Addressing the Multiplicity of Solutions in Optical Lens Design as a Niching Evolutionary Algorithms Computational Challenge

TL;DR

This paper demonstrates that niching evolutionary algorithms, specifically Niching-CMA-ES, effectively identify multiple optimal solutions in complex optical lens design problems, discovering numerous new minima and infeasibility pockets.

Contribution

It applies and validates Niching-CMA-ES for optical lens design, revealing its ability to find known and new optima, and highlights the importance of local search assistance.

Findings

Located 19 of 21 known minima in one run

Discovered 540 new optima with better merit functions

Identified numerous infeasibility pockets

Abstract

Optimal Lens Design constitutes a fundamental, long-standing real-world optimization challenge. Potentially large number of optima, rich variety of critical points, as well as solid understanding of certain optimal designs per simple problem instances, provide altogether the motivation to address it as a niching challenge. This study applies established Niching-CMA-ES heuristic to tackle this design problem (6-dimensional Cooke triplet) in a simulation-based fashion. The outcome of employing Niching-CMA-ES `out-of-the-box' proves successful, and yet it performs best when assisted by a local searcher which accurately drives the search into optima. The obtained search-points are corroborated based upon concrete knowledge of this problem-instance, accompanied by gradient and Hessian calculations for validation. We extensively report on this computational campaign, which overall resulted in (i) the location of 19 out of 21 known minima within a single run, (ii) the discovery of 540 new optima. These are new minima similar in shape to 21 theoretical solutions, but some of them have better merit function value (unknown heretofore), (iii) the identification of numerous infeasibility pockets throughout the domain (also unknown heretofore). We conclude that niching mechanism is well-suited to address this problem domain, and hypothesize on the apparent multidimensional structures formed by the attained new solutions.