Enhancement of the downhill simplex method of optimization

R. John Koshel

doi:10.1117/12.486465

23 December 2002 Enhancement of the downhill simplex method of optimization

R. John Koshel

Proceedings Volume 4832, International Optical Design Conference 2002; (2002) https://doi.org/10.1117/12.486465
Event: International Optical Design Conference 2002, 2002, Tucson, AZ, United States

Abstract

The downhill simplex method of optimization is a "geometric" method to achieve function minimization. The standard algorithm uses arbitrary values for the deterministic factors that describe the "movement" of the simplex in the merit space. While it is a robust method of optimization, it is relatively slow to converge to local minima. However, its stability and the lack of use of derivatives make it useful for optical design optimization, especially for the field of illumination. This paper describes preliminary efforts of optimizing the performance of the simplex optimizer. This enhancement is accomplished by optimizing the various control factors: alpha (reflection), beta (contraction), and gamma (expansion). This effort is accomplished by investigating the "end game" of optimal design, i.e., the shape of the figure of merit space is parabolic in N-dimensions near local minima. The figure of merit for the control factor optimization is the number of iterations to achieve a solution in comparison to the same case using the standard control factors. This optimization is done for parabolic wells of order N equals 2 to 15. In this study it is shown that with the correct choice of the control factors, one can achieve up to a 35% improvement in convergence. Techniques using gradient weighting and the inclusion of additional control factors are proposed.

Citation Download Citation

R. John Koshel "Enhancement of the downhill simplex method of optimization", Proc. SPIE 4832, International Optical Design Conference 2002, (23 December 2002); https://doi.org/10.1117/12.486465

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