The multidimensional merit function space of complex optical systems contains a large number of local minima. We illustrate a method to find new local minima by constructing saddle points, with examples of deep and extreme UV objectives. The central idea of the method is that, at certain positions in a system with N surfaces that is a local minimum, a thin meniscus lens or two mirror surfaces can be introduced to construct a system with N+2 surfaces that is a saddle point. When optimization rolls down on the two sides of the saddle point, two minima are obtained. Often one of these two minima can also be reached from several other saddle points constructed in the same way. With saddle-point construction we can obtain new design shapes from existing ones in a simple, efficient, and systematic manner that is suitable for complex designs such as those for lithographic objectives.