`Optimum' search on an uncertain or fuzzy target function

From: Stephan Lehmke (Stephan.Lehmke@cs.uni-dortmund.de)
Date: Tue Apr 03 2001 - 06:31:31 MET DST

  • Next message: Wolfgang von Hansen: "Software for fuzzy computation"

    Dear colleagues,

    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.

    All pointers and references are gratefully appreciated
    Stephan

    -- 
      Stephan Lehmke     		 Stephan.Lehmke@cs.uni-dortmund.de
      Fachbereich Informatik, LS I	 Tel. +49 231 755 6434 
      Universitaet Dortmund		 FAX 		  6555
      D-44221 Dortmund, Germany             
    

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