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Equivalent minimum linear program

  1. Jul 20, 2014 #1

    Maylis

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    Gold Member

    1. The problem statement, all variables and given/known data

    Suppose you are given the following affine minimization problem.

    \begin{align*} \underset{x \in \mathbb{R}^3} \min \hspace{0.1 in}&c^Tx+3c^Tc \\ &\text{subject to} \hspace{0.1 in} Ax \leq b \end{align*}

    Which of the following is an equivalent linear minimization problem?

    \begin{align*} c^T \big( \underset{x \in \mathbb{R}^3} \min \hspace{0.1 in} &x + 3c \big ) \\ &\text{subject to} \hspace{0.1 in} Ax \leq b \end{align*}

    \begin{align*} \underset{x \in \mathbb{R}^3} \min \hspace{0.1 in}&c^T(x +3c)\\ &\text{subject to} \hspace{0.1 in} Ax \leq b \end{align*}

    \begin{align*}3c^Tc+ \underset{x \in \mathbb{R}^3} \min \hspace{0.1 in}&c^Tx \\ &\text{subject to} \hspace{0.1 in} Ax \leq b \end{align*}

    \begin{align*} \underset{x \in \mathbb{R}^3} \min \hspace{0.1 in}&c^Tx \\ &\text{subject to} \hspace{0.1 in} Ax \leq b \end{align*}



    2. Relevant equations



    3. The attempt at a solution
    I guessed and was right, the answer is


    \begin{align*}3c^Tc+ \underset{x \in \mathbb{R}^3} \min \hspace{0.1 in}&c^Tx \\ &\text{subject to} \hspace{0.1 in} Ax \leq b \end{align*}

    However, I don't feel comfortable and it was more an educated guess, instead of firm understanding. I figured that ##3c^Tc## would not affect the minimum since it is just an additive, but how should I approach these sorts of problems? Why are the other choices incorrect?
     
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