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Math Help - check solution - langrange multipliers

  1. #1
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    Question check solution - langrange multipliers

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

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

    I need to use langrange multitpliers to solve this.

    My solution


    I let L(x,lambda) = (x^t)x + lambda*(k - (v^t)x)
    so therefore grad L = 2x - lambda*v = 0

    so i therefore let x = 1/2*lambda*v and put this into the constraint to get

    (v^t)*1/2*v*lambda = k
    so 1/2*lamda*(v^t)*v = k in terms of v

    To verify this is the min, i differentiated grad L again to get 2 >0 so this is convex and hence a global minimizer.

    Prohlem

    Can some one please check my solution, i was getting whether to differentiate with respect to v as well as it is a vector, but wasnt too sure?.. if some one can give me some advise that would be great thanks
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by dopi View Post
    i need find an expression in terms of v for the minimum value of (x^t)x subject to the constraint (v^t)x = k

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

    I need to use langrange multitpliers to solve this.

    My solution

    I let L(x,lambda) = (x^t)x + lambda*(k - (v^t)x)
    so therefore grad L = 2x - lambda*v = 0

    so i therefore let x = 1/2*lambda*v and put this into the constraint to get

    (v^t)*1/2*v*lambda = k
    so 1/2*lamda*(v^t)*v = k in terms of v

    To verify this is the min, i differentiated grad L again to get 2 >0 so this is convex and hence a global minimizer.

    Prohlem

    Can some one please check my solution, i was getting whether to differentiate with respect to v as well as it is a vector, but wasnt too sure?.. if some one can give me some advise that would be great thanks
    This looks OK, you don't have to differentiate wrt v as v is a constant coefficient matrix.

    CB
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