Minimax Methods for Open-Loop Equilibra in $N$-Person Differential Games Part II: Duality and Penalty Theory (Journal of Computational Mathematics, 2021)
The equilibrium strategy for $N$-person differential games can be obtained from a min-max problem subject to differential constraints. The differential constraints are treated here by the duality and penalty methods. We first formulate the duality theory. This involves the introduction of $N+1$ Lagrange multipliers: one for each player and one commonly shared by all players. The primal min-max problem thus results in a dual problem, which is a max-min problem with no differential constraints. We develop the penalty theory by penalizing $N+1$ differential constraints. We give a convergence proof which generalizes a theorem due to B.T. Polyak.