  This paper studies quasi-Newton methods for solving
strongly-convex-strongly-concave saddle point problems (SPP). We propose greedy
and random Broyden family updates for SPP, which have explicit local
superlinear convergence rate of ${\mathcal
O}\big(\big(1-\frac{1}{n\kappa^2}\big)^{k(k-1)/2}\big)$, where $n$ is
dimensions of the problem, $\kappa$ is the condition number and $k$ is the
number of iterations. The design and analysis of proposed algorithm are based
on estimating the square of indefinite Hessian matrix, which is different from
classical quasi-Newton methods in convex optimization. We also present two
specific Broyden family algorithms with BFGS-type and SR1-type updates, which
enjoy the faster local convergence rate of $\mathcal
O\big(\big(1-\frac{1}{n}\big)^{k(k-1)/2}\big)$. Additionally, we extend our
algorithms to solve general nonlinear equations and prove it enjoys the similar
convergence rate.
