Wednesday, September 23, 2009

How to Not Reach a Global Optimum

It certainly is not anything new, but you don't reach a global optimum by optimizing individual variables [1][2]. That's what consistently irks me about single-interest activists of all creeds. They are simply interested in optimizing one variable and assuming that the global situation will therefore follow.

The new Plastics Scoreboard by Clean Production Action is one such example. You can guess what the targets are (chlorine, bromine, petroleum based monomers) and what the solutions are - agricultural based sources. And all the usual arguments against going to bio-sources polymers can be made here (destroying food, the non-sustainability of these practices...) but in keeping with the introduction above, this effort is taking a multivariable optimization and worrying about just one part of it.

Weighting this one variable above all others [3] is where the arguing should begin. I'm certainly not against considering such a reduction, but the case hasn't even been made that the one variable here is of such importance. But as is usually the case (I can't think of even one counter example), the single interest groups start out with the weighting in place and go from there. Without even attempting to convince anyone of the initial assumption, they argue for the outcome.

[1] As a simple example, consider trying to find the latitude (the x variable) and longitude (the y variable) for the highest point in the continental US. The correct answer is 36 N, 118 W - Mt. Whitney in California. Starting at St. Paul Minnesota, 45 N,93 W, I can try and find the maximum elevation along the 93 W line, which would be somewhere in northern Minnesota, call it 47 N. Then keeping that constant, I would look for the highest elevation along the 47 N line, which would be somewhere in the western Montana about 112 W, quite a ways from California. If I would try and maximize the variables in the other order, I would get a different result. The maximum altitude along the 45 N line is somewhere along the Montana/Wyoming border, about 110W. The maximum elevation along that longitude line is pretty much right at 45 N or a little south. So in the first case, I found 47 N, 112 W, in the second case, I found 45N, 110 W. Two different results and neither one is correct. And in addition, different starting points will yield different results as well.

[2] Yes, it could happen that you do reach the global optimum that way, but the odds of that occurring are very slim, and decrease as the number of variables increase. Obviously for a one-variable problem, one or more of the local optimums will be the global optimum. For a two variable situation, you can already be in trouble, as you can see above.

[3] If certain variables are weighed considerably higher than others the complexity can be reduced. We've all met people - make that children (regardless of age) who weight their happiness above all others. They have managed to take a multivariable problem and reduce it to a single variable.

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