# Fixed-Time Optimization of Perturbed Multi-Agent Systems under the Resource Constraints
## 3 Main Results
### Three-Stage Fixed-Time Robust MSCO Algorithm
When \(t\geq T_{1}\), the dynamic degeneracy of \(x_{i}\) is  
\[\dot{x}_{i}=-\xi_{1}\text{sign}^{n}(e_{i})-\xi_{2}\text{sign}^{v}(e_{i})+\eta_ {i}. \tag{20}\]  
The dynamics of \(e_{i}\) can be simplified by Equation (20) as  
\[\dot{e}_{i}=-\xi_{1}\text{sign}^{n}(e_{i})-\xi_{2}\text{sign}^{v}(e_{i}). \tag{21}\]  
At this point, the error \(e_{i}\) is decoupled from the state \(x_{i}\). Consider the Lyapunov candidate  
\[=u_{i2}(t)+\omega_{i}(t)\]0  
The derivative of \(V_{2,i}(t)\) along the subsystem (21) is  
\[=u_{i2}(t)+\omega_{i}(t)\]1 \[=u_{i2}(t)+\omega_{i}(t)\]2 \[=u_{i2}(t)+\omega_{i}(t)\]3 \[=u_{i2}(t)+\omega_{i}(t)\]4  
According to Lemma 1, By \(V_{2,i=0}\Rightarrow e_{i}=0\), it follows that the error \(e_{i}\) converges to \(0\) in fixed-time \(T_{2}\), and  
\[=u_{i2}(t)+\omega_{i}(t)\]5  
Therefore, when \(t\geq T_{2}\), \(e_{i}=\dot{e}_{i}=0\), there is  
\[=u_{i2}(t)+\omega_{i}(t)\]6  
The graph \(\mathcal{G}\) is an undirected connected graph, then we have  
\[=u_{i2}(t)+\omega_{i}(t)\]7  
From Equations (25) and (26), we have  
\[=u_{i2}(t)+\omega_{i}(t)\]8  
which indicates that the global resource requirement \(D\) is always satisfied when \(t\geq T_{2}\). 1. Based on the design of the existing unconstrained distributed gradient-based optimization algorithm, a fixed-time scheme is applied for driving \(f^{p}(x)\) that reaches the optimal solution \(f^{p}(\hat{\star}^{*})\). We have \(e_{i}=0\) when \(\geq T_{2}\), and the dynamic degeneracy of \(x_{i}\) now is  
\[=u_{i2}(t)+\omega_{i}(t)\]9  
Consider the Lyapunov function  
\[=-\kappa_{0}s_{i}-\kappa_{1}\text{sgn}(s_{i})-\kappa_{2}\text{ sign}^{a}(s_{i})+\omega_{i}(t). \tag{17}\]0  
It is straightforward to see that \(V_{3}(t)\geq 0\), and that \(V_{3}(t)=0\Rightarrow x=\hat{x}^{*}\). The derivative of \(V_{3}(t)\) along the dynamics (28) is  
\[=-\kappa_{0}s_{i}-\kappa_{1}\text{sgn}(s_{i})-\kappa_{2}\text{ sign}^{a}(s_{i})+\omega_{i}(t). \tag{17}\]1 \[=-\kappa_{0}s_{i}-\kappa_{1}\text{sgn}(s_{i})-\kappa_{2}\text{ sign}^{a}(s_{i})+\omega_{i}(t). \tag{17}\]2 \[=-\kappa_{0}s_{i}-\kappa_{1}\text{sgn}(s_{i})-\kappa_{2}\text{ sign}^{a}(s_{i})+\omega_{i}(t). \tag{17}\]3 \[=-\kappa_{0}s_{i}-\kappa_{1}\text{sgn}(s_{i})-\kappa_{2}\text{ sign}^{a}(s_{i})+\omega_{i}(t). \tag{17}\]4  
Let \(\rho_{ij}=a_{ij}|\phi_{j}-\phi_{i}|^{2}\). Based on the conditions \(0<\mu<1\) and \(\
u>1\), Lemma 2 yields  
\[=-\kappa_{0}s_{i}-\kappa_{1}\text{sgn}(s_{i})-\kappa_{2}\text{ sign}^{a}(s_{i})+\omega_{i}(t). \tag{17}\]5 \[=-\kappa_{0}s_{i}-\kappa_{1}\text{sgn}(s_{i})-\kappa_{2}\text{ sign}^{a}(s_{i})+\omega_{i}(t). \tag{17}\]6  
Let the average gradient \(\phi_{c}=\frac{1}{N}\sum_{i=1}^{N}\phi_{i}\), and the gradient error is \(\tilde{\phi}_{i}=\phi_{i}-\phi_{c}\), then  
\[=-\kappa_{0}s_{i}-\kappa_{1}\text{sgn}(s_{i})-\kappa_{2}\text{ sign}^{a}(s_{i})+\omega_{i}(t). \tag{17}\]7  
Let the global gradient vector be \(\phi=(\phi_{1},\cdots,\phi_{N})^{\top}\). The global gradient error vector is \(\tilde{\phi}=(\tilde{\phi}_{1},\cdots,\tilde{\phi}_{N})^{\top}\), noting that \(\phi=\
abla f^{p}(x)\). Additionally, by Step 2 we know that \(1_{N}^{\top}x=D,\ \forall t\geq T_{2}\). The optimal solution \(\hat{x}^{*}\) satisfies the constraint \(1_{N}^{\top}\hat{x}^{*}=D\), then  
\[=-\kappa_{0}s_{i}-\kappa_{1}\text{sgn}(s_{i})-\kappa_{2}\text{ sign}^{a}(s_{i})+\omega_{i}(t). \tag{17}\]8  
Given the \(m\)-strong convexity of \(f^{p}(x)\), let \(f^{p}_{*}=f^{p}(\hat{x}^{*})\), then  
\[=-\kappa_{0}s_{i}-\kappa_{1}\text{sgn}(s_{i})-\kappa_{2}\text{ sign}^{a}(s_{i})+\omega_{i}(t). \tag{17}\]9 \[\dot{V}_{1,i}(t) =2s_{i}\dot{s}_{i}\]0 \[\dot{V}_{1,i}(t) =2s_{i}\dot{s}_{i}\]1  
Denote \(\mathcal{F}(x)=f^{p}(x)-f^{p}_{*}+\tilde{\phi}^{\top}(x^{*}-x)+\frac{m}{2}\|x- \hat{x}^{*}\|^{2}\), and \(\mathcal{F}(x)\) is a positive definite quadratic with respect to \(\hat{x}^{*}-x\).