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Thread: Could not solve optimization problem

  1. #1
    Junior Member
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    Could not solve optimization problem

    I am completely new to optimization. I only know one method of optimization, i.e. optimization using the differentiation (without any constraint).


    I am trying to find the optimal value of $x$. Note the following. $a \in \{1,1/4, 1/2, 3/4\}$ and $n$ is an integer such that $ 50 \leq n \leq 500$. $b$ is also an integer and $5\leq b \leq 44$


    \begin{align}
    f(x) = a\left(1-\dfrac{a}{x}\right)\left(1-\dfrac{a}{bx}\right)^{n-2}
    \end{align}


    \begin{align}
    f'(x) = \dfrac{a^2\left(1-\frac{a}{bx}\right)^{n-2}}{x^2}+\dfrac{a^2\left(n-2\right)\left(1-\frac{a}{x}\right)\left(1-\frac{a}{bx}\right)^{n-3}}{bx^2}
    \end{align}


    letting $ f'(x) = 0$, we get


    \begin{equation}
    x = \dfrac{a(n-1)}{n+b-2} \ \ \ \ \ \ \ \ \ \ \ \ \ \ (1)
    \end{equation}


    Now $x $ is itself an integer and should be $5\leq x \leq 64$. But the roots of $x$ as shown in (1) does not give me the correct result.

    How can I proceed?


    PS. Example. If $a = 1/2, n = 100, b = 10$, then $x < 1$ and
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  2. #2
    Super Member
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    Re: Could not solve optimization problem

    The minimum is at x = \frac{a(n-1)}{b+n-2} but x is a rational number 0<x<a
    Thanks from sjaffry
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