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Little help with min/max problem w/ n-variables

  1. Mar 30, 2004 #1
    I'm given an n-dimensional function such that

    f(x1,x2,...,xn) = a1*x1 + a2*x2 +...+ an*xn

    where a1, a2, ...,an are all positive numbers. This is with the restraint that

    x1^2 + x2^2 + ... + xn^2 = 1

    Using Lagrange multipliers (I'll use 'L' for Lambda):

    a1 = L(2*x1)
    a2 = L(2*x2)
    ...
    an = L(2*xn)

    Solving for L,

    L = a1/(2*x1) = a2/(2*x2) = ... = an(2*xn) [1]

    So, generally, xn = an/(2*L). Plugging into the constraint function, I get:

    (a1/2*L)^2 + (a2/2*L)^2 + ... + (an/2*L)^2 = 1
    (1/4*l^2)*(a1^2 + a2^2 + ... + an^2) = 1
    L = sqrt(a1^2 + a2^2 + ... + an^2)/2

    So, for each x, generally,

    xn = an/sqrt(a1^2 + a2^2 + ... an^2) [2]

    Which implies that a1 = a2 = ... = an from [1]

    a1/(2*L) = a2/(2*L) = ... = an/(2*L)

    So [2] is basically my critical point.

    Plugging back into f(...) I get

    f(x1,x2,...,xn) = a1*(a1/sqrt(a1^2 + a2^2 + ... + an^2)) ...etc

    <=> ... (a1^2 + a2^2 + ... + an^2)/sqrt(a1^2 + a2^2 + ... + an^2)

    Furthermore....

    <=> ... sqrt(a1^2 + a2^2 + ... + an^2)

    Since a1 = a2 = ... = an, then we can say (using an arbitrary (a))

    <=> .... sqrt(n*a^2) <=> a*(sqrt(n)).

    So, to my question (heh), for the minimum value it's obvious that it would be when n = 0. But I'm a little confused as to what the maximum would be.
    Say a constant k = a, then would it be k*(sqrt(n))?? Just a simple question I guess, as it comes down to it. Thanks.

    Actually, wouldn't the min be when n = 1, since there mustb e at least one variable. So the min would be k.
     
    Last edited by a moderator: Mar 30, 2004
  2. jcsd
  3. Mar 31, 2004 #2

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    I'm confused as to what your question really is!

    You start out by saying "I'm given an n-dimensional function" but then you start saying that " for the minimum value it's obvious that it would be when n = 0".

    No, n is a given constant. The problem is to find x1, x2,...., xn so as to minimize the function.

    Also you say
    "So, for each x, generally,

    xn = an/sqrt(a1^2 + a2^2 + ... an^2) [2]

    Which implies that a1 = a2 = ... = an from [1]"

    No, a1, a2, etc. are GIVEN. You can't place any constraints on them.
    [1] implies a1= a2=...= an ONLY if you assume x1= x2=...= xn which is NOT in general true.
     
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