  We propose a hybrid approach that combines Hamilton-Jacobi (HJ) reachability
and mixed-integer optimization for solving a reach-avoid game with multiple
attackers and defenders. The reach-avoid game is an important problem with
potential applications in air traffic control and multi-agent motion planning;
however, solving this game for many attackers and defenders is intractable due
to the adversarial nature of the agents and the high problem dimensionality. In
this paper, we first propose an HJ reachability-based method for solving the
reach-avoid game in which 2 attackers are playing against 1 defender; we derive
the numerically convergent optimal winning sets for the two sides in
environments with obstacles. Utilizing this result and previous results for the
1 vs. 1 game, we further propose solving the general multi-agent reach-avoid
game by determining the defender assignments that can maximize the number of
attackers captured via a Mixed Integer Program (MIP). Our method generalizes
previous state-of-the-art results and is especially useful when there are fewer
defenders than attackers. We validate our theoretical results in numerical
simulations.
