A Lyapunov analysis of Korpelevich's extragradient method with fast and flexible extensions

We present a Lyapunov analysis of Korpelevich's extragradient method andestablish an $\mathcal{O}(1/k)$ last-iterate convergence rate. Building onthis, we propose flexible extensions that combine extragradient steps withuser-specified directions, guided by a line-search procedure derived from thesame Lyapunov analysis. These methods retain global convergence under practicalassumptions and can achieve superlinear rates when directions are chosenappropriately. Numerical experiments highlight the simplicity and efficiency ofthis approach.