In this paper we follow Rodomanov and Nesterov’s work to study quasi-Newton methods. We focus on the common SR1 and BFGS quasi-Newton methods to establish better explicit (local) superlinear convergence. First, based on greedy quasi-Newton update which greedily selects the direction so as to maximize a certain measure of progress, we improve the convergence rate to a condition-number-free superlinear convergence rate. Second, based on random quasi-Newton update that selects the direction randoml…