Obtaining new local minima in lens design by constructing saddle points

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 saddle-point 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.