  We solve large-scale mixed-integer linear programs (MILPs) via distributed
asynchronous saddle point computation. This is motivated by the MILPs being
able to model problems in multi-agent autonomy, e.g., task assignment problems
and trajectory planning with collision avoidance constraints in multi-robot
systems. To solve a MILP, we relax it with a nonlinear program approximation
whose accuracy tightens as the number of agents increases relative to the
number of coupled constraints. Next, we form an equivalent Lagrangian saddle
point problem, and then regularize the Lagrangian in both the primal and dual
spaces to create a regularized Lagrangian that is
strongly-convex-strongly-concave. We then develop a parallelized algorithm to
compute saddle points of the regularized Lagrangian. This algorithm partitions
problems into blocks, which are either scalars or sub-vectors of the primal or
dual decision variables, and it is shown to tolerate asynchrony in the
computations and communications of primal and dual variables. Suboptimality
bounds and convergence rates are presented for convergence to a saddle point.
The suboptimality bound includes (i) the regularization error induced by
regularizing the Lagrangian and (ii) the suboptimality gap between solutions to
the original MILP and its relaxed form. Simulation results illustrate these
theoretical developments in practice, and show that relaxation and
regularization together have only a mild impact on the quality of solution
obtained.
