### Finding integer vertices in an MILP

Posted:

**03 Nov 2021, 12:32**Dear support team,

I am trying to implement a decomposition algorithm and I have to work with extreme points of the MILP. As I asked some questions about defining polyhedron, I have tried to use the same approach to find integer vertices, but I am failing to do that. Also, I am working on lattice and lattice points too. What I am looking for is, in an MILP problem (actually, the defined polyhedron) the integer vertices are $\{(0,0),(4,0),(2,1),(0,1)\}$ respectively. when I am solving the problem the Polymake shows vertices of the linear relaxation of the MILP and it is different from what I am looking for. The syntax I have used is:

***

* without upper bound

$p = new Polytope(INEQUALITIES=>[[18,-4,-9],[4,2,-4],[0,1,0],[0,0,1]]);

***

***

* with upper bound

$p = new Polytope(INEQUALITIES=>[[18,-4,-9],[4,2,-4],[0,1,0],[0,0,1],[-9,-1,0],[-9,0,-1]]);

***

In the first attempt, I got the linear relaxation vertices while, in the second, it seems, the Polymake waits for another argument.

I was wondering if, how can I define/find the integer vertices of the MILP polyhedron?

Regards

Abbas

I am trying to implement a decomposition algorithm and I have to work with extreme points of the MILP. As I asked some questions about defining polyhedron, I have tried to use the same approach to find integer vertices, but I am failing to do that. Also, I am working on lattice and lattice points too. What I am looking for is, in an MILP problem (actually, the defined polyhedron) the integer vertices are $\{(0,0),(4,0),(2,1),(0,1)\}$ respectively. when I am solving the problem the Polymake shows vertices of the linear relaxation of the MILP and it is different from what I am looking for. The syntax I have used is:

***

* without upper bound

$p = new Polytope(INEQUALITIES=>[[18,-4,-9],[4,2,-4],[0,1,0],[0,0,1]]);

***

***

* with upper bound

$p = new Polytope(INEQUALITIES=>[[18,-4,-9],[4,2,-4],[0,1,0],[0,0,1],[-9,-1,0],[-9,0,-1]]);

***

In the first attempt, I got the linear relaxation vertices while, in the second, it seems, the Polymake waits for another argument.

I was wondering if, how can I define/find the integer vertices of the MILP polyhedron?

Regards

Abbas