When an optimization problem is infeasible it is common to ask for the reason for infeasibility. It is very typical that the model is infeasible because it contains a set of mutually contradicting constraints which is relatively small compared to the total size of the problem. Locating and analyzing such a subset is usually helpful in understanding the reasons for infeasibility.
Such considerations have led the OR community to come up with the notion of Irreducible Infeasible Set (IIS) - a subset of constraints which is infeasible but cannot be reduced to a smaller infeasible set (all its proper subsets are already feasible). Every infeasible problem has an IIS - possibly many, possibly of different sizes. For example the problem bgdbg1 (see below) has a few hundred bounds but its infeasibility is caused already by the small segment
X104_JY >= 0.0
X104_JH >= 36.0
X104_JY + X104_JH <= 6.0
consisting of 3 bounds, which is an IIS. In fact the same model has also another IIS
X199_CM1 >= 96.0
X199_CM1 <= 86.0
with just 2 bounds (it doesn't get any better - a single linear bound is always feasible on its own).
An IIS can be found by a simple iterative algorithm which tries to remove one bound at a time and checks if the problem remains infeasible. We have implemented it for linear and mixed-integer linear problems in a recently published python notebook, where you can also read more details of the algorithm and its implementation. You can also download the python code directly.
Below are the results produced by our sample code on the infeasible problems from NETLIB. We report the number of bounds in the problem, the size of the (Farkas) infeasibility certificate (these are the bounds that will appear in the infeasibility report or in the task.getinfeasiblesubproblem() function) and finally the size of the smallest IIS out of 5 runs of the algorithm (the IIS found depends on an ordering of the bounds, which we randomize).
bgdbg1.gz: OK, all bounds = 926, Farkas size = 2, IIS size = 2
bgetam.gz: OK, all bounds = 1577, Farkas size = 31, IIS size = 25
bgindy.gz: OK, all bounds = 14694, Farkas size = 196, IIS size = 154
bgprtr.gz: OK, all bounds = 68, Farkas size = 12, IIS size = 10
box1.gz: OK, all bounds = 723, Farkas size = 10, IIS size = 9
ceria3d.gz: OK, all bounds = 3576, Farkas size = 117, IIS size = 198
chemcom.gz: OK, all bounds = 1416, Farkas size = 86, IIS size = 37
cplex2.gz: numerical issues, all bounds = 733, Farkas size = 177, IIS size = 715
ex72a.gz: OK, all bounds = 609, Farkas size = 61, IIS size = 60
ex73a.gz: OK, all bounds = 597, Farkas size = 28, IIS size = 25
forest6.gz: OK, all bounds = 196, Farkas size = 96, IIS size = 92
galenet.gz: OK, all bounds = 26, Farkas size = 5, IIS size = 5
gosh.gz: OK, all bounds = 15353, Farkas size = 9, IIS size = 9
gran.gz: OK, all bounds = 8386, Farkas size = 2, IIS size = 2
grbeai.gz: OK, all bounds = 10402, Farkas size = 93, IIS size = 54
itest2.gz: OK, all bounds = 13, Farkas size = 3, IIS size = 3
itest6.gz: OK, all bounds = 21, Farkas size = 3, IIS size = 3
klein1.gz: OK, all bounds = 108, Farkas size = 55, IIS size = 54
klein2.gz: OK, all bounds = 531, Farkas size = 57, IIS size = 55
klein3.gz: numerical issues, all bounds = 1082, Farkas size = 97, IIS size = 192
mondou2.gz: OK, all bounds = 1832, Farkas size = 24, IIS size = 23
pang.gz: OK, all bounds = 942, Farkas size = 34, IIS size = 32
pilot4i.gz: OK, all bounds = 1886, Farkas size = 43, IIS size = 47
qual.gz: OK, all bounds = 1360, Farkas size = 217, IIS size = 370
reactor.gz: OK, all bounds = 1652, Farkas size = 15, IIS size = 9
refinery.gz: OK, all bounds = 1360, Farkas size = 206, IIS size = 120
vol1.gz: OK, all bounds = 1360, Farkas size = 275, IIS size = 196
woodinfe.gz: OK, all bounds = 173, Farkas size = 2, IIS size = 2
The work reported here and in the notebook was carried out by Adam Bosák during his student internship at MOSEK. Thanks!
