This week, at the Rmetrics conference, there has been an interesting discussion about heuristic optimization. The starting point was simple: in complex optimization problems (here we mean with a lot of local maxima, for instance), we do not necessarily need extremely advanced algorithms that do converge extremly fast, if we cannot ensure that they reach the optimum. Converging extremely fast, with a great numerical precision to some point (that is not the point we’re looking for) is useless. And some algorithms might be much slower, but at least, it is much more likely to converge to the optimum. Wherever we start from.
We have experienced that with Mathieu, while we were looking for maximum likelihood of our MINAR process: genetic algorithm have performed extremely well. The idea is extremly simple, and natural. Let us consider as a starting point the following algorithm,
- Start from some
- At step
, draw a point 
in a neighborhood of 
, 
- either
then 
- or
then 
This is simple (if you do not enter into details about what such a neighborhood should be). But using that kind of algorithm, you might get trapped and attracted to some local optima if the neighborhood is not large enough. An alternative to this technique is the following: it might be interesting to change a bit more, and instead of changing when we have a maximum, we change if we have almost a maximum. Namely at step ,
- either
then - or
then
for some 
. To illustrate the idea, consider the following function
values arbitrary, we have obtained the following scenarios> x0=15 > MX=matrix(NA,501,2) > MX[1,]=runif(2,-x0,x0) > k=.5 > for(s in 2:501){ + bruit=rnorm(2) + X=MX[s-1,]+bruit*3 + if(X[1]>x0){X[1]=x0} + if(X[1]<(-x0)){X[1]=-x0} + if(X[2]>x0){X[2]=x0} + if(X[2]<(-x0)){X[2]=-x0} + if(f(X[1],X[2])+k>f(MX[s-1,1], + MX[s-1,2])){MX[s,]=X} + if(f(X[1],X[2])+k<=f(MX[s-1,1], + MX[s-1,2])){MX[s,]=MX[s-1,]} +}
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It does not always converge towards the optimum,
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and sometimes, we just missed it after being extremely unlucky
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Note that if we run 10,000 scenarios (with different random noises and starting point), in 50% scenarios, we reach the maxima. Or at least, we are next to it, on top.
What if we compare with a standard optimization routine, like Nelder-Mead, or quasi gradient ?Since we look for the maxima on a restricted domain, we can use the following function,
> g=function(x) f(x[1],x[2]) > optim(X0, g,method="L-BFGS-B", + lower=-c(x0,x0),upper=c(x0,x0))$par
In that case, if we run the algorithm with 10,000 random starting point, this is where we end, below on the right (while the heuristic technique is on the left),
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In only 15% of the scenarios, we have been able to reach the region where the maximum is.
So here, it looks like an heuristic method works extremelly well, if do not need to reach the maxima with a great precision. Which is usually the case actually.
I am a big fan of trees. It is a very nice way to see how financial pricing works, for derivatives. An with a matrix-based language (R for instance), it is extremely simple to compute almost everything. Even options multiple assets. Let us see how it works. But first, I have to assume that everyone knows about trees, and risk neutral probabilities, and is familiar with standard financial derivatives. Just in case, I can upload some old slides of the first course on asset pricing we gave a few years ago at École Polytechnique.
and can go either up, and then have price 
or go down, and have price 
. And the fundamental theorem of asset pricing says that we do not really care about probabilities of going up, or down. Assuming that we can buy or sell that stock, and that a risk free asset is available on the market, it is possible to price any contingent financial product, like a financial option. Since we know the final value of the option when the stock goes either up, or down, it is possible to replicate the payoff of that option using the stock and the risk free asset. And we can prove that the price of the option is simply
Following previous posts on 
