# Linear programming with non-zero constraints?

I'm trying to find references for a linear programming problem where the variables $x_i$ we are searching for have to be above certain thresholds $d_i$.

So that the expression of the problem is:

maximize: $\textbf{c}^{T}\textbf{x}$

subject to: $\textbf{A}\textbf{x} \leq \textbf{b}$

and $\textbf{x}\geq \textbf{d}$ with $d_i > 0$

As opposed to the canonical form:

maximize: $\textbf{c}^{T}\textbf{x}$

subject to: $\textbf{A}\textbf{x} \leq \textbf{b}$

and $\textbf{x}\geq 0$

Does changing the last constraint from $\textbf{x}\geq 0$ to $\textbf{x}\geq \textbf{d}$ change anything in the problem? I assume it does, but I can't find any reference to how to approach it?

• This constraint can also be modelled in the $A$ and $b$ parameters, since you can make a row get something like $-x_i \leq -d_i$, which is the same as $x_i \geq d_i$. Mar 28, 2018 at 16:37

Let $u=x-d$ or equivalently $x=u+d$. Then your problem can be written as
$\max_{u} c^{T}u+c^{T}d$
$Au \leq b+Ad$
$u\geq 0$.