Results 1 to 3 of 3

Math Help - Optimization

  1. #1
    MHF Contributor Mathstud28's Avatar
    Joined
    Mar 2008
    From
    Pennsylvania
    Posts
    3,641

    Optimization

    Hello everyone, while I was at a leadership conference my mind wandered and I found myself pondering the optimization of functions of the form

    f(\theta)=\int_a^{b}f(x,\theta)~dx\quad\text{where }a\ne{b}

    So I basically treated it as a function of one variable, and proceeding as such I arrived at a question.

    f'(\theta)=\frac{d}{d\theta}\int_a^{b}f(x,\theta)~  dx

    Now by Leibniz's rule the above equation is equivalent to

    f'(\theta)=\int_a^{b}\frac{\partial{f(x,\theta)}}{  \partial\theta}~dx

    Now I next looked for critical points.

    If we let F(\theta)=\int_a^{b}\frac{\partial{f(x,\theta)}}{\  partial\theta}~dx

    Then the critical points occur at four points, namely :

    When F(\theta)=0
    Where F(\theta) is undefined
    Less obviously, where \frac{\partial{f(x,\theta)}}{\partial\theta}=0. This obviously arises from the fact that \int_a^{b}0~dx=0
    And also where \frac{\partial{f(x,\theta)}}{d\theta} is undefined.

    Now here is where I ran into an anomaly, I found multiple cases where c\in\mathbb{R}, satisfies two conatradicting critical point criteria, namely.

    F(c) is undefined, but \frac{\partial{f(x,c)}}{\partial\theta}=0.


    So this implies that this is a multi-valued function, so what does one do from here? Does one just sweep this doubly critical point under the carpet and continue? I hope not.

    Any guidance would be appreciated.

    Alex
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Joined
    Aug 2007
    From
    USA
    Posts
    3,110
    Thanks
    2
    Just off the top of my head, Leibniz's Rule does not state equivalence willy-nilly. There are requirements before the differential operator can be moved under the integral. Have you met those requirements? If you do meet them. does this vaporize your anomaly?
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor Mathstud28's Avatar
    Joined
    Mar 2008
    From
    Pennsylvania
    Posts
    3,641
    Quote Originally Posted by TKHunny View Post
    Just off the top of my head, Leibniz's Rule does not state equivalence willy-nilly. There are requirements before the differential operator can be moved under the integral. Have you met those requirements? If you do meet them. does this vaporize your anomaly?
    Thank you for your response, yes I looked at the requirements, but it does not get rid of all of them. Once again, thank you.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. optimization help!
    Posted in the Calculus Forum
    Replies: 1
    Last Post: December 12th 2009, 12:54 AM
  2. Optimization
    Posted in the Calculus Forum
    Replies: 1
    Last Post: December 8th 2009, 02:09 PM
  3. Optimization
    Posted in the Calculus Forum
    Replies: 2
    Last Post: November 29th 2009, 10:56 AM
  4. optimization
    Posted in the Calculus Forum
    Replies: 1
    Last Post: November 12th 2008, 10:47 AM
  5. Optimization
    Posted in the Pre-Calculus Forum
    Replies: 0
    Last Post: October 13th 2008, 06:44 PM

Search Tags


/mathhelpforum @mathhelpforum