Thursday, November 17, 2011

Good & Popular Algorithms are Simple

I recently tried to minimise a function according to some constraints. One popular method to minimise a function in several dimensions is Nelder-Mead Simplex. It is quite simple, so simple that I programmed it in Java in 1h30, including a small design and a test. It helped that the original paper from Nelder-Mead is very clear:
However the main issue is that it works only for unconstrained problems. Nelder and Mead suggested to add a penalty, but in practice this does not work so well. For constrained problems, there is an adaptation of the original idea by Box (incredible name for a constrained method) that he called the Complex method, a deliberate pun to the Nelder-Mead Simplex method. The basic idea is to reset the trial point near the fixed boundary if it goes outside. Now this took me much longer to program, as the paper is not as clear, even if the method is still relatively simple. But worst, after a day of deciphering the paper and programming the complex method, I find out that it does not works so well: it does not manage to minimise a simple multidimensional quadratic with a simple bound constraint: f(X)=sum(X)^2 with X >= 0 where X is an N-dimensional vector. In the end, I don't want to work with such a simple function, but it is a good simple test to see if the method is really working or not. Here is how the function looks with N=2:
And if we restrict to x,y >= 0 it becomes:

I suspected an error in my program, so I decided to try with scilab, that has also the Box method as part of their neldermead_search functionality. Scilab also failed to compute the minimum in 8 dimensions of my simple quadratic.
I tried various settings, without ever obtaining a decent result (I expect to find a value near 0).

There is another algorithm that can find a global minimum, also very popular: the Differential Evolution. At first, being a genetic algorithm, you would think it would be complicated to write. But no, the main loop is around 20 lines.

Those 20 lines are a bit more difficult than Nelder-Mead, but still, when I saw that, I understood that "this is a classic algorithm". And it does work with constraints easily. How to do this is explained well in K. Price book "Differential Evolution", and takes only a few lines of code. Here is the result I got in dimension 29:
dim=29 min=1.2601854176573729E-12 genMax=412 x=[3.340096901536317E-8, 8.889252404343621E-8, 7.163904251807348E-10, 9.71847877381699E-9, 2.7423324674150668E-8, 2.4022269439114537E-9, 1.7336434478718816E-11, 7.244238163901778E-9, 1.0013136274729337E-8, 7.412154679865083E-9, 5.4694460144807974E-9, 2.3682413086417524E-9, 4.241739250073559E-7, 4.821920889534676E-10, 2.115396281722523E-9, 8.750883007882899E-8, 2.512011485133975E-9, 4.811507109129279E-9, 1.0752997894113096E-7, 5.120475258343548E-9, 8.404448964497456E-9, 4.1062290228305595E-9, 1.7030766521603753E-8, 5.589430643552073E-9, 8.237098544820173E-10, 3.5796523161196554E-9, 5.186299547466997E-9, 2.430326342762937E-7, 5.493850433494286E-9]

It works well, although there is quite a bit of parameters. I noticed that the strategy was especially important.

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