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

arXiv:2609.20327v1 Announce Type: cross Abstract: 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 $\varepsilon$ fraction of its initial value requires $O(\sqrt{\kappa_x\kappa_y}\log(2\kappa_x\kappa_y/\varepsilon))$ full-gradient queries, where $\kappa_x=L/\mu_x$ and $\kappa_y=L/\mu_y$. This bound attains the optimal condition-number order up to logarithmic factors through fixed explicit updates. Numerical experiments demonstrate the effectiveness of the method.