Stackelberg and Nash Equilibrium Computation in Non-Convex Leader-Follower Network Aggregative Games

This paper considers Stackelberg equilibrium (SE) and Nash equilibrium (NE) computation in a class of non-convex network aggregative games with one leader and multiple followers. The cost function of each follower is influenced by its strategy, the leader's strategy, and its neighbors' aggregative strategies. Also, the structured non-convex cost function of the leader is the composition of a canonical function and a vector-valued geometrical operator that relies on its strategy and followers' strategies. In the leader-follower scheme, when the leader has knowledge of the best responses of the followers in a closed form, the SE strategy will be the optimal choice due to its relatively low cost. When the leader does not know the exact expression of followers' best responses or the leader's dominance is threatened, NE will be what all players are committed to achieving. The widespread existence of nonconvexity creates a significant challenge for computing the above equilibria in different circumstances. The results in existing convex games are not directly applicable to such a non-convex case, as they get trapped in local equilibria or stationary points rather than global equilibria. Here, we adopt the canonical transformation to reformulate the non-convex games and present the existence condition based on the canonical duality theory. Then two projection gradient algorithms are designed to pursue the SE and the NE, followed by proving the convergence of the algorithms.