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