We show that in the lens design landscape saddle points exist that are closely related to local minima of simpler problems. On the basis of this new theoretical insight we develop a systematic and efficient saddlepoint method that uses a-priori knowledge for obtaining new local minima. In contrast with earlier saddle-point methods, the present method can create both positive and negative lenses. As an example, by successively using the method a good-quality local minimum is obtained from a poor-quality one. The method could also be applicable in other global optimization problems that satisfy the requirements discussed in this paper.