The traditional BFGS algorithm has been proved very efficient. It is convergent for convex non
linear 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.