  We describe a novel approach for computing collision-free \emph{global}
trajectories for $p$ agents with specified initial and final configurations,
based on an improved version of the alternating direction method of multipliers
(ADMM). Compared with existing methods, our approach is naturally
parallelizable and allows for incorporating different cost functionals with
only minor adjustments. We apply our method to classical challenging instances
and observe that its computational requirements scale well with $p$ for several
cost functionals. We also show that a specialization of our algorithm can be
used for {\em local} motion planning by solving the problem of joint
optimization in velocity space.
