# Fixed-Time Optimization of Perturbed Multi-Agent Systems under the Resource Constraints
## 1 Introduction  
With the increasing number of clustered and networked engineering applications, many tasks have been transformed from monolithic to multi-agent systems collaboration, and MSCO is an important problem arising from it. The ultimate aim of a MSCO problem is to minimize the global objective function of a group of agents under global resource requirements. Each agent has access solely to its own state and goal information, communicating exclusively with neighboring agents. MSCO problems arise in many decision-making tasks, such as economic dispatches in smart grids [1; 2; 3; 4; 5], data transmission in wireless sensor networks [6; 7; 8], and path planning in multiple unmanned vehicles [9; 10], etc. For a MSCO problem, the global objective function is often expressed as a sum of the local objective functions that are generally convex and only accessible by local agents. Different from conventional distributed optimization problems, the agents are always assumed to be constrained by some convex sets, which brings many difficulties in dealing with such problems. For solving such a problem, many discrete-time distributed optimization algorithms [11; 12; 13] have been proposed. The distributed subgradient descent method [11; 12] is proposed based on the consistency algorithm, which solves the unconstrained convex optimization problem under the condition of consistent joint connectivity of time-varying undirected networks. The algorithm requires that each agent keeps a backup copy of the global decision variables, which undoubtedly increases the burden of communication and storage. The exact first-order algorithm (EXTRA) [13] for distributed exact convergence is proposed as a distributed accelerated optimization algorithm that achieves a linear convergence property under undirected graph connectivity and certain algorithm parameter settings. Since many dynamic systems are modeled as continuous-time models, manycontinuous-time distributed optimization algorithms have been proposed [14; 15; 16]. The distributed PI algorithm [14] is proposed under the equilibrium-directed graph based on the idea of PI control, and gives the stability analysis of the algorithm based on the Lyapunov theorem. The paper analyzes the convergence characteristics of distributed optimization from the perspective of a continuous-time control system and gives the design framework of a distributed optimization algorithm. A distributed extended monotropic optimization algorithm is proposed in [16] using the extended Lagrangian method, which considers a specific class of separable structures without the need for the same number of dimensions of each agent. Compared to exponential convergence algorithms, fixed-time convergence algorithms gained increasing attention due to their initial state-independent convergence property [17]. Fixed-time convergence algorithms for multi-agent systems have achieved some results, as evidenced by [18; 19; 20; 21; 22] and related references. In [18], a distributed fixed-time convex optimization algorithm is developed using the projected gradient and penalty function methods, which can be easily implemented for practical usage. Similarly, a fixed-time economic scheduling algorithm for smart grids is proposed in [19] under a directed communication network. In [20], the authors employ the integral sliding mode control method to eliminate external perturbations in multi-agent systems, thereby achieving a distributed fixed-time economic scheduling algorithm. However, the proposed algorithm does not take into account the local constraints of the agents. In [21], a class of heterogeneous agents with random noise is considered to achieve the finite-time consistent control using a power-integrator approach, which provides a reference for designing distributed control strategies for multi-agent systems with nonlinear dynamic structures. A fixed-time consensus proposal by a fuzzy adaptive method is proposed in [22] for the consensus tracking problem.