The global convergence of a modified BFGS method for nonconvex functions

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The standard BFGS method plays an important role among the quasi-Newton algorithms for constrained/un-constrained optimization problems. However, Dai (2003) constructed a counterexample to demonstrate that this method may fail for non-convex functions with inexact Wolfe line searches, and Mascarenhas (2004) showed the nonconvergence of this method and other methods in the Broyden family, even with exact line search techniques. These works motivate us to try to find another way to obtain another quasi-Newton method with better convergence based on the standard BFGS formula. In this paper, four approaches are used in the designed algorithm: (1) a modified weak Wolfe–Powell line search technique is introduced; (2) if one defined condition is satisfied, then the search direction and its associated step length are accepted, and the next point is designed; (3) otherwise, a parabolic will be presented, and it is regarded as the projection surface to avoid using the failed direction, and the next point x k + 1 is generated by a projection technique; and (4) to easily obtain the global convergence of the proposed algorithm, the projection point is not used in the current BFGS update but rather for all the next iterations. The new algorithm possesses global convergence for general functions under the inexact modified weak Wolfe–Powell line search technique, and it is shown that other methods in the Broyden class also have this property. Numerical results are reported to show the performance of the given algorithm and other similar methods.