Attached is a mixed-integer program in .lp format "problemIP.lp" The optimal solution is 178 which I compute using CPLEX software.

I enumerate all feasible points of this mixed integer program using my own method outside of polymake and it lists 3104 points. Then, using polymake, I compute the facets of these set of 3104 points. There are 31 facets reported for this problem by polymake as:

Code: Select all

```
Facets:
0: 210 C1 + 30 C2 + 20 C4 - 20 C5 + 20 C7 + 120 x0_1_2 - 40 x0_1_3 - 40 x0_2_3 - 40 x0_6_3 - 80 x0_4_5 - 5 V0_7_8 - 8 V0_7_9 >= 950
1: -21 C1 - 3 C2 + C3 - 12 x0_1_2 + 6 x0_1_3 + 6 x0_2_3 + 2 x0_6_3 >= -75
2: 3 C1 + C2 - C3 - 6 x0_1_3 - 6 x0_2_3 - 2 x0_6_3 >= 1
3: 7 C1 + C2 + 4 x0_1_2 >= 29
4: -16 x0_5_7 + V0_7_8 >= 0
5: -6 C1 - 2 C2 - 4 C3 - 4 C4 - 12 C5 - 8 C6 - 4 C7 + 32 x0_5_7 - 3 V0_7_8 >= -426
6: 15 C1 + 5 C2 + 10 C3 + 10 C4 + 10 C5 + 20 C6 + 10 C7 - 2 V0_7_9 >= 745
7: -210 C1 - 30 C2 - 20 C7 - 120 x0_1_2 + 40 x0_1_3 + 40 x0_2_3 + 40 x0_6_3 - 40 x0_4_5 + 5 V0_7_8 + 8 V0_7_9 >= -1110
8: 210 C1 + 30 C2 - 20 C4 + 20 C5 + 20 C7 + 120 x0_1_2 - 40 x0_1_3 - 40 x0_2_3 - 40 x0_6_3 + 80 x0_4_5 - 5 V0_7_8 - 8 V0_7_9 >= 1190
9: 700 C1 + 100 C2 + 40 C7 + 400 x0_1_2 - 160 x0_1_3 - 160 x0_2_3 - 160 x0_6_3 - 80 x0_3_4 - 5 V0_7_8 - 12 V0_7_9 >= 3220
10: 40 C5 - 160 x0_5_7 + 15 V0_7_8 + 12 V0_7_9 >= 640
11: 7 C1 + C2 - C3 + 4 x0_1_2 - 6 x0_1_3 - 2 x0_2_3 - 2 x0_6_3 >= 17
12: -700 C1 - 100 C2 - 40 C3 + 40 C4 - 40 C7 - 400 x0_1_2 + 160 x0_1_3 + 160 x0_2_3 + 160 x0_6_3 + 160 x0_3_4 + 5 V0_7_8 + 12 V0_7_9 >= -3060
13: x0_1_3 >= 0
14: x0_5_7 >= 0
15: 168 C1 + 24 C2 + 8 C5 + 16 C7 + 96 x0_1_2 - 32 x0_1_3 - 32 x0_2_3 - 32 x0_6_3 + 32 x0_4_5 - V0_7_8 - 4 V0_7_9 >= 1016
16: -140 C1 - 20 C2 + 40 C4 + 40 C7 - 80 x0_1_2 + 80 x0_1_3 + 80 x0_2_3 + 80 x0_6_3 + 160 x0_3_4 - 5 V0_7_8 - 12 V0_7_9 >= 540
17: -700 C1 - 100 C2 + 40 C3 - 40 C4 - 40 C7 - 400 x0_1_2 + 160 x0_1_3 + 160 x0_2_3 + 160 x0_6_3 - 160 x0_3_4 + 5 V0_7_8 + 12 V0_7_9 >= -3540
18: x0_6_3 >= 0
19: -7 C1 - C2 + C3 + 6 x0_1_3 + 2 x0_2_3 + 2 x0_6_3 >= -17
20: -x0_9_3 >= -1
21: -x0_9_4 >= -1
22: -x0_9_5 >= -1
23: -x0_9_8 >= -1
24: -C9 >= -20
25: -7 C1 - C2 - 2 C7 - 4 x0_1_2 + 2 C9 >= -29
26: x0_9_5 >= 0
27: -3 C1 - C2 + C3 + 2 x0_1_3 + 2 x0_2_3 - 2 x0_6_3 >= -5
28: x0_9_3 >= 0
29: C1 - x0_1_3 >= 3
30: x0_9_8 >= 0
31: x0_9_4 >= 0
```

This is highly surprising for me. I expected the optimal LP solution to "problem.lp" to be exactly 178 since we have added the 31 facets that approximate the integer convex hull.

I am using polymake 3.0 for enumerating the facets.

Also, when I use lp2poly function of polymake, on the MIP file, to enumerate the facets, as specified by

https://polymake.org/doku.php/tutorial/optimization

I get "INPUT polytope must be bounded error."

Is it possible to clarify why these discrepancies occur?

Thanks.