Alexander Serebriakov

Networks of local minima in optical system optimization

We discuss a surprising new feature of the merit function landscape in optical system design. When certain conditions are satisfied, the local minima form a network in which all nodes are connected. Each link between two neighboring minima contains a saddle point with a Morse index of 1. For a simple global optimization search (the symmetric Cooke triplet), the network of the corresponding set of local minima is presented.

Network structure of the set of local minima in optical system optimization

We discuss a surprising new feature of the merit function landscape in optical system design. When certain conditions are satisfied, the set of local minima forms a network in which all nodes are connected. Each link between two neighboring minima contains a special type of saddle point (more precisely, a saddle point having a Morse index. On this basis, a new global optimization method that takes advantage of this feature is proposed. The central component of the new method, the algorithm for saddle point detection, works in a parameter space of arbitrary dimensionality, and uses only the local optimization engine of the optical design program. For a simple global optimization search (the symmetric Cooke triplet) the network of the corresponding set of local minima is presented.

Finding new local minima by switching merit functions in optical system optimization

A strategy to escape from poor local minima by switching merit functions during local optimization is discussed. As a switching partner, we define a new auxiliary merit function, which also tends to zero for ideal systems, but differs significantly from traditional merit functions. The examples include high-dimensional optimization problems.

Local optimization strategies to escape from poor local minima

Three local optimization strategies to escape from a local minimum are discussed. The first strategy is to radically modify the error function. The old and new error function should both tend to zero for ideal systems but must differ sufficiently from another. The second strategy is to temporarily over-design the system, i.e. to make available for optimization more system parameters than a designer would normally use for the given aperture and field specifications. Finally, it is shown that, with a small enhancement of the local optimization algorithm, it is possible to move from one local minimum into a neighboring one by locating a saddle point between them.