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Generalized optimization under uncertainty problem

  1. Jan 23, 2012 #1
    Hi,

    I have formulated what I believe to be a generalized(to some degree) optimization under uncertainty problem. The write up is included in the attached file. I would appreciate any and all input, help or guidance as to how this problem could be solved. If you have any questions please feel free to ask.

    My understanding is that this problem doesn't easily fall into any one branch of mathematics, so there are no limitations of methods/ideas to be used in its solution. I have pondered this problem myself for a while but I will not post my ideas just yet because I do NOT want to create a predisposition to a particular solution, i.e. I want to hear fresh unbiased (by me) ideas.

    Thanks in advance for your time.
     

    Attached Files:

  2. jcsd
  3. Jan 24, 2012 #2

    chiro

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    Hey scrypter and welcome to the forums.

    This looks like a standard optimization problem.

    I'm not even acquainted with a fraction of the theory of optimization (yet), but there is a lot out there in terms of the theory and application of optimization (both minimization and maximization) for both linear and non-linear optimization problems with variable constraint classes.

    In terms of your problem you are trying to maximize a multi-linear function under some set of constraints. The actual classification of the constraints is important because if they are linear or fit in a certain class, then you will be able to use a general algorithm that will be able to be solved rather quickly. If the constraints however are more complicated, then other methods will be required.

    In terms of theory for existing of global solutions for optimization problems, as far as I am aware you will need to know about convexity conditions. I haven't gone into enough depth yet for this to give you adequate advice.

    What kind of background do you have? Have you taken any optimization courses or done any work/personal related experience with optimization techniques?
     
  4. Jan 24, 2012 #3
    Hi chiro,

    Thank you for responding. If this is indeed a standard optimization problem (which would be great), I would really appreciate if you could point me to a few good books on subject matter.

    I am not particularly well versed in optimization techniques, I have certainly not taken a course dedicated solely to optimization, although I am familiar with basic techniques. In my understanding the solution to the problem really depends on the output of function M, which even though known and constrained (from empirical evidence) is at least pseudo random. Does that not complicate things?

    Thanks!
     
  5. Jan 25, 2012 #4

    chiro

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    Hey scrypter.

    I'm not really an expert on Optimization Theory in any means and like you know I know just the basics myself and wish to explore specifics at a later date.

    I have notes from an Australian University which seem to be a good introductory text that outline not only the ideas themselves, but the reasons why they are considered and built upon. An example of this is the study of convexity as it relates to global minimimums and maximums in terms of uniqueness (i.e. these are somewhat gauranteed if convexity holds).

    The other thing is the constraints.

    With the constraints, just like with mathematics different constraints may have different frameworks and different algorithms to solve the particular problem. The analogy that I would make is to consider matrix inversion: we have quite a few techniques to do this, but if you know the classification of a matrix beforehand, then the inversion might be really really trivial or some specialized algorithm might exist to do it faster than a general purpose algorithm which usually always turns out to be the case in mathematics.

    Also if you are introducing any kind of randomness through a random variable, things just get way more complicated. When you are doing calculus with random variables you have to do so with respect to a particular measure which is based on the distribution and is described through more advanced integration methods. Once you see computer code for these kinds of things it makes a tonne more sense, but its still a pain in the arse.

    I'm sorry I can't give you any more specific advice. I think I might be doing you a disservice recommending something I haven't had personal experience with, but hopefully there should be someone like an engineer, scientist or applied mathematician that can give you specific advice.
     
  6. Jan 25, 2012 #5

    chiro

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    Also I just remembered you said "pseudo-random" instead of random. It might help to define what you meant by this.
     
  7. Jan 27, 2012 #6
    Hi,

    By pseudo-random I mean that the values of z-components of I are produced systematically from a distribution rather than from a truly random source. I figure this fact would come in handy since one could generate z's to provide the randomness component. Does that make sense?

    Regards
     
  8. Jan 27, 2012 #7

    chiro

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    Do you mean from a computer program? As in a random number generator?

    If they are from a pseudo-random generator chances are they will still have the properties of randomness in a probabilitistic sense which means you need to treat them like ordinary random variables.

    If you can specify the function or relationship between variables completely in an analytic fashion then the above doesn't apply, but otherwise you will need to use stochastic analysis.

    If however you have something like say x^2 + y^2 + z = w and you have to specify three variables to get the fourth then that is analytic. If however for some reason you provided three variables and still couldn't predict the fourth or if for some reason you can't reduce your system or all of its variables down to a deterministic fashion then you have to use stochastic analysis.
     
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