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(L2DYΔˉ0ε−3), where
L is the gradient Lipschitz constant,
DY bounds the diameter of the dual feasible set, and
Δˉ0 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(L2DYΔφε−3), up to an additive lower-order cost, where
Δφ:=φ(x0)−infxφ(x). We further establish a lower bound of
Ω(L2DYΔφε−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(L3/2DY1/2Δφε−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(κLΔφε−2) leading complexity for both stationarity criteria, where
κ=L/μ is the dual condition number, up to an additive initialization cost. The
ε−2 accuracy dependence is optimal under fixed regularity and initialization bounds.