Near-Optimal Pure Single-Loop Extragradient Method for Strongly Convex--Strongly Concave Minimax Optimization

We study smooth strongly convex–strongly concave minimax optimization with general nonlinear coupling in the deterministic unconstrained setting. We propose a pure single-loop damped extragradient method with fixed parameters and two new full-gradient evaluations per iteration after one initialization query. The method uses an auxiliary feedback recursion and requires no inner solves, accuracy schedules, or staged restarts. We establish last-iterate linear convergence and show that reducing the squared Euclidean distance to the saddle point to an $\epsilon$ fraction of its initial value requires $O(\sqrt{{\kappa }_x{\kappa }_y}\log (2{\kappa }_x{\kappa }_y\mathrm{/}\epsilon ))$ full-gradient queries, where ${\kappa }_x=L\mathrm{/}{\mu }_x$ and ${\kappa }_y=L\mathrm{/}{\mu }_y$. This bound attains the optimal condition-number order up to logarithmic factors through fixed explicit updates. Numerical experiments demonstrate the effectiveness of the method.