  Minimax problems have gained tremendous attentions across the optimization
and machine learning community recently. In this paper, we introduce a new
quasi-Newton method for minimax problems, which we call $J$-symmetric
quasi-Newton method. The method is obtained by exploiting the $J$-symmetric
structure of the second-order derivative of the objective function in minimax
problem. We show that the Hessian estimation (as well as its inverse) can be
updated by a rank-2 operation, and it turns out that the update rule is a
natural generalization of the classic Powell symmetric Broyden (PSB) method
from minimization problems to minimax problems. In theory, we show that our
proposed quasi-Newton algorithm enjoys local Q-superlinear convergence to a
desirable solution under standard regularity conditions. Furthermore, we
introduce a trust-region variant of the algorithm that enjoys global
R-superlinear convergence. Finally, we present numerical experiments that
verify our theory and show the effectiveness of our proposed algorithms
compared to Broyden's method and the extragradient method on three classes of
minimax problems.
