In the following, we will use the Global Search and Hammer algorithms. The Global Search algorithm used a mixture of genetic optimization, random starting points and dampled least squares (DLS) optimization and is good at efficiently searching multi-dimensional parameter spaces for regions of low merit function. The Hammer optimizer also uses genetic and DLS optimizers to exhaustively refine a design once Global Search has found a promising region of parameter space.

The starting value of the merit function is 0.036, and the brightness at zero angle is 25 Cd.

First of all, we know that the simple parameter scan of the radius parameter shows a strong peak at around -100, so we set the radius to this value, and then run the Global optimizer. After a little while, the optimizer sets the radius to be around -44mm, and the merit function falls to 0.009, corresponding to a brightness of just over 100 Cd.

Then, make the conic constant of the extended polynomial mirror variable also, and repeat the optimization. The merit function falls to around 0.0085, corresponding to a brightness of about 117 Cd.

This then has given us the best-form conicoid mirror. Now make the polynomials variable, and run the Global optimizer again. With so many variables (23), it may take some time to find the best solution, but a merit function of around 0.004 and a peak brightness of 250 Cd should be achieved. Then, a final Hammer optimization is run. This yields a final merit function of 0.003, and a peak brightness of  326 Cd. The zip file at the end of this article contains this mirror.

So, the results of this optimization are as follows:

Variables	Merit Function	On-axis Brightness (Candela)
None (flat mirror)	0.036	25
Radius	0.009	100
Radius and Conic	0.0085	117
All polynomials to order 5, Global Search	0.004	251
Final Hammer optimization	0.003	326

It can be seent that a 13x increase in the on-axis brightness of the LED has resulted.

Note that the optimization could have been performed on all variables right at the start of the process. However, the stepwise process used here is usually faster, and also yields more physical insight. Almost all curved surfaces of optical interest can be considered as deviations from conicoid surfaces.