  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).
