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In this paper, a new inexact line search rule is presented, which is a modified version of the classical Armijo line search rule. With lower cost of computation, a larger descent magnitude of objective function is obtained at every iteration. In addition, the initial step size in the modified line search is adjusted automatically for each iteration. On the basis of this line search, a new cautious Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm is developed. Under some mild assumptions, the global convergence of the algorithm is established for nonconvex optimization problems. Numerical results demonstrate that the proposed method is promising, especially in comparison with the existent methods.