The Linear Programming Solver

Irreducible Infeasible Set

For a linear programming problem, an irreducible infeasible set (IIS) is an infeasible subset of constraints and variable bounds that will become feasible if any single constraint or variable bound is removed. It is possible to have more than one IIS in an infeasible LP. Identifying an IIS can help isolate the structural infeasibility in an LP.

The presolver in the LP solvers can detect infeasibility, but it identifies only the variable bound or constraint that triggers the infeasibility.

The IIS=ON option directs the LP solver to search for an IIS in a specified LP. You should specify the OPTMODEL option PRESOLVER=NONE when you specify IIS=ON; otherwise the IIS results can be incomplete. The LP solver does not apply the LP presolver to the problem during the IIS search. If the LP solver detects an IIS, it updates the .status suffix of the decision variables and constraints, and then it stops. The number of iterations that are reported in the macro variable and the ODS table is the total number of simplex iterations. This total includes the initial LP solve and all subsequent iterations during the constraint deletion phase.

The IIS= option can add special values to the .status suffixes of variables and constraints. (For more information, see the section Variable and Constraint Status.) For constraints, a status of “I_L”, “I_U”, or “I_F” indicates that the “GE” ( $\geq$ ), “LE” ( $\leq$ ), or “EQ” (=) constraint, respectively, is part of the IIS. For range constraints, a status of “I_L” or “I_U” indicates that the lower or upper bound, respectively, of the constraint is needed for the IIS, and “I_F” indicates that the bounds in the constraint are conflicting. For variables, a status of “I_L”, “I_U”, or “I_F” indicates that the lower, upper, or both bounds, respectively, of the variable are needed for the IIS. From this information, you can identify both the names of the constraints (variables) in the IIS and the corresponding bound where infeasibility occurs.

Making any one of the constraints or variable bounds in the IIS nonbinding removes the infeasibility from the IIS. In some cases, changing a right-hand side or bound by a finite amount removes the infeasibility. However, the only way to guarantee removal of the infeasibility is to set the appropriate right-hand side or bound to $\infty$ or $-\infty$ . Because it is possible for an LP to have multiple irreducible infeasible sets, simply removing the infeasibility from one set might not make the entire problem feasible. To make the entire problem feasible, you can specify IIS=ON and rerun the LP solver after removing the infeasibility from an IIS. Repeating this process until the LP solver no longer detects an IIS results in a feasible problem. This approach to infeasibility repair can produce different end problems depending on which right-hand sides and bounds you choose to relax.

The IIS= option in the LP solver uses two different methods to identify an IIS:

Based on the result of the initial solve, the sensitivity filter removes several constraints and variable bounds immediately while still maintaining infeasibility. This phase is quick and dramatically reduces the size of the IIS.
Next, the deletion filter removes each remaining constraint and variable bound one by one to check which of them are needed to obtain an infeasible system. This second phase is more time consuming, but it ensures that the IIS set that the LP solver returns is indeed irreducible. The progress of the deletion filter is reported at regular intervals. The sensitivity filter might be called again during the deletion filter to improve performance.

See Example 6.4 for an example that demonstrates the use of the IIS= option in locating and removing infeasibilities.