# Equivalent minimum linear program

1. Jul 20, 2014

### Maylis

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?