Re: `Optimum' search on an uncertain or fuzzy target function

From: predictr@bellatlantic.net
Date: Mon Apr 02 2001 - 12:50:28 MET DST

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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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