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Math Help - Finding a minimum for a "vector" function

  1. #1
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    Finding a minimum for a "vector" function

    Suppose I have a "scalar" function H which takes a "vector" ( L_i) for an argument:

    H(L_i) = \sum_{i=1}^D [\alpha (L_{i+1} - L_i)^2 + P \cdot L_i]

    The vector L_i (D dimensional) takes only integer values.

    Also suppose that D, \alpha and P are known.
    To close the system we choose quasi-periodic boundaries L_{D+1} = L_1.

    I wish to find a "vector" L_i such that H takes its minimum value, under an assumption that L_i \geqslant X_i, where the integer values of X_i are known.

    I'm looking for two kind's of solutions:
    1) When I'm simply given a vector X_i and asked to find Min(H), with some arbitrary initial guess for L_i.
    2) When I have found a solution to this problem, and then one of the X_i's is changed by 1.

    Any advice on how to economically solve this problem would be greatly appreciated!
    Last edited by Marril; May 4th 2011 at 08:07 AM.
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  2. #2
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    What does it mean to square a vector? Do you mean the square of its magnitude?
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  3. #3
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    I seem to have left out the thing asked by Prove It:

    This creature L_i is not really a proper vector, and all calculations are done "element by element".
    Following this, (L_{i+1} - L_i)^2 would mean the subtraction of L's ith element from the (i+1)th element, squared.
    This subtraction is just the discrete version of the gradient of L at point i, where now L can be viewed as a discrete function.

    Same goes for L_i \geqslant X_i which must be true element by element.


    Now that I think about it, this problem is a discrete form of finding the minimum of a function H depending on the function f(x) like:
    H[f(x)] = \int_{0}^d [\alpha \nabla f(x)^2 + P \cdot f(x)]dx
    With the condition: f(x) \geqslant g(x)
    Last edited by Marril; May 4th 2011 at 08:20 AM.
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