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Math Help - Optimization problem

  1. #1
    Senior Member
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    Optimization problem

    Hello everybody!

    I could really use some help here:

     a \in \mathbb{R}^n, \ e^T = (1,...,1) \in \mathbb{R}^n, \ x \in \mathbb{R}^n
    b \in \mathbb{R}

    Solve the following optimization problem

    \max \ e^T x, where a^T x \le b \mbox{ and } x \ge 0

    I hope, someone knows how to do this.

    Best wishes,
    Rapha
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by Rapha View Post
    Hello everybody!

    I could really use some help here:

     a \in \mathbb{R}^n, \ e^T = (1,...,1) \in \mathbb{R}^n, \ x \in \mathbb{R}^n
    b \in \mathbb{R}

    Solve the following optimization problem

    \max \ e^T x, where a^T x \le b \mbox{ and } x \ge 0

    I hope, someone knows how to do this.

    Best wishes,
    Rapha
    This is a linear program and the maximum occurs at a vertex of the simplex defined by the constraints:

    a^T x \le b \mbox{ and } x \ge 0

    (it may occur at multiple vertices, and so on the line, surface or whatever conecting those vertices, but that is not needed to find the maximum)

    CB
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  3. #3
    Senior Member
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    Hello CaptainBlack,
    thanks for your comment.
     \max \ e^T x = \max \ (1,Ö,1)^T \begin{pmatrix} x_1 ; x_2 ; ... ; x_n \end{pmatrix} = \max \ (x_1 + x_2 + ... +x_n)

    a^T x  = a_1 x_1 + a_2 x_2 + Ö a_n x_n \le b
    I still do not know how to solve it, Iím sorry, but I donít understand your hint. :-(

    More help would be appreciated.
    Rapha
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  4. #4
    Senior Member
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    I am still working on this problem, but did not find a solution yet. Did anyone else?

    Rapha
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