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Stephan Lehmke wrote:

"optimization algorithms (at least those I'm interested in here) search

for an optimum on a multi-dimensional target (or fitness, or

desirability, or preference) function.

What now if the target function is not known with certainty?

In particular, there might be a probabilistic error involved in the

mapping from "settings" (which can be influenced by the user) and

properties on which the desirability function is based, or the

desirability of certain properties is known only vaguely.

In fact, this leads to a probability distribution or a fuzzy set on the

set of all possible target functions.

How to find an optimum in this case?

One possibility is of course to defuzzify or find the expected value

before optimizing, so that the optimization is carried out on a `crisp'

target function.

But depending on the amount of uncertainty involved, the optimum found

this way may be far off the `real' optimum, if the optimum of the

`expected value' function lies in an area of high variance.

Is it possible to find an `optimum' _directly_ on the probability

distribution on the set of all possible target functions, taking the

known uncertainty into account?

Of course, it has to be specified what `optimum' means in this context,

but at least there should be a provable bound on the probability that

the `optimum' found is in fact very bad.

Sorry for the vagueness in stating this problem, and for the wild

mixture of paradigms (especially for mixing up fuzziness and

probability) and news groups, but I'm trying not to miss any promising

approach by unduely restricting the statement of the problem."

Ultimately, to specify this problem completely, one would need to define

when one evaluation is preferred to another, regardless of intermediate

fuzzy or probabilistic evaluation function results. Collapsing the

distribution (of whatever type) to a single value (whether mean,

centroid, 90th percentile, etc.) and feeding the result to a

conventional optimization process is one way of doing this, but

regardless of the approach taken, being able to clearly indicate

preference of one point over another would seem to be crucial.

Will Dwinnell

predictor@dwinnell.com

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