Consider the following original LP: $\mathit{min}$ c'$x$ s.t: $Ax=0 \wedge 0\le x\le 1$ . This is my original LP which has to be solved. Now, using some reductions, I reduced the original LP to the following converted LP: $\mathit{min}$ d'$y$ s.t: $By=0 \wedge 0\le y\le 1$. Using some relations between $x$ and $y$, I proved that these two LP are equivalent. This means the optimal solution for one of them gives the optimal solution for the other and visa verse. For optimal solutions the objective values are the same as well. Now, the converted LP cannot be solved exactly. Now, I have got an approximate optimal solution vector for the converted LP, in which values for each pairs of variables are within some $\epsilon$ of each other. This solution is not feasible for the original LP but still is optimal based on the variable transformation rules that I have used to show the equivalence of these two LP's. My problem is how to push this solution to be feasible by not ruining the objective value too much. How can I find the best lower bound?
LP infeasibility
linear programming