A robust BFGS algorithm for unconstrained nonlinear optimization problems

The traditional BFGS algorithm has been proved very efficient. It is convergent for convex nonlinear optimization problems. However, for non-convex nonlinear optimization problems, it is known that the BFGS algorithm may not be convergent. This paper proposes a robust BFGS algorithm in the sense that the algorithm superlinearly converges to a local minimum under some mild assumptions for both convex and non-convex nonlinear optimization problems. Numerical test on the CUTEst test set is reported to demonstrate the merit of the proposed robust BFGS algorithm. This result shows that the robust BFGS algorithm is very efficient and effective.