  We propose adaptive, line search-free second-order methods with optimal rate
of convergence for solving convex-concave min-max problems. By means of an
adaptive step size, our algorithms feature a simple update rule that requires
solving only one linear system per iteration, eliminating the need for line
search or backtracking mechanisms. Specifically, we base our algorithms on the
optimistic method and appropriately combine it with second-order information.
Moreover, distinct from common adaptive schemes, we define the step size
recursively as a function of the gradient norm and the prediction error in the
optimistic update. We first analyze a variant where the step size requires
knowledge of the Lipschitz constant of the Hessian. Under the additional
assumption of Lipschitz continuous gradients, we further design a
parameter-free version by tracking the Hessian Lipschitz constant locally and
ensuring the iterates remain bounded. We also evaluate the practical
performance of our algorithm by comparing it to existing second-order
algorithms for minimax optimization.
