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Math Help - minimum value subject to the constraint

  1. #1
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    Question minimum value subject to the constraint

    Hi i was wondering if anyone can help me with this problem.. i was thinking of using Lagrangian's multipliers method to solve this., but i alsos get confused with transpose part i.e. (^t)

    i need to find an expression in terms of s for the minumum value of (x^t)x subject to the constraint (s^t)x = k

    s is a fixed vector in R^n
    k is a real constant

    and i also need to justify this is a minimum

    if anyone can help..that would be great thanks
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  2. #2
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    Quote Originally Posted by dopi View Post
    Hi i was wondering if anyone can help me with this problem.. i was thinking of using Lagrangian's multipliers method to solve this., but i alsos get confused with transpose part i.e. (^t)

    i need to find an expression in terms of s for the minumum value of (x^t)x subject to the constraint (s^t)x = k

    s is a fixed vector in R^n
    k is a real constant

    and i also need to justify this is a minimum
    You don't need Lagrange multipliers, it's just simple algebra!

    (\mathbf{s}-\mathbf{x})^{\textsc{t}}(\mathbf{s}-\mathbf{x}) = \mathbf{s}^{\textsc{t}}\mathbf{s} + \mathbf{x}^{\textsc{t}}\mathbf{x} - 2\mathbf{s}^{\textsc{t}}\mathbf{x}, and therefore \mathbf{x}^{\textsc{t}}\mathbf{x} = 2\mathbf{s}^{\textsc{t}}\mathbf{x} - \mathbf{s}^{\textsc{t}}\mathbf{s} + (\mathbf{s}-\mathbf{x})^{\textsc{t}}(\mathbf{s}-\mathbf{x}). But (\mathbf{s}-\mathbf{x})^{\textsc{t}}(\mathbf{s}-\mathbf{x})\geqslant0. So the minimum value of \mathbf{x}^{\textsc{t}}\mathbf{x} is 2k - \mathbf{s}^{\textsc{t}}\mathbf{s}, and it occurs when \mathbf{x} = \mathbf{s}.
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  3. #3
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    Question verifying the min

    Hi thanks for the reply..however as i am learning about lagrangian multipliers and kuhn tucker methods, i sujested using langrangian method, so then i could use hessian matrix to justify my minimum, by checking the leading minor principles. Like i said when i expand or differentiate equations with the the transpose sign,,..i get confused where to put the t.

    But with what you have shown i dont know how to justify the minumum..any suggestions??
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