Well, there is no universally best way. But one approach I like to suggest is: use (or construct) a known feasible solution and fix decision variables. This approach is not often mentioned. Most algorithm suppliers say: just use IIS [1]. In my opinion (and based on experience), fixing is often more reliable: it will pin-point infeasible constraints. In addition, fixing is conceptually much simpler than looking at IIS output. The results of fixing does not require any explanation.
Example: a transportation problem
Let's use a simple transportation problem as an example: \[\bbox[lightcyan,10px,border:3px solid darkblue]{\begin{align}\min&\sum_{i,j} c_{i,j} x_{i,j}\\ & \sum_i x_{i,j} \ge d_j& \forall j&\text{ (demand)}\\ & \sum_j x_{i,j} \le s_i& \forall i&\text{ (supply)}\\ &x_{i,j} \ge 0\end{align}}\]
The data looks like:---- 45 PARAMETER c transport cost
new-york chicago topeka
seattle 0.2250.1530.162
san-diego 0.2250.1620.126
---- 45 PARAMETER d demand at market j
new-york 325.000, chicago 300.000, topeka 275.000
---- 45 PARAMETER s capacity of plant i
seattle 350.000, san-diego 600.000
When we deal with infeasibilities, we can ignore the objective: this will never have a role. So in the data above, we don't need to worry about parameter \(c\).
A feasible solution is easily constructed:
 |
| Feasible solution |
--- 70 PARAMETER s capacity of plant i
seattle 280.000, san-diego 480.000
The model is now infeasible. The reason is the total supply is too small to meet total demand.
The GAMS equation listing will show the individual equations (before solving):
---- demand =G= satisfy demand at market j
demand(new-york).. x(seattle,new-york) + x(san-diego,new-york) =G= 325 ; (LHS = 0, INFES = 325 ****)
demand(chicago).. x(seattle,chicago) + x(san-diego,chicago) =G= 300 ; (LHS = 0, INFES = 300 ****)
demand(topeka).. x(seattle,topeka) + x(san-diego,topeka) =G= 275 ; (LHS = 0, INFES = 275 ****)
---- supply =L= observe supply limit at plant i
supply(seattle).. x(seattle,new-york) + x(seattle,chicago) + x(seattle,topeka) =L= 280 ; (LHS = 0)
supply(san-diego).. x(san-diego,new-york) + x(san-diego,chicago) + x(san-diego,topeka) =L= 480 ; (LHS = 0)
This is a useful listing to look at. The initial values for the variables are zero here, so the demand equations are infeasible at the initial point. The supply constraints are feasible at the initial point. It is noted that the variables are always feasible with respect to their bounds. GAMS makes sure the variable levels are always within their bounds at this initial point.
One good way to prevent even to pass the model on to the solver is to add a data check:
abort$(sum(i,s(i)) < sum(j,d(j)) - 0.01) "Supply too small for demand";
- Solve as is, and see how different solver report a solution for this infeasible model. We'll see there is little uniformity.
- Run with IIS (Irreducible Infeasible Set). This is often advised as good tool to inspect and diagnose infeasibilities. I am usually less successful when using this approach.
- Fix decision variables to a known feasible solution, and see what is reported. This is often my preferred approach.
Solve and look at the solution
The best solution will minimize the sum of the infeasibilities (this sum is 140) and properly mark the infeasible rows and columns. Let's see how the solver do on this.
Cplex
---- EQU demand satisfy demand at market j
LOWER LEVEL UPPER MARGINAL
new-york 325.0000325.0000 +INF 0.2250
chicago 300.0000300.0000 +INF 0.1530
topeka 275.0000275.0000 +INF 0.1260
---- EQU supply observe supply limit at plant i
LOWER LEVEL UPPER MARGINAL
seattle -INF 280.0000280.0000 EPS
san-diego -INF 620.0000480.0000 . INFES
---- VAR x shipment quantities in cases
LOWER LEVEL UPPER MARGINAL
seattle .new-york . -20.0000 +INF . INFES
seattle .chicago . 300.0000 +INF .
seattle .topeka . . +INF 0.0360
san-diego.new-york . 345.0000 +INF .
san-diego.chicago . . +INF 0.0090
san-diego.topeka . 275.0000 +INF .
**** REPORT SUMMARY : 0 NONOPT
2 INFEASIBLE (INFES)
SUM 160.0000
MAX 140.0000
MEAN 80.0000
0 UNBOUNDED
We see there are two reported infeasibilities. One of them is an infeasible variable. Interestingly, this is not the solution with the minimum phase 1 objective (i.e. minimum sum of infeasibilities). Here we have a total infeasibility of 160 while the optimal phase 1 objective is 140.
MINOS, CONOPT
A proper minimum phase 1 solution with a total infeasiblity of 140 is reported by MINOS and Conopt:
---- EQU demand satisfy demand at market j
LOWER LEVEL UPPER MARGINAL
new-york 325.0000325.0000 +INF 0.0039
chicago 300.0000300.0000 +INF 0.0039
topeka 275.0000135.0000 +INF 0.0039INFES
---- EQU supply observe supply limit at plant i
LOWER LEVEL UPPER MARGINAL
seattle -INF 280.0000280.0000 -0.0039
san-diego -INF 480.0000480.0000 -0.0039
---- VAR x shipment quantities in cases
LOWER LEVEL UPPER MARGINAL
seattle .new-york . . +INF EPS
seattle .chicago . 145.0000 +INF .
seattle .topeka . 135.0000 +INF .
san-diego.new-york . 325.0000 +INF .
san-diego.chicago . 155.0000 +INF .
san-diego.topeka . . +INF EPS
**** REPORT SUMMARY : 0 NONOPT
1 INFEASIBLE (INFES)
SUM 140.0000
MAX 140.0000
MEAN 140.0000
0 UNBOUNDED
0 ERRORS
In both cases the solver distributes infeasibilities in a way that is unrelated to the original problem: we see a demand equation is marked (while both supply equations are the real culprit here).
Gurobi
---- EQU demand satisfy demand at market j
LOWER LEVEL UPPER MARGINAL
new-york 325.0000 . +INF -1.0000
chicago 300.0000 . +INF -1.0000
topeka 275.0000 . +INF -1.0000
---- EQU supply observe supply limit at plant i
LOWER LEVEL UPPER MARGINAL
seattle -INF . 280.00001.0000
san-diego -INF . 480.00001.0000
---- VAR x shipment quantities in cases
LOWER LEVEL UPPER MARGINAL
seattle .new-york . . +INF .
seattle .chicago . . +INF .
seattle .topeka . . +INF .
san-diego.new-york . . +INF .
san-diego.chicago . . +INF .
san-diego.topeka . . +INF .
**** REPORT SUMMARY : 0 NONOPT
0 INFEASIBLE
0 UNBOUNDED
This is just a bogus solution: all levels are zero. Infeasibilities are not properly marked. The number and sum of infeasibilities is also wrong.
The conclusion must be: many solvers do not do a good job of returning a proper phase 1 solution that minimizes the sum of the infeasibilities. I always find this somewhat disappointing.
IIS: Irreducable Infeasible Set
Many modern solvers have a facility to use an IIS (Irreducable Infeasible Set) algorithm to return groups of equations such that when one equation is removed thit set of equations becomes feasible.
Here is the IIS reported by Gurobi:
LP status(3): Model was proven to be infeasible.
Computing IIS...
An IIS is a set equations and variables (ie a submodel) which is
infeasible but becomes feasible if any one equation or variable bound
is dropped.
A problem may contain several independent IISs but only one will be
found per run.
Number of equations in the IIS: 5
demand(new-york) > 325
demand(chicago) > 300
demand(topeka) > 275
supply(seattle) < 280
supply(san-diego) < 480
Number of variables in the IIS: 0
Cplex gives the same thing:
Minimal conflict found.
A conflict is a set equations and variables (ie a submodel) which is
infeasible but becomes feasible if any one equation or variable bound
is dropped.
A problem may contain several independent conflicts but only one will be
found per run.
Number of equations in the conflict: 5.
lower: demand(new-york) > 325
lower: demand(chicago) > 300
lower: demand(topeka) > 275
upper: supply(seattle) < 280
upper: supply(san-diego) < 480
Number of variables in the conflict: 0.
The IIS set has all constraints of the model. This actually means: if we drop any of the constraints of our model, the resulting model will be feasible. Well, that is interesting, but it does not bring me closer to the conclusion that the supply capacities are wrong.
Fixing: the best approach for this problem
The best approach is to fix the variable to our feasible solution. In GAMS, we can do this with:
table solx(i,j)
new-york chicago topeka
seattle 325
san-diego 300 275
;
x.fx(i,j) = solx(i,j);
The equation listing tells us exactly what is wrong:
---- demand =G= satisfy demand at market j
demand(new-york).. x(seattle,new-york) + x(san-diego,new-york) =G= 325 ; (LHS = 325)
demand(chicago).. x(seattle,chicago) + x(san-diego,chicago) =G= 300 ; (LHS = 300)
demand(topeka).. x(seattle,topeka) + x(san-diego,topeka) =G= 275 ; (LHS = 275)
---- supply =L= observe supply limit at plant i
supply(seattle).. x(seattle,new-york) + x(seattle,chicago) + x(seattle,topeka) =L= 280 ; (LHS = 325, INFES = 45 ****)
supply(san-diego).. x(san-diego,new-york) + x(san-diego,chicago) + x(san-diego,topeka) =L= 480 ;
(LHS = 575, INFES = 95 ****)
The initial point with the fixed levels for the variables, yields infeasibilities for the supply equation. The solver will also pinpoint a single infeasibility in the log:
Infeasibility row 'supply(seattle)': 0 <= -45.Only one infeasibility is reported here, This is because the presolver handled this. When a presolver finds the model is infeasible most often it can produce a good message, but it will only report one infeasibility.
Note:
However, the presolver can also lead to the dreaded:
Model was proven to be either infeasible or unbounded.
The solver puts premium on minimization of its time. One could say, the real problem a solver solves is: minimize the time spent on solving the problem. In case of this message, the problem to find out whether the model is really unbounded or infeasible is off-loaded to the user. Apparently the user's time is less important.Some notes on fixing:
- Often you don't need to fix all variables in a model. Just fixing the central variables may be enough: the other variables can be calculated from them in an unambiguous way. If you only fix a few central variables, you need to rely on the solver messages to pinpoint the infeasibility. As the initial point is not complete, we cannot rely on the GAMS equation listing.
- Make sure you don't destroy any bounds on the variables.
- Bounds can play an important role in the model being infeasible.
- This method also works when the problem is integer infeasible (but LP feasible).
Conclusion
If you know (or can construct) a feasible point, fixing variables to this point is an extremely useful tool to find out what caused the problem to be infeasible. Of course, constructing a feasible solution is not always easy, so this method is not suited for all cases. For other models it may be surprisingly easy to generate a feasible solution. For instance, for a machine scheduling model, just order jobs by their release date on a single machine.
The GAMS output for different solvers is unfortunately difficult to interpret: some of the output is contradictory, and output is spread around in log and listing files. Solvers are not very consistent in what they report. Depending on whether the presolver or the solver itself detects infeasibilities, the reporting may be totally differently. This is all somewhat messy.
If you want to protect your models against infeasibilities, it may help to formulate an elastic version. This means that you allow to violate constraints, but at a cost. In other words: convert hard constraints into soft ones. For example, a manpower constraint can be made elastic by allowing to hire temporary workers (at a higher cost).
Finally I want to mention that unbounded models are easier to debug. Just put a bound on the objective, e.g.: \[\begin{align}\min\>&z \\ & z = c^Tx \\ & z \ge -1000000\\ &Ax=b \\ & \ell \le x \le u\end{align}\] and inspect the solution to for large (negative) values.
References
- encountering INFEASIBLE status in Gurobi Matlab interface, https://stackoverflow.com/questions/51743892/encountering-infeasible-status-in-gurobi-matlab-interface