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predictr@bellatlantic.net wrote:

*>
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*> Stephan Lehmke wrote:
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*> "optimization algorithms (at least those I'm interested in here) search
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*> for an optimum on a multi-dimensional target (or fitness, or
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*> desirability, or preference) function.
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*>
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*> What now if the target function is not known with certainty?
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*>
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*> In particular, there might be a probabilistic error involved in the
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*> mapping from "settings" (which can be influenced by the user) and
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*> properties on which the desirability function is based, or the
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*> desirability of certain properties is known only vaguely.
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*>
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*> In fact, this leads to a probability distribution or a fuzzy set on the
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*> set of all possible target functions.
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*>
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*> How to find an optimum in this case?
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*>
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*> One possibility is of course to defuzzify or find the expected value
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*> before optimizing, so that the optimization is carried out on a `crisp'
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*> target function.
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*>
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*> But depending on the amount of uncertainty involved, the optimum found
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*> this way may be far off the `real' optimum, if the optimum of the
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*> `expected value' function lies in an area of high variance.
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*>
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*> Is it possible to find an `optimum' _directly_ on the probability
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*> distribution on the set of all possible target functions, taking the
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*> known uncertainty into account?
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*>
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*> Of course, it has to be specified what `optimum' means in this context,
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*> but at least there should be a provable bound on the probability that
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*> the `optimum' found is in fact very bad.
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********************************************

Dear Will

I have found this out from two sources:==>

[1] Discovered during researching fuzziness and pricing using Ultra

Fuzzy Function representation for the marketplace. What I call the

Ultra Lower and Ultra Upper Bounds. (As you hint, you must work your

butt off to find these!!) *****Most important.*****

[2] A general estimating idea from Prof. Michel Poloujadoff of Paris,

FRANCE is to now use this information to establish a likely candidate

(near optimal??) to test and that is simply the Geometric Mean; i.e.

optimal ~= SQRT( ABS(Ultra Upper Bound * ABS(Ultra Lower Bound)) for

any variable on the planet for which you have some >>reasonable<< idea

for the strict bounds.

BTW:==> Sometimes this works and sometimes it doesn't!! But it strictly

depends on your estimation of these bounds. A small story. I have used

the formula to astonish people. But trying on my wife once was

interesting when she got a bonus. I said, I will predict what the

number is within a few 10s of $. I guessed the lower bound as something

near trivial, something the company would have to expend more money to

make the check than warranted. This was ~ 50$ I said. For the upper,

I used what I thought was a reasonable limit given past efforts here by

the Company and what other of her friends may have gotten and took the

maximum to be about 2000 $.

She actually received 5600$ so the method failed because the Ultra

Upper Limit failed miserably. But if you get both either rather well or

slightly off, the SQRT function seems to work well in many

circumstances and may be worthy of a look here. Thanks again to the

Prof. who is one of the best Electrical Engineers in the world and a

fine gentleman and fantastic ambassador for France. I spent about 1

year with Michel on a special project when at Westinghouse in the

1980s. Merci Michel!!

all the best

Paul

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