# Fixed-Time Optimization of Perturbed Multi-Agent Systems under the Resource Constraints
## 2 Preliminary and Problem Description
### Problem Description  
Consider a multi-agent system consisting of \(N\) first-order agents, the communication topology of the system is given by the graph \(\mathcal{G}=(A,\mathcal{V})\). The state of agent \(i\) is \(x_{i}\in\mathbb{R}\), and the global state of the entire system is \(x=(x_{1},\cdots,x_{n})^{\top}\in\mathbb{R}^{N}\). Agent \(i\)'s local objective function \(f_{i}:\mathbb{R}^{|\mathcal{N}_{i}|+1}\rightarrow\mathbb{R}\) is only related to the local state \(x_{i}\) and the state of the neighboring agents \(\{x_{j}\}_{j\in\mathcal{N}_{i}}\). Denote the local augmented state of the agent \(i\) as \(\bar{x}_{i}=(x_{i},\{x_{j}\}_{j\in\mathcal{N}_{i}})^{\top}\). The global objective function of the system can be separated into the sum of the local objective functions of the agents, i.e., \(f(x)=\sum_{i=1}^{N}f_{i}(\bar{x}_{i})\). Consider the following MSCO problem:  
\[\min_{x\in\mathbb{R}^{N}} f(x)=\sum_{i=1}^{N}f_{i}(\bar{x}_{i}),\] s.t. \[\sum_{i=1}^{N}x_{i}=\sum_{i=1}^{N}d_{i}=D,\] \[(x_{i}-x_{i}^{\mathrm{d}})(x_{i}-x_{i}^{\mathrm{u}})\leq 0,i\in \mathcal{N}, \tag{5}\]  
where, \(d_{i}\in\mathbb{R}\) denotes the local resource requirements of the agent \(i\), \(D\in\mathbb{R}\) is the global resource requirement of the multi-agent system, and \(x_{i}^{\mathrm{u}},x_{i}^{\mathrm{u}}\in\mathbb{R}_{\geq 0}\) are the known upper and lower local resource constraints of the agent \(i\), respectively. The distributed solution of the problem (5) requires that the agent \(i\) store only their local states \(x_{i}\), local objective function \(f_{i}\), local resource constraints \(x_{i}^{\mathrm{u}}\) and \(x_{i}^{\mathrm{u}}\), and local resource requirements \(d_{i}\), and only interacts with information from its neighboring agents. The set of feasible solutions to the problem (5) is denoted as  
\[X_{\mathrm{fe}}=\{x\in\mathbb{R}^{N}|x_{1}+\cdots+x_{N}=D,x_{i}^{\mathrm{u}} \leq x_{i}\leq x_{i}^{\mathrm{u}},i\in\mathcal{N}\}. \tag{6}\]  
The optimal set of solutions to the problem (5) is denoted as \(\Omega^{\star}\), and the corresponding objective function value is \(f^{\star}\). The dynamics of the first-order agents can be described as  
\[\dot{x}_{i}(t)=u_{i}(t)+\omega_{i}(t),i=1,2,\cdots,N, \tag{7}\]  
where, \(u_{i}(t)\in\mathbb{R}\) denotes the control input of the agent \(i\), \(w_{i}(t)\in\mathbb{R}\) is the unknown external disturbance. Now, it is necessary to design the control law \(u_{i}(t)\) in the presence of external perturbation \(w_{i}(t)\) such that \(\Omega^{\star}\) is a global fixed-time attraction domain of the MSCO problem (5). **Assumption 1**. : _Graph \(\mathcal{G}\) is a static undirected connected graph._  
**Assumption 2**. : _The local objective function \(f_{i}(\bar{x}_{i})\) is continuously differentiable, and the global objective function \(f(x)\in C^{1}(\mathbb{R}^{N},\mathbb{R})\) is m-strongly convex._  
**Assumption 3** (relaxed Slater condition). : _There exists_  
\[\bar{x}=(\bar{x}_{1},\cdots,\bar{x}_{N})^{\top},\]  
_such that \((\bar{x}_{i}-x_{i}^{\mathrm{l}})(\bar{x}_{i}-x_{i}^{\mathrm{u}})\leq 0,i\in \mathcal{N}\), \(\sum_{i=1}^{N}\bar{x}_{i}=D\) holds simultaneously._  
**Assumption 4**. : _The \(l_{2}\) norm of the external perturbation \(\omega_{i}(t)\) is bounded, i.e., there exists a constant \(\delta_{i}\in\mathbb{R}^{>0}\) such that \(\|\omega_{i}\|\leq\delta_{i}\), and we write \(\overline{\delta}=\max(\delta_{i})\)._  
**Remark 1**. : _Assumptions 1-3 are commonly used assumptions in the design of distributed fixed-time optimization algorithms. Assumptions 1 are used to simplify the description of the algorithm, which is also applicable in time-varying topologies and directed strongly connected graphs as seen later. Assumption 2 and 3 guarantee the existence and uniqueness of the problem (5), which can be verified by the dual problem of the original problem (5), i.e., there exists a unique \(x^{\star}\in\mathbb{R}^{N}\) such that \(f(x^{\star})=f^{\star}\) is a minimal value point.