Linear programming formulation of if-then constraint Consider an LP for which you want to add the restriction that 

Only if $x_1\geq 3$ then $x_2$ and $x_3$ are allowed to be larger than $0$; otherwise $x_2$ and $x_3$ are $0$.

Demonstrate how to formulate this.
 A: The only way I can think to do this and preserve this being a linear programming problem is to split it into two problems. In one, have the constraints $x_1<3$ and $x_2 = x_3 = 0$. In the other, have the constraints $x_1 \geq 3$, $x_2\geq 0$ and $x_3 \geq 0$.
A: Let the column vectors $c^T=(c_1,c_2,c_3, \ldots, c_n)\in\mathbb{R}^n$, $b^T=(b_1,b_2,b_3, \ldots, b_m)\in\mathbb{R}^m$ and a matrix $A=(A_{ij})_{\substack{1\leq i\leq m\\ 1\leq j\leq n}}\in \mathbb{R}^{n\times m}$. Consider a typical linear programming problem
$$
\min_{x\in \mathscr{F}} c^Tx 
\hspace{1cm} or \hspace{1cm}
\begin{array}{rl}
\min & c^Tx\\
\mbox{such that} & x\in \mathscr{F}
\end{array}
\hspace{1cm} or \hspace{1cm}
\begin{array}{rl}
\min & c^Tx\\
\mbox{such that} & Ax\geq b\\
\end{array}
$$
for $\mathscr{F}=\{x\in\mathbb{R}^n:Ax\geq b \}$. For your restriction consider the sets
$$
B=\left\{x\in \mathbb{R}^n\left| \begin{array}{l} x_1\geq 3 \\ x_2> 0\\ x_3> 0 \end{array}\right.\right\}
\mbox{ and }
C=\left\{x\in \mathbb{R}^n\left| \begin{array}{l} x_1< 3 \\ x_2= 0\\ x_3= 0 \end{array}\right.\right\}.
$$ 
So by adding the above restrictions we have
$$
\begin{array}{rl}
\min & c^Tx\\
\mbox{such that} & x\in \mathscr{F} \\
                 & x\in (B\cup C)
\end{array}
\hspace{0.5cm} or \hspace{0.5cm}
\begin{array}{rl}
\min & c^Tx\\
\mbox{such that} & x\in (B\cup C)\cap \mathscr{F} 
\end{array}
$$
But $B\cup C=\{x\in\mathbb{R}: x_1\geq 0, x_2\geq 0\}$. Then your linear programming problem comes to be
$$
\begin{array}{rl}
\min & c^Tx\\
\mbox{such that} & x\in (B\cup C)\cap \mathscr{F} 
\end{array}
$$
Which in block matrix notation comes to be written as
$$
\begin{array}{rl}
\min & c^Tx\\
\mbox{ such that} & 
\left\lgroup 
\begin{array}{c}  
A             \\
\tilde{A}     
\end{array}
\right\rgroup
\left\lgroup
\begin{array}{c}
x_1\\
x_2\\
\vdots\\
x_n
\end{array}
\right\rgroup \geq 
\left\lgroup
\begin{array}{c}  
b\\
0\\
0     
\end{array}
\right\rgroup
\end{array}
$$
for 
$$
\tilde{A}=
\left\lgroup
\begin{matrix}
1&0&0&\ldots &0\\
0&1&0&\ldots &0\\
\end{matrix}
\right\rgroup_{2\times n}
$$
