Chaotic behavior in an algorithm to escape from poor local minima in lens design

Area of Science:

• Optical Engineering
• Computational Optics
• Applied Mathematics

Background:

• Damped least-squares (DLS) methods are standard for local optimization in lens design.
• These methods typically converge to the nearest local minimum in the merit function landscape.
• Exploring DLS beyond local optimization is crucial for complex design challenges.

Purpose of the Study:

• To investigate the application of DLS methods for escaping local minima in lens design.
• To analyze the complex dynamics and phenomena associated with DLS when used for global optimization.
• To enhance the understanding of DLS behavior in both local and non-local optimization tasks.

Main Methods:

• Utilizing low damping parameters in the DLS algorithm.
• Allowing temporary increases in the merit function during iteration.
• Analyzing chaotic transients, complex dynamics, and crisis phenomena.

Main Results:

• Successfully overcoming merit function barriers to escape poor local minima.
• Observed chaotic behavior, including chaotic transients and crisis phenomena.
• Demonstrated that successful escapes correlate with the transformation of chaotic attractors into chaotic saddles.
• Gained insights into peculiar behaviors of DLS in conventional local optimization.

Conclusions:

• DLS methods, with modified damping, can be used to escape local minima in lens design.
• The optimization process exhibits complex dynamics, offering a richer understanding of DLS.
• This study provides a deeper comprehension of DLS peculiarities in both local and global optimization contexts.