Matching Multi-Loop Complexities with a Single Loop: Optimal Optimization Stationarity and Best-Known Game Stationarity in Nonconvex--Concave Minimax Optimization

We introduce a new single-loop algorithmic framework for smooth nonconvex--concave minimax optimization. The resulting projected damped extragradient method combines projected extragradient updates, dual momentum, and a moving proximal center. Under both the optimization-stationarity and game-stationarity criteria, our method achieves the best-known complexity among single-loop first-order methods. For optimization stationarity, our method achieves a gradient complexity of O(L2DYbarDelta0varepsilon-3), where L is the gradient Lipschitz constant, DY bounds the diameter of the dual feasible set, and barDelta0 is an initialization quantity involving the value-function gap and the initial gradients. Moreover, by incorporating a fixed-center warm-up phase, the complexity can be improved to O(L2DYDeltaφvarepsilon-3), up to an additive lower-order cost, where Deltaφ:=φ(x0)-infxφ(x). We further establish a lower bound of Omega(L2DYDeltaφvarepsilon-3) for optimization stationarity over projected zero-respecting first-order methods. This lower bound proves that the warm-started version of our algorithm is optimal up to a constant factor for optimization stationarity within this oracle class. For game stationarity, our method achieves O!(L3/2DY1/2Deltaφvarepsilon-5/2) gradient complexity. This matches the best-known complexity of multi-loop first-order methods, thereby establishing the same complexity with a single-loop algorithmic structure. Under dual strong concavity, the proposed framework achieves O!(√kappa,LDeltaφvarepsilon-2) leading complexity for both stationarity criteria, where kappa=L/μ is the dual condition number, up to an additive initialization cost. The varepsilon-2 accuracy dependence is optimal under fixed regularity and initialization bounds.