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

From: Paul Victor Birke (nonlinear@home.com)
Date: Tue Apr 03 2001 - 21:10:47 MET DST

  • Next message: Silvia Muzzioli: "fuzzy equations and linear systems of equations"

    predictr@bellatlantic.net wrote:
    >
    > 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.
    ********************************************
    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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