  In this paper, we present a computationally efficient trajectory optimizer
that can exploit GPUs to jointly compute trajectories of tens of agents in
under a second. At the heart of our optimizer is a novel reformulation of the
non-convex collision avoidance constraints that reduces the core computation in
each iteration to that of solving a large scale, convex, unconstrained
Quadratic Program (QP). We also show that the matrix factorization/inverse
computation associated with the QP needs to be done only once and can be done
offline for a given number of agents. This further simplifies the solution
process, effectively reducing it to a problem of evaluating a few matrix-vector
products. Moreover, for a large number of agents, this computation can be
trivially accelerated on GPUs using existing off-the-shelf libraries. We
validate our optimizer's performance on challenging benchmarks and show
substantial improvement over state of the art in computation time and
trajectory quality.
