\le L,\]
where
$L=(n-f)\underset{j\in {{\mathcal{V}}_{\mathcal{N}}}}{\mathop{\max }}\,\underset{x\in \mathcal{X}}{\mathop{\max }}\,{{\left\| \nabla {{Q}_{j}}(x) \right\|}_{2}}$.
Therefore, we have
\begin{equation}\label{PlusTwo}
\left|c_{j}^{l}[k] \right|\le \left\| {{c}_{j}}[k] \right\|_2\le L.
\end{equation}
Combining (\ref{PlusOne}) with (\ref{PlusTwo}) yields that
\[\left|(u_{j}^{l}[k]-x_{i}^{l}[k])c_{j}^{l}[k] \right|\le L\underset{s\in {{\mathcal V}_{\mathcal N}}}{\mathop{\max }}\,\left| x_{s}^{l}[k]-x_{i}^{l}[k] \right|.\]
This concludes the proof.
\end{proof}

Next, we will give the main result of this paper.
\begin{theorem}\label{MainResult}
Under Assumptions
\ref{BoundedSolutionAssumption}, \ref{GradientContinuousAssumption}, \ref{StrongConvexityAssumption}, \ref{CommunicationTopologyAssumption},
\ref{StepSizeAssumption},
if the parameter
\[f < \frac{{\lambda n}}{{3\sqrt d \mu }} - \frac{2}{3},\]
then the proposed MSR-DCGE optimization algorithm is resilient to the $f$-total Byzantine attacks.
\end{theorem}
\begin{proof}
Under Assumption \ref{StrongConvexityAssumption}, the unique minimum point belonging to $\mathcal X$ of problem (\ref{OptimizationModel}) is denoted as $a$, i.e.,
\[\{a\}=\mathcal X\cap \underset{x\in {{\mathbb R}^{d}}}{\mathop{\arg \min }}\,\sum\nolimits_{i\in {{\mathcal V}_{\mathcal N}}}{{{Q}_{i}}}(x).\]
Furthermore, we define
\[{{x}^{l}}[k]\triangleq \underset{y\in \{x_{i}^{l}[k]\left| i\in {{\mathcal V}_{\mathcal N}}\} \right.}{\mathop{\arg \max }}\,\left| y-{{a}^{l}} \right|\]
as the $l$-th component of $x[k]$, $l=1,2,\ldots,d$. Correspondingly,
we denote the index of the normal agent whose $l$-th component of local estimate at instant $k$ equal to $x^l[k]$ as $\sigma_{k,l}$, i.e., $x_{{{\sigma }_{k,l}}}^{l}[k]={{x}^{l}}[k]$.
Denote
\[{{V}^{l}}[k]\triangleq{{(x_{{{\sigma }_{k,l}}}^{l}[k]-{{a}^{l}})}^{2}}={{({{x}^{l}}[k]-{{a}^{l}})}^{2}}.\]
Furthermore, we denote
\begin{equation}\label{VKDefinition}
V[k]\triangleq\sum\limits_{l=1}^{d}{{{V}^{l}}[k]}=\left\| x[k]-a \right\|_{2}^{2}.
\end{equation}
If we can prove that $\mathop {\lim }\nolimits_{k\to \infty}V[k]=0$, then the proof is completed.

Based on the analysis of the projection ${{P}_{[-\xi ,\xi ]}}(x^l)$, we have
\begin{equation}\label{FarestNodeQuantityProjection}
x_{{{\sigma }_{k+1,l}}}^{l}[k+1]={{P}_{[-\xi ,\xi ]}}(u_{{{\sigma }_{k+1,l}}}^{l}[k]-{{\alpha }_{k}}c_{{{\sigma }_{k+1,l}}}^{l}[k]).
\end{equation}
Therefore, it is obtained that
\begin{equation}\label{MinusFarestNodeQuantityProjection}
x_{{{\sigma }_{k+1,l}}}^{l}[k+1]-{{a}^{l}}={{P}_{[-\xi ,\xi ]}}(u_{{{\sigma }_{k+1,l}}}^{l}[k]-{{\alpha }_{k}}c_{{{\sigma }_{k+1,l}}}^{l}[k])-{{a}^{l}}.
\end{equation}
According to $a^l \in [-\xi ,\xi ]$ and the definition of ${{P}_{[-\xi ,\xi ]}}(x^l)$, we can know that
\begin{equation}\label{AbsoluteProjectionBound}
\left| {{P}_{[-\xi ,\xi ]}}(x_{i}^{l}[k])-{{a}^{l}} \right|\le \left| x_{i}^{l}[k]-{{a}^{l}} \right|.
\end{equation}
Combining (\ref{MinusFarestNodeQuantityProjection}) and (\ref{AbsoluteProjectionBound}) yields that
\begin{equation}\label{VQuantityBound}
\begin{array}{l}
{V^l}[k + 1] = {(x_{{\sigma _{k + 1,l}}}^l[k + 1] - {a^l})^2}\\
\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; = {\left({P_{[ - \xi ,\xi ]}}(u_{{\sigma _{k + 1,l}}}^l[k] - {\alpha _k}c_{{\sigma _{k + 1,l}}}^l[k]) - {a^l}\right)^2}\\
\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \le {(u_{{\sigma _{k + 1,l}}}^l[k] - {\alpha _k}c_{{\sigma _{k + 1,l}}}^l[k] - {a^l})^2}\\
\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; = {(u_{{\sigma _{k + 1,l}}}^l[k] - {a^l})^2} + \alpha _k^2{(c_{{\sigma _{k + 1,l}}}^l[k])^2}\\
\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; - 2{\alpha _k}(u_{{\sigma _{k + 1,l}}}^l[k] - {a^l})c_{{\sigma _{k + 1,l}}}^l[k].
\end{array}
\end{equation}
Due to (\ref{ConvexCombination}), we can obtain that
\[u_{{{\sigma }_{k+1,l}}}^{l}[k]-{{a}^{l}}=\sum\limits_{j\in {{\mathcal V}_{\mathcal N}}}{\beta_{{{\sigma }_{k+1,l}}j}^{l}}[k](x_{j}^{l}[k]-{{a}^{l}}).\]
Therefore, for the first term of the right side of (\ref{VQuantityBound}), we have
\begin{equation}\label{VLK}
\begin{array}{l}
{(u_{{\sigma _{k + 1,l}}}^l[k] - {a^l})^2} = {\left(\sum\limits_{j \in {{\cal V}_{\cal N}}} {\beta _{{\sigma _{k + 1,l}}j}^l} [k](x_j^l[k] - {a^l})\right)^2}\\
\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \le \sum\limits_{j \in {{\cal V}_{\cal N}}} {\beta _{{\sigma _{k + 1,l}}j}^l[k]{{(x_j^l[k] - {a^l})}^2}} \\
\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \le {(x_{{\sigma _{k,l}}}^l[k] - {a^l})^2}\sum\limits_{j \in {{\cal V}_{\cal N}}} {\beta _{{\sigma _{k + 1,l}}j}^l[k]} \\
\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; = {(x_{{\sigma _{k,l}}}^l[k] - {a^l})^2}\\
\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; = {V^l}[k].
\end{array}
\end{equation}
Substituting (\ref{VLK}) into (\ref{VQuantityBound}) yields that
\begin{equation}\label{VInequality}
\begin{array}{l}
{V^l}[k + 1] \le {V^l}[k] + \alpha _k^2{(c_{{\sigma _{k + 1,l}}}^l[k])^2}\\
\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; - 2{\alpha _k}(u_{{\sigma _{k + 1,l}}}^l[k] - {a^l})c_{{\sigma _{k + 1,l}}}^l[k].
\end{array}
\end{equation}
Due to $V[k]=\sum\nolimits_{l=1}^{d}{{{V}^{l}}[k]}$, we sum (\ref{VInequality}) on $l=1,\ldots,d$, it can be obtained that
\[\begin{array}{l}
V[k + 1] \le V[k] + \alpha _k^2\sum\nolimits_{l = 1}^d {{{(c_{{\sigma _{k + 1,l}}}^l[k])}^2}} \\
\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; - 2{\alpha _k}\sum\nolimits_{l = 1}^d {(u_{{\sigma _{k + 1,l}}}^l[k] - {a^l})c_{{\sigma _{k + 1,l}}}^l[k]} .
\end{array}\]
Denote
\[U[k]\triangleq \sum\limits_{l=1}^{d}{(u_{{{\sigma }_{k+1,l}}}^{l}[k]-{{a}^{l}})c_{{{\sigma }_{k+1,l}}}^{l}[k]},\]
it follows that
\begin{equation}\label{VkUk}
V[k+1]\le V[k]+\alpha _{k}^{2}\sum\limits_{l=1}^{d}{{{(c_{{{\sigma}_{k+1,l}}}^{l}[k])}^{2}}}-2{{\alpha }_{k}}U[k].
\end{equation}
Decompose $U[k]$ into the sum of the following two terms:
\[\begin{array}{*{20}{l}}
{U[k] = \sum\limits_{l = 1}^d {(x_i^l[k] - {a^l} + u_{{\sigma _{k + 1,l}}}^l[k] - x_i^l[k])c_{{\sigma _{k + 1,l}}}^l[k]} }\\
\begin{array}{l}
\;\;\;\;\;\;= \sum\limits_{l = 1}^d {(x_i^l[k] - {a^l})c_{{\sigma _{k + 1,l}}}^l[k]} \\
\;\;\;\;\;\;\;\;\; + \sum\limits_{l = 1}^d {(u_{{\sigma _{k + 1,l}}}^l[k] - x_i^l[k])c_{{\sigma _{k + 1,l}}}^l[k]},
\end{array}
\end{array}\]
where $x_i^l[k]$ corresponds to any normal agent $i \in \mathcal V_\mathcal N$.
Denote
\[{{S}^{l}}[k]\buildrel \Delta \over = (x_{i}^{l}[k]-{{a}^{l}})c_{{{\sigma }_{k+1,l}}}^{l}[k],\]
\[{{W}^{l}}[k]\buildrel \Delta \over =(u_{{{\sigma }_{k+1,l}}}^{l}[k]-x_{i}^{l}[k])c_{{{\sigma }_{k+1,l}}}^{l}[k],\]
then we have
\[U[k] = \sum\limits_{l = 1}^d {{S^l}[k]}  + \sum\limits_{l = 1}^d {{W^l}[k]}.\]
For $\sum\nolimits_{l = 1}^d {{W^l}[k]}$, it is known from Proposition \ref{WAbsoluteBoundedness} that
\[\left| {{W}^{l}}[k] \right|\le L\underset{s\in {{\mathcal V}_{\mathcal N}}}{\mathop{\max }}\,\left| x_{s}^{l}[k]-x_{i}^{l}[k] \right|.\]
Therefore, we have
\begin{equation}\label{AbsoluteWUpper}
\left|\sum\limits_{l=1}^{d}{{{W}^{l}}[k]} \right|\le \sum\limits_{l=1}^{d}{\left|{{W}^{l}}[k] \right|}\le L\sum\limits_{l=1}^{d}{\underset{s\in {{\mathcal V}_{\mathcal N}}}{\mathop{\max }}\,\left| x_{s}^{l}[k]-x_{i}^{l}[k] \right|}.
\end{equation}
Furthermore, we can obtain a lower bound on $\sum\nolimits_{l=1}^{d}{{{W}^{l}}[k]}$, i.e.,
\begin{equation}\label{WLKLowerBound}
\sum\limits_{l=1}^{d}{{{W}^{l}}[k]}\ge -L\sum\limits_{l=1}^{d}{\underset{s\in {{\mathcal V}_{\mathcal N}}}{\mathop{\max }}\,\left| x_{s}^{l}[k]-x_{i}^{l}[k] \right|}.
\end{equation}

Next, we will give a lower bound on $\sum\nolimits_{l=1}^{d}{{{S}^{l}}[k]}$.

Based on the MSR-DCGE resilient optimization algorithm, it is established for any normal agent $j \in \mathcal V_\mathcal N$ that
\begin{equation}\label{FiletringGradientTransformation}
\begin{array}{*{20}{l}}
{c_j^l[k] = \sum\limits_{s \in {\mathcal M_j}[k] \cup \{ j\} } {g_{js}^l[k]} }\\
{\;\;\;\;\;\;\;\; = \sum\limits_{s \in {\mathcal M_j}[k] \cap {\mathcal V_\mathcal N}} {g_{js}^l[k]}  + \sum\limits_{s \in {\mathcal M_j}[k] \cap {\mathcal V_\mathcal B}} {g_{js}^l[k]}  + g_{jj}^l[k]}\\
{\;\;\;\;\;\;\;\; = \sum\limits_{s \in {\mathcal V_\mathcal N}} {g_{js}^l[k] - {\sum\limits_{s \in ({\mathcal N_j} \cap {\mathcal V_\mathcal N})\backslash ({\mathcal M_j}[k] \cap {\mathcal V_\mathcal N})}}g_{js}^l[k]} }\\
\begin{array}{l}
\;\;\;\;\;\;\;\;\;\;\; - {\sum\limits_{s \in {\mathcal V_\mathcal N}\backslash ({\mathcal N_j} \cap {\mathcal V_\mathcal N})}}g_{js}^l[k] + \sum\limits_{s \in {\mathcal M_j}[k] \cap {\mathcal V_\mathcal B}} {g_{js}^l[k]}\\
\;\;\;\;\;\;\;\;\;\;\;+ g_{jj}^l[k].
\end{array}
\end{array}
\end{equation}
Multiplying $(x_i^l[k]-a^l)$ on both sides of (\ref{FiletringGradientTransformation}) yields that
\begin{equation}\label{SmallerOne}
\begin{array}{*{20}{l}}
{(x_i^l[k] - {a^l})c_j^l[k]}\\
\begin{array}{l}
 \ge \sum\limits_{s \in {\mathcal V_\mathcal N}} {(x_i^l[k] - {a^l})g_{js}^l[k]} \\
\;\;\;\;\;\;\;\; - {\sum\limits_{s \in ({\mathcal N_j} \cap {\mathcal V_\mathcal N})\backslash ({\mathcal M_j}[k] \cap {\mathcal V_\mathcal N})}}\left| {x_i^l[k] - {a^l}} \right|\left| {g_{js}^l[k]} \right|
\end{array}\\
\begin{array}{l}
\;\;\;\;\;\;\;\; - {\sum\limits_{s \in {\mathcal V_\mathcal N}\backslash ({\mathcal N_j} \cap {\mathcal V_\mathcal N})}}\left| {x_i^l[k] - {a^l}} \right|\left|{g_{js}^l[k]} \right|\\
\;\;\;\;\;\;\;\; - \sum\limits_{s \in {\mathcal M_j}[k] \cap {\mathcal V_\mathcal B}} {\left| {x_i^l[k] - {a^l}} \right|\left| {g_{js}^l[k]} \right|}
\end{array}\\
{\;\;\;\;\;\;\;\;\;\; - \left| {x_i^l[k] - {a^l}} \right|\left|{g_{jj}^l[k]} \right|}\\
\begin{array}{l}
 \ge \sum\limits_{s \in {\mathcal V_\mathcal N}} {(x_i^l[k] - {a^l})g_{js}^l[k]} \\
\;\;\;\;\;\;\;\; - \left| {x_i^l[k] - {a^l}} \right|({\sum\limits_{s \in ({\mathcal N_j} \cap {\mathcal V_\mathcal N})\backslash ({\mathcal M_j}[k] \cap {\mathcal V_\mathcal N})}}{\left\| {{g_{js}}[k]} \right\|_2}
\end{array}\\
\begin{array}{l}
\;\;\;\;\;\;\;\;+ {\sum\limits_{s \in {\mathcal V_\mathcal N}\backslash ({\mathcal N_j} \cap {\mathcal V_\mathcal N})}}{\left\| {{g_{js}}[k]} \right\|_2}\\
\;\;\;\;\;\;\;\; + \sum\limits_{s \in {\mathcal M_j}[k] \cap {\mathcal V_\mathcal B}} {{{\left\| {{g_{js}}[k]} \right\|}_2}}  + {\left\| {{g_{jj}}[k]} \right\|_2}),
\end{array}
\end{array}
\end{equation}
where $i \in \mathcal V_\mathcal N$.
Based on the MSR-DCGE resilient optimization algorithm, it holds that
\begin{equation}\label{S1S2GradientNormComparision}
{{\left\| {{g}_{j{{s}_{1}}}[k]} \right\|}_{2}}\le {{\left\| {{g}_{j{{s}_{2}}}[k]} \right\|}_{2}}
\end{equation}
for any ${{s}_{1}}\in {{\mathcal M}_{j}}[k]\cap {{\mathcal V}_{\mathcal B}}$ and ${{s}_{2}}\in ({{\mathcal N}_{j}}\cap {{\mathcal V}_{\mathcal N}})\backslash ({{\mathcal M}_{j}}[k]\cap {{\mathcal V}_{\mathcal N}})$.
Moreover, due to
\[\begin{array}{l}
\left| {({{\cal N}_j} \cap {{\cal V}_{\cal N}})\backslash ({{\cal M}_j}[k] \cap {{\cal V}_{\cal N}})} \right|\\
\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;+ \left| {({{\cal N}_j} \cap {{\cal V}_{\cal B}})\backslash ({{\cal M}_j}[k] \cap {{\cal V}_{\cal B}})} \right| = f
\end{array}\]
and
\[\begin{array}{l}
\left| {{{\cal M}_j}[k] \cap {{\cal V}_{\cal B}}} \right| + \left| {({{\cal N}_j} \cap {{\cal V}_{\cal B}})\backslash ({{\cal M}_j}[k] \cap {{\cal V}_{\cal B}})} \right|\\
\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;
= \left| {{{\cal N}_j} \cap {{\cal V}_{\cal B}}} \right| \le f,
\end{array}\]
it is derived that
\begin{equation}\label{SetNumberComparision}
\left| {{\mathcal{M}}_{j}}[k]\cap {{\mathcal{V}}_{\mathcal B}} \right|\le \left| ({{\mathcal{N}}_{j}}\cap {{\mathcal{V}}_{\mathcal{N}}})\backslash ({{\mathcal{M}}_{j}}[k]\cap {{\mathcal{V}}_{\mathcal{N}}}) \right|.
\end{equation}
Combining (\ref{S1S2GradientNormComparision}) with (\ref{SetNumberComparision}) yields that
\begin{equation}\label{SetGradientComparisionResult}
\sum\limits_{s\in {{\mathcal{M}}_{j}}[k]\cap {{\mathcal{V}}_{\mathcal B}}}{{{\left\| {{g}_{js}}[k] \right\|}_{2}}}\le \sum\limits_{s\in ({{\mathcal{N}}_{j}}\cap {{\mathcal{V}}_{\mathcal{N}}})\backslash ({{\mathcal{M}}_{j}}[k]\cap {{\mathcal{V}}_{\mathcal{N}}})}{{{\left\| {{g}_{js}}[k] \right\|}_{2}}}.
\end{equation}
Therefore, substituting (\ref{SetGradientComparisionResult}) into (\ref{SmallerOne}) yields that
\begin{equation}\label{InnovationA}
\begin{array}{l}
(x_i^l[k] - {a^l})c_j^l[k]\\\ge \sum\limits_{s \in {{\cal V}_{\cal N}}} {(x_i^l[k] - {a^l})g_{js}^l[k]}\\
\;\;\;\;\;- 2\left| {x_i^l[k] - {a^l}} \right|{\sum\limits_{s \in ({{\cal N}_j} \cap {{\cal V}_{\cal N}})\backslash ({{\cal M}_j}[k] \cap {{\cal V}_{\cal N}})}}{\left\| {{g_{js}}[k]} \right\|_2}\\
\;\;\;\;\;{\rm{ - }}\left| {x_i^l[k] - {a^l}} \right|{\sum\limits_{s \in {{\cal V}_{\cal N}}\backslash ({{\cal N}_j} \cap {{\cal V}_{\cal N}})}}{\left\| {{g_{js}}[k]} \right\|_2}\\
\;\;\;\;\;-\left| {x_i^l[k] - {a^l}} \right|{\left\| {{g_{jj}}[k]} \right\|_2}\\
{\rm{ = }}\sum\limits_{s \in {{\cal V}_{\cal N}}}(x_i^l[k] - {a^l})\left( {g_{js}^l({x_i}[k]) + g_{js}^l({x_s}[k]) - g_{js}^l({x_i}[k])} \right)\\
\;\;\;\;\;- 2\left|{x_i^l[k] - {a^l}} \right|{\sum\limits_{s \in ({{\cal N}_j} \cap {{\cal V}_{\cal N}})\backslash ({{\cal M}_j}[k] \cap {{\cal V}_{\cal N}})}}{\left\| {{g_{js}}[k]} \right\|_2}\\
\;\;\;\;\;-\left|{x_i^l[k] - {a^l}} \right|{\sum\limits_{s \in {{\cal V}_{\cal N}}\backslash ({{\cal N}_j} \cap {{\cal V}_{\cal N}})}}{\left\|{{g_{js}}[k]} \right\|_2}\\
\;\;\;\;\;-\left| {x_i^l[k] - {a^l}} \right|{\left\| {{g_{jj}}[k]} \right\|_2}.
\end{array}
\end{equation}
Furthermore, due to
\[\begin{array}{l}
(x_i^l[k] - {a^l})\left( {g_{js}^l({x_s}[k]) - g_{js}^l({x_i}[k])} \right)\\
\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \le \left| {x_i^l[k] - {a^l}} \right|{\left\| {{g_{js}}({x_s}[k]) - {g_{js}}({x_i}[k])} \right\|_2},
\end{array}\]
it is derived from (\ref{InnovationA}) that
\begin{equation}\label{InnovationB}
\begin{array}{*{20}{c}}
\begin{array}{l}
(x_i^l[k] - {a^l})c_j^l[k]\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\\
 \ge \sum\limits_{s \in {\mathcal V_\mathcal N}} {(x_i^l[k] - {a^l})g_{js}^l({x_i}[k])}
\end{array}\\
\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{ - \left| {x_i^l[k] - {a^l}} \right|\sum\limits_{s \in {\mathcal V_\mathcal N}} {{{\left\| {{g_{js}}({x_s}[k]) - {g_{js}}({x_i}[k])} \right\|}_2}} }\\
\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{ - 2\left| {x_i^l[k] - {a^l}} \right|{\sum\limits_{s \in ({\mathcal N_j} \cap {\mathcal V_\mathcal N})\backslash ({\mathcal M_j}[k] \cap {\mathcal V_\mathcal N})}}{{\left\| {{g_{js}}[k]} \right\|}_2}}\\
\;\;\;\begin{array}{l}
 - \left| {x_i^l[k] - {a^l}} \right|{\sum\limits_{s \in {\mathcal V_\mathcal N}\backslash ({\mathcal N_j} \cap {\mathcal V_\mathcal N})}}{\left\| {{g_{js}}[k]} \right\|_2}\\
 - \left| {x_i^l[k] - {a^l}} \right|{\left\| {{g_{jj}}[k]} \right\|_2}.
\end{array}
\end{array}
\end{equation}
Based on the triangle inequality, it holds for any $s \in \mathcal V_\mathcal N$ that
\begin{equation}\label{TriangleInequality}
{\left\| {{g_{js}}({x_s}[k])} \right\|_2} \le {\left\| {{g_{js}}({x_i}[k])} \right\|_2} + {\left\| {{g_{js}}({x_s}[k]) - {g_{js}}({x_i}[k])} \right\|_2}.
\end{equation}
Substituting (\ref{TriangleInequality}) into (\ref{InnovationB}) yields that
\begin{equation}\label{InnovationC}
\begin{array}{l}
(x_i^l[k] - {a^l})c_j^l[k]\\
\ge \sum\limits_{s \in {{\cal V}_{\cal N}}} {(x_i^l[k] - {a^l})g_{js}^l({x_i}[k])}
\\- \left| {x_i^l[k] - {a^l}} \right|\sum\limits_{s \in {{\cal V}_{\cal N}}} {{{\left\| {{g_{js}}({x_s}[k]) - {g_{js}}({x_i}[k])} \right\|}_2}} \\
- 2\left| {x_i^l[k] - {a^l}} \right|{\sum\limits_{s \in ({{\cal N}_j} \cap {{\cal V}_{\cal N}})\backslash ({{\cal M}_j}[k] \cap {{\cal V}_{\cal N}})}}({\left\| {{g_{js}}({x_i}[k])} \right\|_2} + {\left\| {{g_{js}}({x_s}[k]) - {g_{js}}({x_i}[k])} \right\|_2})\\
- \left| {x_i^l[k] - {a^l}} \right|{\sum\limits_{s \in {{\cal V}_{\cal N}}\backslash ({{\cal N}_j} \cap {{\cal V}_{\cal N}})}}({\left\| {{g_{js}}({x_i}[k])} \right\|_2} + {\left\| {{g_{js}}({x_s}[k]) - {g_{js}}({x_i}[k])} \right\|_2})\\
- \left| {x_i^l[k] - {a^l}} \right|({\left\| {{g_{jj}}({x_i}[k])} \right\|_2} + {\left\| {{g_{jj}}({x_j}[k]) - {g_{jj}}({x_i}[k])} \right\|_2})\\
= \sum\limits_{s \in {{\cal V}_{\cal N}}} {(x_i^l[k] - {a^l})g_{js}^l({x_i}[k])} \\
- (2\left| {x_i^l[k] - {a^l}} \right|{\sum\limits_{s \in ({{\cal N}_j} \cap {{\cal V}_{\cal N}})\backslash ({{\cal M}_j}[k] \cap {{\cal V}_{\cal N}})}}{\left\| {{g_{js}}({x_i}[k])} \right\|_2}\\
+ \left| {x_i^l[k] - {a^l}} \right|{\sum\limits_{s \in {{\cal V}_{\cal N}}\backslash ({{\cal N}_j} \cap {{\cal V}_{\cal N}})}}{\left\| {{g_{js}}({x_i}[k])} \right\|_2}\\
+\left| {x_i^l[k] - {a^l}} \right|{\left\| {{g_{jj}}({x_i}[k])} \right\|_2})\\
- (\left| {x_i^l[k] - {a^l}} \right|\sum\limits_{s \in {{\cal V}_{\cal N}}} {{{\left\| {{g_{js}}({x_s}[k]) - {g_{js}}({x_i}[k])} \right\|}_2}}\\
+ 2\left|{x_i^l[k] - {a^l}} \right|{\sum\limits_{s \in ({{\cal N}_j} \cap {{\cal V}_{\cal N}})\backslash ({{\cal M}_j}[k] \cap {{\cal V}_{\cal N}})}}{\left\| {{g_{js}}({x_s}[k]) - {g_{js}}({x_i}[k])} \right\|_2}\\
 + \left|{x_i^l[k] - {a^l}} \right|{\sum\limits_{s \in {{\cal V}_{\cal N}}\backslash ({{\cal N}_j} \cap {{\cal V}_{\cal N}})}}{\left\| {{g_{js}}({x_s}[k]) - {g_{js}}({x_i}[k])} \right\|_2}\\
+ \left| {x_i^l[k] - {a^l}} \right|{\left\| {{g_{jj}}({x_j}[k]) - {g_{jj}}({x_i}[k])} \right\|_2}).
\end{array}
\end{equation}
Due to
\[({\mathcal N_j} \cap {\mathcal V_\mathcal N})\backslash ({\mathcal M_j}[k] \cap {\mathcal V_\mathcal N}) \cap \left( {{{\cal V}_{\cal N}}\backslash ({{\cal N}_j} \cap {{\cal V}_{\cal N}})} \right)
= \emptyset\]
and
\[\begin{array}{l}
({{\cal N}_j} \cap {{\cal V}_{\cal N}})\backslash ({{\cal M}_j}[k] \cap {{\cal V}_{\cal N}}) \cup ({{\cal V}_{\cal N}}\backslash ({{\cal N}_j} \cap {{\cal V}_{\cal N}}))\\
\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; = {{\cal V}_{\cal N}}\backslash ({{\cal M}_j}[k] \cap {{\cal V}_{\cal N}}),
\end{array}\]
it is obtained from (\ref{InnovationC}) that
\begin{equation}\label{InnovationD}
\begin{array}{l}
(x_i^l[k] - {a^l})c_j^l[k]\\ \ge \sum\limits_{s \in {\mathcal V_\mathcal N}} {(x_i^l[k] - {a^l})g_{js}^l({x_i}[k])} \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\\
\;\;\;\;\;\;\;\; - \left| {x_i^l[k] - {a^l}} \right|({\sum\limits_{s \in ({\mathcal N_j} \cap {\mathcal V_\mathcal N})\backslash ({\mathcal M_j}[k] \cap {\mathcal V_\mathcal N})}}{\left\| {{g_{js}}({x_i}[k])} \right\|_2}\\
\;\;\;\;\;\;\;\; + {\sum\limits_{s \in {\mathcal V_\mathcal N}\backslash ({\mathcal M_j}[k] \cap {\mathcal V_\mathcal N})}}{\left\| {{g_{js}}({x_i}[k])} \right\|_2} + {\left\| {{g_{jj}}({x_i}[k])} \right\|_2})\\
\;\;\;\;\;\;\;\; - \left|{x_i^l[k] - {a^l}} \right|(\sum\limits_{s \in {\mathcal V_\mathcal N}} {{{\left\| {{g_{js}}({x_s}[k]) - {g_{js}}({x_i}[k])} \right\|}_2}} \\
\;\;\;\;\;\;\;\; + {\sum\limits_{s \in ({\mathcal N_j} \cap {\mathcal V_\mathcal N})\backslash ({\mathcal M_j}[k] \cap {\mathcal V_\mathcal N})}}{\left\| {{g_{js}}({x_s}[k]) - {g_{js}}({x_i}[k])} \right\|_2}\\
\;\;\;\;\;\;\;\; + {\sum\limits_{s \in {\mathcal V_\mathcal N}\backslash ({\mathcal M_j}[k] \cap {\mathcal V_\mathcal N})}}{\left\| {{g_{js}}({x_s}[k]) - {g_{js}}({x_i}[k])} \right\|_2}\\
\;\;\;\;\;\;\;\; + {\left\| {{g_{jj}}({x_j}[k]) - {g_{jj}}({x_i}[k])} \right\|_2}).
\end{array}
\end{equation}
For any $s \in \mathcal V_\mathcal N$, we have
\begin{equation}\label{NormalGradientNoVarying}
{g_{js}}(x) = \nabla {Q_s}(x),
\end{equation}
Also, based on Assumption \ref{GradientContinuousAssumption}, it is established for any $s,i \in \mathcal V_\mathcal N$ that
\begin{equation}\label{BiggerGradientContinuousAssumption}
\begin{array}{l}
{\left\| {\nabla {Q_s}({x_s}[k]) - \nabla {Q_s}({x_i}[k])} \right\|_2} \le \mu {\left\| {{x_s}[k] - {x_i}[k]} \right\|_2}\\
\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \le \mu \mathop {\max }\limits_{p \in {{\cal V}_{\cal N}}} {\left\| {{x_p}[k] - {x_i}[k]} \right\|_2}.
\end{array}
\end{equation}
Furthermore, it is known from Assumption \ref{CommunicationTopologyAssumption} and the MSR-DCGE resilient optimization algorithm that
\begin{equation}\label{InnovationE}
\left| {{{\cal V}_{\cal N}}\backslash ({{\cal M}_j}[k] \cap {{\cal V}_{\cal N}})} \right| + \left| {\{ j\} } \right| \le \left|\frac{\lambda }{{\sqrt d \mu }}\left| {{\mathcal V_\mathcal N}} \right| - f - 1\right|
\end{equation}
and
\begin{equation}\label{InnovationF}
\left| {({\mathcal N_j} \cap {\mathcal V_\mathcal N})\backslash ({\mathcal M_j}[k] \cap {\mathcal V_\mathcal N})} \right| \le f.
\end{equation}
Therefore, based on
(\ref{InnovationD}), (\ref{NormalGradientNoVarying}), (\ref{BiggerGradientContinuousAssumption}), (\ref{InnovationE}) and (\ref{InnovationF}), we can derive that
\begin{equation}\label{InnovationG}
\begin{array}{l}
(x_i^l[k] - {a^l})c_j^l[k]\\ \ge \sum\limits_{s \in {{\cal V}_{\cal N}}} {(x_i^l[k] - {a^l}){\nabla ^l}{Q_s}({x_i}[k])} \;\\
\;- \left| {x_i^l[k] - {a^l}} \right|({\sum\limits_{s \in ({{\cal N}_j} \cap {{\cal V}_{\cal N}})\backslash ({{\cal M}_j}[k] \cap {{\cal V}_{\cal N}})}}{\left\| {\nabla {Q_s}({x_i}[k])} \right\|_2}\\
\;\;\;\;\;\;\;+ {\sum\limits_{s \in {{\cal V}_{\cal N}}\backslash ({{\cal M}_j}[k] \cap {{\cal V}_{\cal N}})}}{\left\| {\nabla {Q_s}({x_i}[k])} \right\|_2} + {\left\| {\nabla {Q_j}({x_i}[k])} \right\|_2})\\
\;- \mu (2\left| {{\mathcal V_\mathcal N}} \right| + f)\left| {x_i^l[k] - {a^l}} \right|\mathop {\max }\limits_{p \in {\mathcal V_\mathcal N}} {\left\| {{x_p}[k] - {x_i}[k]} \right\|_2},
\end{array}
\end{equation}
where ${\nabla ^l}{Q_s}({x_i}[k])$ represents the $l$-th component of $\nabla {Q_s}({x_i}[k])$.
Under the $2b$-redundancy condition \cite{gupta2020fault}, we have
\[\nabla {Q_s}(a) = 0\]
for any $s \in \mathcal V_\mathcal N$.
Therefore, it is obtained from Assumption \ref{GradientContinuousAssumption} that
\begin{equation}\label{ApplyingGradientContinuousAssumption}
{\left\| {\nabla {Q_s}({x_i}[k])} \right\|_2} \le \mu {\left\| {{x_i}[k] - a} \right\|_2}
\end{equation}
for any $s \in \mathcal V_\mathcal N$.
Based on (\ref{InnovationG}) and (\ref{ApplyingGradientContinuousAssumption}), we have
\begin{equation}\label{InnovationH}
\begin{array}{*{20}{l}}
\begin{array}{l}
(x_i^l[k] - {a^l})c_j^l[k]\\
\ge (x_i^l[k] - {a^l})\sum\limits_{s \in {{\cal V}_{\cal N}}} {{\nabla ^l}{Q_s}({x_i}[k])} \;
\end{array}\\
{- \mu \left( {f + 1 + \left| {{{\cal V}_{\cal N}}\backslash ({{\cal M}_j}[k] \cap {{\cal V}_{\cal N}})} \right|} \right)\left| {x_i^l[k] - {a^l}} \right|{{\left\| {{x_i}[k] - a} \right\|}_2}}\\
{- \mu (2\left| {{{\cal V}_{\cal N}}} \right| + f)\left| {x_i^l[k] - {a^l}} \right|\mathop {\max }\limits_{p \in {{\cal V}_{\cal N}}} {{\left\| {{x_p}[k] - {x_i}[k]} \right\|}_2}.}
\end{array}
\end{equation}
The definition
$Q(x) = \frac{1}{{\left| {{\mathcal V_\mathcal N}} \right|}}\sum_{s \in {\mathcal V_\mathcal N}} {{Q_s}(x)}$ yields that
\begin{equation}\label{AverageCostGradient}
\sum\limits_{s \in {\mathcal V_\mathcal N}} {{\nabla ^l}{Q_s}({x_i}[k]) = \left| {{\mathcal V_\mathcal N}} \right|{\nabla ^l}Q({x_i}[k])}.
\end{equation}
Furthermore, substituting (\ref{AverageCostGradient}) into (\ref{InnovationH}) yields that
\begin{equation}\label{InnovationI}
\begin{array}{*{20}{l}}
\begin{array}{l}
(x_i^l[k] - {a^l})c_j^l[k]\\
 \ge \left| {{\mathcal V_\mathcal N}} \right|\;(x_i^l[k] - {a^l}){\nabla ^l}Q({x_i}[k])
\end{array}\\
{ - \mu (f + 1 + \left| {{\mathcal V_\mathcal N}\backslash ({\mathcal M_j}[k] \cap {\mathcal V_\mathcal N})} \right|)\left| {x_i^l[k] - {a^l}} \right|{{\left\| {{x_i}[k] - a} \right\|}_2}}\\
{ - \mu (2\left| {{\mathcal V_\mathcal N}} \right| + f)\left| {x_i^l[k] - {a^l}} \right|\mathop {\max }\limits_{p \in {\mathcal V_\mathcal N}} {{\left\| {{x_p}[k] - {x_i}[k]} \right\|}_2}.}
\end{array}
\end{equation}
Under Assumption \ref{CommunicationTopologyAssumption}, it holds that
\[\left| {{\mathcal N_j}} \right| \ge 3f + 2 + (1 - \frac{\lambda }{{\sqrt d \mu }})\left| {{\mathcal V_\mathcal N}} \right|\]
for any $j \in \mathcal V_\mathcal N$.
Therefore, it can be derived that
\[\left| {{{\cal M}_j}[k] \cap {{\cal V}_{\cal N}}} \right| \ge \left| {{\mathcal N_j}} \right| - 2f \ge f + 2 + (1 - \frac{\lambda }{{\sqrt d \mu }})\left| {{{\cal V}_{\cal N}}} \right|\]
under the $b$-total-$f$-local Byzantine attacks and the MSR-DCGE resilient optimization algorithm.
It thereby holds that
\begin{equation}\label{InnovationJ}
\begin{array}{l}
\left| {{{\cal V}_{\cal N}}\backslash ({{\cal M}_j}[k] \cap {{\cal V}_{\cal N}})} \right| = \left| {{{\cal V}_{\cal N}}} \right| - \left| {{{\cal M}_j}[k] \cap {{\cal V}_{\cal N}}} \right|\\
\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \le \dfrac{\lambda }{{\sqrt d \mu }}\left| {{{\cal V}_{\cal N}}} \right| - f - 2
\end{array}
\end{equation}
for any $j \in \mathcal V_\mathcal N$.
Based on (\ref{InnovationI}), we have
\begin{equation}\label{SingleSLKGreater}
\begin{array}{l}
{S^l}[k] = (x_i^l[k] - {a^l})c_{{\sigma_{k + 1,l}}}^l[k]\\
\begin{array}{*{20}{l}}
{\ge \left| {{\mathcal V_\mathcal N}} \right|\;(x_i^l[k] - {a^l}){\nabla ^l}Q({x_i}[k])}\\
{-\mu (f + 1 + \left| {{\mathcal V_\mathcal N}\backslash ({\mathcal M_{{\sigma_{k + 1,l}}}}[k] \cap {\mathcal V_\mathcal N})} \right|)\left| {x_i^l[k] - {a^l}} \right|{{\left\| {{x_i}[k] - a} \right\|}_2}}\\
{-\mu (2\left| {{\mathcal V_\mathcal N}} \right| + f)\left| {x_i^l[k] - {a^l}} \right|\mathop {\max }\limits_{p \in {\mathcal V_\mathcal N}} {{\left\| {{x_p}[k] - {x_i}[k]} \right\|}_2}.}
\end{array}
\end{array}
\end{equation}
Denote
\[{Y^l}[k] \buildrel \Delta \over = \left| {{\mathcal V_\mathcal N}} \right|\;(x_i^l[k] - {a^l}){\nabla ^l}Q({x_i}[k]),\]
\[{Z^l}[k] \buildrel \Delta \over = \mu (f + 1 + \left| {{\mathcal V_\mathcal N}\backslash ({\mathcal M_{{\sigma_{k + 1,l}}}}[k] \cap {\mathcal V_\mathcal N})} \right|)\left| {x_i^l[k] - {a^l}} \right|{\left\| {{x_i}[k] - a} \right\|_2}.\]
Based on Assumption \ref{StrongConvexityAssumption}, it can be obtained that
\begin{equation}\label{YLKGreater}
\begin{array}{l}
\sum\limits_{l = 1}^d {{Y^l}[k] = } \left| {{\mathcal V_\mathcal N}} \right|\;\sum\limits_{l = 1}^d {(x_i^l[k] - {a^l}){\nabla ^l}Q({x_i}[k])} \\
\;\;\;\;\;\;\;\;\;\;\;\;\;\; = \left| {{\mathcal V_\mathcal N}} \right|\left\langle {{x_i}[k] - a,\nabla Q({x_i}[k])} \right\rangle \\
\;\;\;\;\;\;\;\;\;\;\;\;\;\; = \left| {{\mathcal V_\mathcal N}} \right|\left\langle {{x_i}[k] - a,\nabla Q({x_i}[k]) - \nabla Q(a)} \right\rangle \\
\;\;\;\;\;\;\;\;\;\;\;\;\;\; \ge \lambda \left| {{\mathcal V_\mathcal N}} \right|\left\| {{x_i}[k] - a} \right\|_2^2.
\end{array}
\end{equation}
From (\ref{InnovationJ}), we have
\[\begin{array}{*{20}{l}}
{{Z^l}[k] = \mu (f + 1 + \left| {{{\cal V}_{\cal N}}\backslash ({{\cal M}_{{\sigma_{k + 1,l}}}}[k] \cap {{\cal V}_{\cal N}})} \right|)\left| {x_i^l[k] - {a^l}} \right|{{\left\| {{x_i}[k] - a} \right\|}_2}}\\
{\;\;\;\;\;\;\;\;\; \le (\dfrac{\lambda }{{\sqrt d }}\left| {{{\cal V}_{\cal N}}} \right|{\rm{ - }}\mu )\left| {x_i^l[k] - {a^l}} \right|{{\left\| {{x_i}[k] - a} \right\|}_2}.}
\end{array}\]
Therefore, it is obtained that
\begin{equation}\label{ZLKGreater}
\begin{array}{*{20}{l}}
{\sum\limits_{l = 1}^d {{Z^l}[k]}  \le \sum\limits_{l = 1}^d {(\frac{\lambda }{{\sqrt d }}\left| {{{\cal V}_{\cal N}}} \right| - \mu )\left| {x_i^l[k] - {a^l}} \right|{{\left\| {{x_i}[k] - a} \right\|}_2}} }\\
{\;\;\;\;\;\;\;\;\;\;\;\;\;\; = (\dfrac{\lambda }{{\sqrt d }}\left| {{{\cal V}_{\cal N}}} \right| - \mu ){{\left\| {{x_i}[k] - a} \right\|}_2}\sum\limits_{l = 1}^d {\left| {x_i^l[k] - {a^l}} \right|} }\\
{\;\;\;\;\;\;\;\;\;\;\;\;\;\; \le (\dfrac{\lambda }{{\sqrt d }}\left| {{{\cal V}_{\cal N}}} \right| - \mu ){{\left\| {{x_i}[k] - a} \right\|}_2}\cdot\sqrt d {{\left\| {{x_i}[k] - a} \right\|}_2}}\\
{\;\;\;\;\;\;\;\;\;\;\;\;\;\; = (\lambda \left| {{{\cal V}_{\cal N}}} \right| - \sqrt d \mu )\left\| {{x_i}[k] - a} \right\|_2^2.}
\end{array}
\end{equation}
A lower bound on $\sum\nolimits_{l = 1}^d {{S^l}[k]}$ can be derived from (\ref{SingleSLKGreater}), (\ref{YLKGreater}) and (\ref{ZLKGreater}). Specifically, we have
\begin{equation}\label{SLKLowerBound}
\begin{array}{*{20}{l}}
\begin{array}{l}
\sum\limits_{l = 1}^d {{S^l}[k]}\\
\ge \sum\limits_{l = 1}^d {{Y^l}[k]}  - \sum\limits_{l = 1}^d {{Z^l}[k]} \\
\;\;\; - \mu (2\left| {{{\cal V}_{\cal N}}} \right| + f)\mathop {\max }\limits_{p \in {{\cal V}_{\cal N}}} {\left\| {{x_p}[k] - {x_i}[k]} \right\|_2}\sum\limits_{l = 1}^d {\left| {x_i^l[k] - {a^l}} \right|}
\end{array}\\
{\;\; \ge \lambda \left| {{{\cal V}_{\cal N}}} \right|\left\| {{x_i}[k] - a} \right\|_2^2 - (\lambda \left| {{{\cal V}_{\cal N}}} \right| - \sqrt d \mu )\left\| {{x_i}[k] - a} \right\|_2^2}\\
{\;\;\;\;\; - \mu (2\left| {{{\cal V}_{\cal N}}} \right| + f)\mathop {\max }\limits_{p \in {{\cal V}_{\cal N}}} {{\left\| {{x_p}[k] - {x_i}[k]} \right\|}_2}\sqrt d {{\left\| {{x_i}[k] - a} \right\|}_2}}\\
{\;\; = \sqrt d \mu \left\| {{x_i}[k] - a} \right\|_2^2}\\
{\;\;\;\;\; - \sqrt d \mu (2\left| {{{\cal V}_{\cal N}}} \right| + f){{\left\| {{x_i}[k] - a} \right\|}_2}\mathop {\max }\limits_{p \in {{\cal V}_{\cal N}}} {{\left\| {{x_p}[k] - {x_i}[k]} \right\|}_2}.}
\end{array}
\end{equation}

To sum up, we know from (\ref{WLKLowerBound}) and (\ref{SLKLowerBound}) that
\begin{equation}\label{UKLowerBound}
\begin{array}{l}
U[k] \\\;\;\;\;= \sum\limits_{l = 1}^d {{S^l}[k]}  + \sum\limits_{l = 1}^d {{W^l}[k]} \\
\;\;\;\; \ge \sqrt d \mu \left\| {{x_i}[k] - a} \right\|_2^2\\
\;\;\;\;\;\;- \sqrt d \mu (2\left| {{\mathcal V_\mathcal N}} \right| + f){\left\| {{x_i}[k] - a} \right\|_2}\mathop {\max }\limits_{p \in {\mathcal V_\mathcal N}} {\left\| {{x_p}[k] - {x_i}[k]} \right\|_2}\\
\;\;\;\;\;\; - L\sum\limits_{l = 1}^d {\mathop {\max }\limits_{p \in {\mathcal V_\mathcal N}} \left| {x_p^l[k] - x_i^l[k]} \right|}.
\end{array}
\end{equation}
Based on (\ref{VkUk}) and (\ref{UKLowerBound}), we have
\[\begin{array}{l}
V[k + 1]\\
 \le V[k] + \alpha _k^2\sum\limits_{l = 1}^d {{{(c_{{\sigma _{k + 1,l}}}^l[k])}^2}}  - 2{\alpha _k}U[k]\\
\le V[k] + \alpha _k^2\sum\limits_{l = 1}^d {{{(c_{{\sigma _{k + 1,l}}}^l[k])}^2}}
- 2\sqrt d \mu {\alpha _k}\left\| {{x_i}[k] - a} \right\|_2^2\\
\;+ 2\sqrt d \mu {\alpha _k}(2\left| {{{\cal V}_{\cal N}}} \right| + f){\left\| {{x_i}[k] - a} \right\|_2}\mathop {\max }\limits_{p \in {{\cal V}_{\cal N}}} {\left\| {{x_p}[k] - {x_i}[k]} \right\|_2}\\
\;+ 2L{\alpha _k}\sum\limits_{l = 1}^d {\mathop {\max }\limits_{p \in {{\cal V}_{\cal N}}} \left| {x_p^l[k] - x_i^l[k]} \right|}.
\end{array}\]
Denote
\begin{equation}\label{OmegaDefinition}
\begin{array}{l}
\Omega [k] \buildrel \Delta \over = \alpha _k^2\sum\limits_{l = 1}^d {{{(c_{{\sigma _{k + 1,l}}}^l[k])}^2}} \\
\;+ 2\sqrt d \mu {\alpha _k}(2\left| {{{\cal V}_{\cal N}}} \right| + f){\left\| {{x_i}[k] - a} \right\|_2}\mathop {\max }\limits_{p \in {{\cal V}_{\cal N}}} {\left\| {{x_p}[k] - {x_i}[k]} \right\|_2}\;\\
\;+ 2L{\alpha _k}\sum\limits_{l = 1}^d {\mathop {\max }\limits_{p \in {{\cal V}_{\cal N}}} \left| {x_p^l[k] - x_i^l[k]} \right|}.
\end{array}
\end{equation}
Therefore, for any $k\geq 0$, it is established that
\begin{equation}\label{VIterativeInequality}
V[k + 1] \le V[k] - 2\sqrt d \mu {\alpha _k}\left\| {{x_i}[k] - a} \right\|_2^2 + \Omega [k],
\end{equation}
Thus, we have $V[k + 1] \le V[k] + \Omega [k]$, i.e.,
\[V[k + 1] - V[k] \le \Omega [k].\]
Due to $\Omega [k] \ge 0$, it is obtained that
\[{(V[k + 1] - V[k])_ + } \le \Omega [k]\]
for any $k\geq 0$.
Therefore, we have
\begin{equation}\label{SumThanSumSmaller}
\sum\limits_{k = 0}^\infty {{{(V[k + 1] - V[k])}_ + }} \le \sum\limits_{k = 0}^\infty {\Omega [k]}.
\end{equation}

Next, we prove that $\sum\nolimits_{k = 0}^\infty  {\Omega [k]}  < \infty$.

According to Proposition \ref{FilteringGradientBoundedness}, we can know that
\[{(c_{{\sigma _{k + 1,l}}}^l[k])^2} \le {\left\| {{c_{{\sigma _{k + 1,l}}}}[k]} \right\|_2^2} \le {L^2}\]
for any $l \in \{1,2,\ldots, d\}$. Therefore, based on (\ref{OmegaDefinition}), we have
\begin{equation}\label{ScalingFirstStep}
\begin{array}{l}
\Omega [k] \le d{L^2}\alpha_k^2 \\+ 2\sqrt d \mu {\alpha _k}(2\left| {{\mathcal V_\mathcal N}} \right| + f){\left\|{{x_i}[k] - a} \right\|_2}\mathop {\max }\limits_{p \in {\mathcal V_\mathcal N}} {\left\| {{x_p}[k] - {x_i}[k]} \right\|_2}\\
+ 2L{\alpha _k}\sum\limits_{l = 1}^d {\mathop {\max }\limits_{p \in {\mathcal V_\mathcal N}}}\left| {x_p^l[k] - x_i^l[k]} \right|.
\end{array}
\end{equation}
Also, it holds that ${x_i}[k] \in {[ - \xi ,\xi ]^d}$ and $a \in {[ - \xi ,\xi ]^d}$. Based on the triangle inequality, we have
\begin{equation}\label{DistanceOptimalPoint}
{\left\| {{x_i}[k] - a} \right\|_2} \le {\left\| {{x_i}[k]} \right\|_2} + {\left\| a \right\|_2} \le \sqrt d \xi  + \sqrt d \xi  = 2\sqrt d \xi.
\end{equation}
At the same time, due to
\[{\left\| {{x_p}[k] - {x_i}[k]} \right\|_2} \le \sum\limits_{l = 1}^d {\left| {x_p^l[k] - x_i^l[k]} \right|},\]
it is obtained that
\begin{equation}\label{MaxDistanceTwoNorm}
\mathop {\max }\limits_{p \in {\mathcal V_\mathcal N}} {\left\| {{x_p}[k] - {x_i}[k]} \right\|_2} \le \sum\limits_{l = 1}^d {\mathop {\max }\limits_{p \in {\mathcal V_\mathcal N}} \left| {x_p^l[k] - x_i^l[k]} \right|}.
\end{equation}
Combining (\ref{ScalingFirstStep}), (\ref{DistanceOptimalPoint}) and (\ref{MaxDistanceTwoNorm}) yields that
\begin{equation}\label{OmegaScaling}
\begin{array}{l}
\Omega [k] \le d{L^2}\alpha _k^2\\
+ \left( {4d\mu \xi (2\left| {{{\cal V}_{\cal N}}} \right| + f) + 2L} \right)\left( {\sum\limits_{l = 1}^d {\mathop {\max }\limits_{p \in {{\cal V}_{\cal N}}} } \left| {x_p^l[k] - x_i^l[k]} \right|} \right){\alpha _k}
\end{array}
\end{equation}
for any $k\geq 0$.
Therefore, it can be obtained that
\begin{equation}\label{SumOmegeScaling}
\begin{array}{l}
\sum\limits_{k = 0}^\infty  {\Omega [k]}  \le d{L^2}\sum\limits_{k = 0}^\infty  {\alpha _k^2} \\
 + (4d\mu \xi (2\left| {{{\cal V}_{\cal N}}} \right| + f) + 2L)\sum\limits_{k = 0}^\infty  {\left( {{\alpha _k}\sum\limits_{l = 1}^d {\mathop {\max }\limits_{p \in {{\cal V}_{\cal N}}} } \left| {x_p^l[k] - x_i^l[k]} \right|} \right)}.
\end{array}
\end{equation}
For the first term on the right side of (\ref{SumOmegeScaling}), it is known from Assumption \ref{StepSizeAssumption} that
\begin{equation}\label{StepSizeSquareFinite}
\sum\limits_{k = 0}^\infty  {\alpha _k^2}  < \infty.
\end{equation}
For the second term on the right side of (\ref{SumOmegeScaling}), it is established from Proposition \ref{StepEstimationErrorFinite} that
\[\sum\limits_{k=0}^{\infty }{{{\alpha }_{k}}}\underset{i,j\in {{\mathcal V}_{\mathcal N}}}{\mathop{\max }}\,\left| x_{i}^{l}[k]-x_{j}^{l}[k] \right|<\infty.\]
Therefore, we have
\begin{equation}\label{SumStepEstimationErrorFinite}
\begin{array}{l}
\sum\limits_{k = 0}^\infty  {\left( {{\alpha _k}\sum\limits_{l = 1}^d {\mathop {\max }\limits_{p \in {{\cal V}_{\cal N}}} } \left| {x_p^l[k] - x_i^l[k]} \right|} \right)} \\
 = \sum\limits_{l = 1}^d {\sum\limits_{k = 0}^\infty  {{\alpha _k}\mathop {\max }\limits_{p \in {{\cal V}_{\cal N}}} } } \left| {x_p^l[k] - x_i^l[k]} \right| < \infty .
\end{array}
\end{equation}
Combining (\ref{SumOmegeScaling}), (\ref{StepSizeSquareFinite}) and (\ref{SumStepEstimationErrorFinite}) yields that
\begin{equation}\label{SumOmegaFinite}
\sum\limits_{k = 0}^\infty  {\Omega [k]} < \infty.
\end{equation}

Based on (\ref{SumThanSumSmaller}) and (\ref{SumOmegaFinite}), we have
\[\sum\limits_{k = 0}^\infty  {{{(V[k + 1] - V[k])}_ + }}< \infty.\]
Due to $V[k]\geq 0$, there exists some non-negative real number $\phi< \infty$ such that
\begin{equation}\label{VKLimitFinite}
\mathop {\lim }\limits_{k \to \infty } V[k] = \phi
\end{equation}
based on Lemma \ref{FiniteLimitLemma}.
Furthermore, we have
\begin{equation}\label{SumVFrontBehindError}
\sum\limits_{k = 0}^\infty {V[k + 1]} - \sum\limits_{k = 0}^\infty {V[k]} = \phi - V[0] < \infty.
\end{equation}
From (\ref{VIterativeInequality}), we have
\[\begin{array}{l}
\sum\limits_{k = 0}^\infty  {V[k + 1]}  \le \sum\limits_{k = 0}^\infty  {V[k]}  - 2\sqrt d \mu \sum\limits_{k = 0}^\infty  {{\alpha _k}} \left\| {{x_i}[k] - a} \right\|_2^2\\
\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; + \sum\limits_{k = 0}^\infty  {\Omega [k]}.
\end{array}\]
That is to say, we have
\[\phi  - V[0] + 2\sqrt d \mu \sum\limits_{k = 0}^\infty  {{\alpha _k}} \left\| {{x_i}[k] - a} \right\|_2^2 \le \sum\limits_{k = 0}^\infty  {\Omega [k]}<\infty.\]
Therefore, we have
\begin{equation}\label{ImportantResult}
\sum\limits_{k = 0}^\infty {{\alpha_k}} \left\| {{x_i}[k] - a} \right\|_2^2 < \infty.
\end{equation}
Because (\ref{ImportantResult}) is established for any $i \in \mathcal V_\mathcal N$, we have
\begin{equation}\label{SumSumStepSizeDistance}
\sum\limits_{i \in {\mathcal V_\mathcal N}} {\sum\limits_{k = 0}^\infty  {{\alpha _k}} \left\| {{x_i}[k] - a} \right\|_2^2 < \infty.}
\end{equation}
According to the definition of $V[k]$, it is known that
\begin{equation}\label{VKUpperBound}
V[k] \le \sum\limits_{i \in {\mathcal V_\mathcal N}} {\left\|{{x_i}[k] - a} \right\|_2^2.}
\end{equation}
Multiplying $\alpha_k$ on both sides of (\ref{VKUpperBound}) yields that
\begin{equation}\label{AlphaVKUpperBound}
{\alpha_k}V[k] \le \sum\limits_{i \in {\mathcal V_\mathcal N}} {{\alpha_k}\left\| {{x_i}[k] - a} \right\|_2^2.}
\end{equation}
Furthermore, combining (\ref{SumSumStepSizeDistance}) with (\ref{AlphaVKUpperBound}) yields that
\begin{equation}\label{SumAlphaKVK}
\begin{array}{l}
\sum\limits_{k = 0}^\infty  {{\alpha_k}V[k]}  \le \sum\limits_{k = 0}^\infty  {\sum\limits_{i \in {{\cal V}_{\cal N}}} {{\alpha _k}\left\| {{x_i}[k] - a} \right\|_2^2} } \\
\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;= \sum\limits_{i \in {{\cal V}_{\cal N}}} {\sum\limits_{k = 0}^\infty  {{\alpha _k}\left\| {{x_i}[k] - a} \right\|_2^2} }  < \infty .
\end{array}
\end{equation}
Based on (\ref{VKLimitFinite}), (\ref{SumAlphaKVK}) and Assumption \ref{StepSizeAssumption}, it is obtained from Lemma \ref{SequenceLimitZero} that
\[\mathop {\lim }\limits_{k \to \infty } V[k] = \phi  = 0.\]
This concludes the proof.
\end{proof}

























\section{Conclusions}\label{SectionVI}
In this paper, we have studied the resilient optimization problem for $2f$-redundancy under the $f$-total Byzantine attacks.
Compared with the previous work, we propose the MSR-DCGE resilient optimization algorithm only with local information.
Under the assumption on the communication topology, we have successfully proved that the proposed algorithm 
can make all normal agents achieve consensus on the minimum point of the optimization problem.


\footnotesize
\bibliographystyle{IEEEtran}
