A first-order Lagrangian penalty method is claimed to reach an ε-stationary multi-follower Stackelberg equilibrium in O(k²ε^{-6-α}) gradient evaluations.
Finite--Dimensional Variational Inequalities and Complementarity Problems, volume 1 of Springer Series in Operations Research and Financial Engineering
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Finding a Multiple Follower Stackelberg Equilibrium: A Fully First-Order Method
A first-order Lagrangian penalty method is claimed to reach an ε-stationary multi-follower Stackelberg equilibrium in O(k²ε^{-6-α}) gradient evaluations.