I need to find a solution (all solutions, or at least upper and lower bounds) in positive integer numbers to the system $Ax \ge f$, where $A$ is an integer matrix.


With SageMath, I solved it with the function



```
Polyhedron.integral_points()

```

But, this is very slow and can take about 2-3 hours for a matrix which is about $30 \times 10$.


Is there another, faster way to do this? Preferably with SageMath or Python.



How to find a solution of a large system of linear diophantine inequalities?
Essentially this is an integer linear programming problem (e.g. for finding bounds on a variable, your objective could be to maximize or minimize that variable). Although integer linear programming is NP-complete, there is well-developed software for this which should be quite fast for a problem the size you mentioned. In SageMath you can use the MILP class.


