Comp. Appl. Math. https://doi.org/10.1007/s40314-018-0620-8 A new modified BFGS method for unconstrained optimization problems Razieh Dehghani 1 · Narges Bidabadi 1 · Mohammad Mehdi Hosseini 1 Received: 6 June 2017 / Revised: 10 April 2018 / Accepted: 12 April 2018 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2018 Abstract Using chain rule, we propose a modified secant equation to get a more accurate approximation of the second curvature of the objective function. Then, based on this modified secant equation we present a new BFGS method for solving unconstrained optimization problems. The proposed method makes use of both gradient and function values, and utilizes information from two most recent steps, while the usual secant relation uses only the latest step information. Under appropriate conditions, we show that the proposed method is globally convergent without convexity assumption on the objective function. Comparative numerical results show computational efficiency of the proposed method in the sense of the Dolan–Moré performance profiles. Keywords Unconstrained optimization · Nonlinear function · Two-step secant equation · Global convergence Mathematics Subject Classification 90C53 · 65K05 1 Introduction Consider the unconstrained nonlinear optimization problem min f (x ), x ∈ R n , (1) Communicated by Joerg Fliege. B Narges Bidabadi n_bidabadi@yazd.ac.ir Razieh Dehghani rdehghani@stu.yazd.ac.ir Mohammad Mehdi Hosseini hosse_m@yazd.ac.ir 1 Department ofMathematical Sciences, Yazd University, Yazd, Iran 123