  We present a batch trajectory optimizer that can simultaneously solve
hundreds of different instances of the problem in real-time. We consider
holonomic robots but relax the assumption of circular base footprint. Our main
algorithmic contributions lie in: (i) improving the computational tractability
of the underlying non-convex problem and (ii) leveraging batch computation to
mitigate initialization bottlenecks and improve solution quality. We achieve
both goals by deriving a multi-convex reformulation of the kinematics and
collision avoidance constraints. We exploit these structures through an
Alternating Minimization approach and show that the resulting batch operation
reduces to computing just matrix-vector products that can be trivially
accelerated over GPUs. We improve the state-of-the-art in three respects.
First, we improve quality of navigation (success-rate, tracking) as compared to
baseline approach that relies on computing a single locally optimal trajectory
at each control loop. Second, we show that when initialized with trajectory
samples from a Gaussian distribution, our batch optimizer outperforms
state-of-the-art cross-entropy method in solution quality. Finally, our batch
optimizer is several orders of magnitude faster than the conceptually simpler
alternative of running different optimization instances in parallel CPU
threads. \textbf{Codes:} \url{https://tinyurl.com/a3b99m8}
