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Math Help - Constrained Optimization

  1. #1
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    Constrained Optimization



    (a) The reference they were referring is : "Find a lower triangular matrix L, and a diagonal matrix D such that A = LDL^T". I don't see how this is relevant though.

    Here's what I've done so far

    L= x+y+z- \lambda(x^2+y^2+z^2-k)

    L_x = 1-2 \lambda x = 0
    L_y = 1-2 \lambda y = 0
    L_z = 1-2 \lambda z = 0

    From this, critical points are x=y=z = \frac{1}{2 \lambda}

    Since f_{xx}=f_{yy}=f_{zz}=-2\lambda
    The Hessian matrix will be

    H = \begin{bmatrix} -2 \lambda & 0 & 0 \\ 0& -2 \lambda & 0 \\ 0&0& -2 \lambda \end{bmatrix}

    So what else do I need to do here to solve this? (I can't find any similar examples from my textbook)

    Also, can anyone show me how to prove part (b). I don't understand it...
    Last edited by demode; May 19th 2010 at 02:16 AM.
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