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Maximize nonlinear function

  1. Oct 21, 2016 #1
    Hi to everyone,

    I'm optimizing a nonlinear function but I'm struggling to achieve it. The function is the following:
    eq.PNG

    X and i are relationed so i doesn't go to infinite. Do you have any idea how to maximize this function?

    Thanks in advance,

    Eric
     
  2. jcsd
  3. Oct 21, 2016 #2

    mfb

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    What do you mean by "related"? i is a summation index, not a free variable. The starting value is missing.

    The sum has an explicit formula, this should be easy to simplify.
     
  4. Oct 21, 2016 #3

    Mark44

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    What does this mean? Your summation appears to be from i = <something> to ##\infty##.
    The summation in the formula for F(x) is unclear. Is this what you have in mind for the summation?
    $$\sum_{i = 0}^{\infty}e^{-i(x + 2)}$$
     
  5. Oct 21, 2016 #4
    Yes, it is from i to ∞, but if I want to optimize it I guess that I will have to set a bound.
    And yes, the summation corresponds to what you have posted. Let's say that i is related to x as follows: i=T/x, T is a time. Therefore, x determines i.
     
    Last edited by a moderator: Oct 21, 2016
  6. Oct 21, 2016 #5

    mathman

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    How can the summation index (i) start at i?
     
  7. Oct 21, 2016 #6

    Mark44

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    This makes no sense -- "from i to ∞". i is merely the index. You have to give a starting value, such as 0 or 1 or whatever, and an ending value, such as 10 or N or ∞.
    This also makes no sense. x is presumably a real number, and a summation index is usually an integer.

    The summation that I wrote can be expanded like so:
    ##1 + e^{-(x + 2)} + e^{-2(x + 2)} + e^{-3(x + 2)} + \dots##

    It's not at all clear to me or the other people replying in this thread what you're trying to do.
     
  8. Oct 21, 2016 #7
    Apologies, I wanted to say that it goes from i=1 to inifinite. I will reformulate my question so you can understand it better.
    I want to maximize this function:

    Sin título.jpg
    And x is related to i as follows; i=T/x, where T is a constant. Due to the nature of the problem, i is an integer.

    Thanks for your patience.

    Eric
     
  9. Oct 22, 2016 #8

    Mark44

    Staff: Mentor

    This still doesn't make sense. i is an index of the summation, and x occurs outside the summation (as well as being part of the things being summed).
    i takes on an infinite number of values: 1, 2, 3, ... in the summation, but the x that multiplies the summation can't change with the change in index values.

    Based on what you said, you can write the summation as ##\sum_{i = 1}^{\infty}e^{-i(T/i + 2)}##, but you can't replace either x outside the summation by T/i.

    It seems that you're trying to come up with a formula for a function that involves a summation, without understanding how a summation works.
     
  10. Oct 22, 2016 #9
    If we forget about the relationship between i and T, how would you optimize it?
     
  11. Oct 22, 2016 #10

    Ssnow

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    Your question is related to find the max of this function "in general" or with constraints ?
    I don't understand well how this function is defined, ##i## is the integer index in the sum but what about ##x##?, is real or integer? Has this function a domain?
    If it is a real function try with the derivative...

    Ssnow
     
  12. Oct 22, 2016 #11

    mfb

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    If that is your function, it is identical to (just a different notation)
    $$F(x)=\frac{1}{x+x \left( e^{-1(x+2)} + e^{-2(x+2)} + e^{-3(x+2)} + ... \right)}$$
    You see how i disappears just by rewriting it? i cannot depend on anything.

    You can use$$\sum_{i=1}^\infty e^{-i(x+2)} = \sum_{i=1}^\infty \left(e^{-(x+2)}\right)^i$$
    The sum on the right (a sum over qi for some q) can be evaluated with a well-known formula.

    Afterwards, you can use that a maximum of your function (which does not occur at 0) is a minimum of the inverse, 1/F(x). From there it should not be hard to look for minima.
     
  13. Oct 22, 2016 #12
    Hi Ssnow,

    x is a real, i'm working on a domain and i do not have any constrains more.
     
  14. Oct 22, 2016 #13

    Ssnow

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    Hi, ok thanks for the clarification, I suggest you to follow the suggestions of @mfb.
     
  15. Oct 22, 2016 #14
    Thanks for your advice, so far the best I've received.

    Can I use this formula?
    Sin título.png

    By the way, how do you introduce formulas in this forum?
     
  16. Oct 22, 2016 #15

    mfb

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    You can use this formula, to take the limit n->infinity you have to check if z is in the correct range for that.

    You can use LaTeX for formulas. The quote in your post has two examples.
     
  17. Oct 22, 2016 #16
    In this formula, the summaton begins at i=0, while in my case it's i=1. How can I solve that?

    Thanks
     
  18. Oct 22, 2016 #17

    mfb

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    You can replace i by i+1 everywhere (!) in your expression, then simplify.

    There is a German Wikipedia article about - no English version, but the formulas are international.
     
  19. Oct 22, 2016 #18
    In fact, my function is this one:

    Sin título.png

    I posted a simplified one before, can I still express this one with a well-known formula?

    Thanks
     
  20. Oct 23, 2016 #19

    mfb

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    No.

    Where was the point of the other function if that is not what you actually want to solve?
     
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