This paper investigates the collision-free control problem for multi-agent systems. For such multi-agent systems, it is the typical situation where conventional methods using either the usual centralized model predictive control (MPC), or even the distributed counterpart, would suffer from substantial difficulty in balancing optimality and computational efficiency. Additionally, the non-convex characteristics that invariably arise in such collision-free control and optimization problems render it difficult to effectively derive a reliable solution (and also to thoroughly analyze the associated convergence properties). To overcome these challenging issues, this work establishes a suitably novel parallel computation framework through an innovative mathematical problem formulation; and then with this framework and formulation, a parallel algorithm based on alternating direction method of multipliers (ADMM) is presented to solve the sub-problems arising from the resulting parallel structure. Furthermore, an efficient and intuitive initialization procedure is developed to accelerate the optimization process, and the optimum is thus determined with significantly improved computational efficiency. As supported by rigorous proofs, the convergence of the proposed ADMM iterations for this non-convex optimization problem is analyzed and discussed in detail. Finally, a simulation with a group of unmanned aerial vehicles (UAVs) serves as an illustrative example here to demonstrate the effectiveness and efficiency of the proposed approach. Also, the simulation results verify significant improvements in accuracy and computational efficiency compared to other baselines, including primal quadratic mixed integer programming (PQ-MIP), non-convex quadratic mixed integer programming (NC-MIP), and non-convex quadratically constrained quadratic programming (NC-QCQP).

翻译：本文研究多智能体系统的无碰撞控制问题。对于此类系统，传统方法——无论是采用常见的集中式模型预测控制(MPC)还是其分布式变体——都难以在最优性与计算效率之间取得平衡。此外，此类无碰撞控制与优化问题中不可避免的非凸特性，使得有效推导可靠解（以及全面分析相关收敛性质）变得困难。为克服这些挑战，本研究通过创新性的数学问题建模构建了一个新颖的并行计算框架；基于该框架与建模，提出了一种基于交替方向乘子法(ADMM)的并行算法，用于求解并行化结构产生的子问题。此外，还开发了一种高效直观的初始化过程以加速优化进程，从而显著提升计算效率确定最优解。通过严格数学证明，本文详细分析并讨论了所提ADMM迭代在非凸优化问题中的收敛性。最后，以一组无人机(UAV)仿真作为示例，验证了所提方法的有效性与高效性。与其他基准方法（包括原始二次混合整数规划(PQ-MIP)、非凸二次混合整数规划(NC-MIP)及非凸二次约束二次规划(NC-QCQP)）相比，仿真结果证实了该方法在精度与计算效率上的显著提升。