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Minimize function subject to constraint

  1. Feb 5, 2014 #1
    Suppose given a function of the form:

    f(x,y,z) = ax + by + cz
    with the constrain x+y=k

    My book minimizes this function by a way I am not completely familiar with:

    dF = adx + bdy + cdz 0

    and since dy=-dx we can write:

    dF = (a-b)dx + cdz = 0
    =>
    a-b = c dz/dx (1)

    How I would minimize is simply plug y=k-x into the definition of f:

    f(x,z) = (a-b)x + bk + cz

    And take partial derivatives

    df/dx = a-b + cdz/dx

    df/dz = (a-b) dx/dz + c

    And seting both equal to zero yields a system of equations which does not reduce to (1).
    What is wrong?
     
  2. jcsd
  3. Feb 5, 2014 #2

    lurflurf

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    (1) should be
    a-b+c dz/dx=0
    all the equations are equivalent
     
  4. Feb 5, 2014 #3
    What about the dx/dz term i get?
     
  5. Feb 5, 2014 #4

    Ray Vickson

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    I hope you have left out some important information, because if we take your problem exactly as you have written it, it has no solution; that is, if a, b, c are all non-zero, you can find a sequence of ##x_n, y_n, z_n## giving ##x_n + y_n = k## for all ##n##, but ##ax_n + by_n+ cz_n \to -\infty ## as ##n \to \infty##. In other words, there is no finite minimum.

    Are you sure you have stated the problem completely and exactly?
     
    Last edited: Feb 5, 2014
  6. Feb 5, 2014 #5

    CompuChip

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    Which has level sets at z = Ax + B for some suitable constants A and B. I.e. the function is linear and does not have a minimum.

    I would generally tackle these problems using Lagrange multipliers by the way, if you haven't seen them check them out, they are cool ;)
     
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