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Convergence properties of the augmented Hessian (AH) method when searching for stationary points of an arbitrary fixed index are investigated. It is shown that the displacement vector of this method is proportional to one of the Hessian eigenvectors if the current point is far from a stationary one of the required index. A simple and reliable criterion for nearness of the current point to a stationary one of the desired index is proposed. The efficiency of a new one-dimensional optimization scheme that uses this criterion is studied. The case of coincidence of Hessian eigenvalues, which is a bottleneck of the standard AH method, is analyzed. A relation of the AH method to those by Poppinger and Wales is outlined.