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Quasi-Newton methods were introduced by Charles Broyden [A class of methods for solving nonlinear simultaneous equations, Math Comp. 19 (1965), pp. 577–593] as an alternative to Newton's method for solving nonlinear algebraic systems; in 1970 Broyden [The convergence of a class of double rank minimization algorithms, IMA J Appl Math. 6, part I and II (1970), pp. 76–90, 222–231] extended them to nonlinear unconstrained optimization as a generalization of the DFP method which is proposed by Davidon [Variable metric method for minimization (revised), Technical Report ANL-5990, Argonne National Laboratory, USA, 1959] and investigated by Fletcher and Powell [A rapidly convergent descent method for minimization, Comput J. 6 (1963), pp. 163–168]. Such methods (in particular, the BFGS (Broyden–Fletcher–Goldfarb–Shanno) method) are very useful in practice and have been subject to substantial …