I need to solve the following linear program:
$$\displaystyle{\min_{\bar{X},t}} \hspace{0.1in}t$$
such that: $$A\bar{X}=\tilde{x} + td$$
where $A$ is $N\times N$ and known, $t$ is scalar, $\tilde{x}$ is a known $N\times 1$ vector, $d$ is a known $N\times 1$ vector and $\bar{X}$ lies in the set $\mathcal{X}$ where: $$\mathcal{X}=\{X | \underline{X} <X < \overline{X}, X \in \mathbb{R}^N\}$$ and $\overline{X}$ and $\overline{X}$ are known $N\times 1$ vectors.
To solve this I am essentially taking the $td$ to the left hand side, adding $-d$ as a column to the given A and adding $t$ as an additional unknown on the LHS.
The problem is that when implementing this in Gurobi, I get the error that the problem is infeasible or unbounded. Removing gurobi's DualReduction approximation gives us that the algorithm is infeasible.
I was thinking about writing the dual problem and figuring out if the issue becomes more apparent there, but I am unable to do so. Can anyone help me figure out how to write the dual problem?
Example:
https://www.dropbox.com/sh/weljct2fepvz3o9/AAAwgBCVoVz_C9z72ETtU_h1a?dl=0
X_L file corresponds to $\underline{X}$ and X_U file corresponds to $\overline{X}$. Q file corresponds to $A$ in the notation above. $x$ corresponds to $\tilde{x}$.
When I use lpSolve in R for this problem, I get weird solutions that are not within the bounds (I use -Inf and Inf as bounds for $t$).
When I use gurobi, it gives me a unbounded or infeasbile error. If I set DualReductions=0 (link), it says that the model is infeasible. However, if I solve this system with the objective set to 0, I get a solution. Again the bounds on $t$ are set to -Inf and Inf.
Therefore I am very confused as to what is happening with this problem. Is this problem infeasible? If so, then how come setting objective equal to 0 produces solutions? If not, then why won't gurobi or lpSolve work?