Please recall that projecting a polytope down by

**one** dimension might (asymptotically) square the number of facets. Loosing nine dimension this might result in an enormous number of facets. As a side comment: exploiting the reverse is the idea behind the so-called "extended formulations"; see, e.g.,

https://arxiv.org/abs/1107.0371. So it is perfectly normal that polymake may require a lot of memory.

In your case I recommend the following:

Code: Select all

```
> $p=lp2poly("fcnf.lp");
> $p->VERTICES;
> q = projection($p, [1, 2, 3, 4, 5, 6, 7, 8, 9],nofm=>1);
```

The second command triggers the dual convex hull computation which is pretty simple in this case. The projection with the option "nofm=>1" avoids computing the facets (via iterated projections, i.e., Fourier-Motzkin elimination). Again this becomes a simple computation which will, e.g. allows you to find out (without knowing the facets yet):

Yet, finding the facets of the projection is a bit more tricky. However, this is where polymake shows its real strength. Your observation that the naive projection is a bit hard means that finding the facets via Fourier-Motzkin is not easy (maybe impossible; don't know). Here is how polymake does it in an instant:

Code: Select all

```
> prefer_now "lrs"; print $q->FACETS;
-5 7 0 0 0 0 0 0 0 0
-1 0 0 0 0 6 0 0 0 0
1 0 -1 0 0 0 0 0 0 0
1 0 0 -1 0 0 0 0 0 0
1 0 0 0 -1 0 0 0 0 0
1 0 0 0 0 0 -1 0 0 0
-3 0 0 0 0 0 0 0 4 5
1 0 0 0 0 0 0 -1 0 0
-1 0 0 0 0 0 0 5 4 0
0 0 0 0 0 0 1 0 0 0
0 0 0 0 1 0 0 0 0 0
1 0 0 0 0 0 0 0 0 -1
-3 0 0 0 0 6 5 0 0 5
1 -1 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 1 0 0
0 0 0 1 0 0 0 0 0 0
1 0 0 0 0 0 0 0 -1 0
0 0 0 0 0 0 0 0 1 0
0 0 0 0 0 0 0 0 0 1
-2 0 5 0 0 0 5 0 0 5
0 0 1 0 0 0 0 0 0 0
1 0 0 0 0 -1 0 0 0 0
```

That is, we employ the reverse search method (implemented in lrslib), and this can do the computation easily.

There was nothing wrong, neither in your way of using polymake, nor with the software itself. You just hit upon a problem which requires some more geometric insight. If you want to know more about such things, please, look into our paper

Computing convex hulls and counting integer points with polymake and the associated

web page.