Feasibility and Infeasibility in Optimization pp 159-195 | Cite as

# Finding the Maximum Feasible Subset of Linear Constraints

When a linear program is infeasible the usual first tactic is to isolate an IIS via the methods described in the previous chapter. However there is a complementary approach that has analytic value: find the smallest number of constraints to remove such that the remaining constraints constitute a feasible set. The removed constraints in some sense contribute to the infeasibility most heavily. Consider two overlapped IISs: {A,B,C} and {C,D,E} in a larger model with some number *m* of constraints in total. To eliminate all infeasibility from the model, we can remove one constraint from each IIS, say A from the first IIS and E from the second, leaving a feasible set of size *m* – 2. However, we can also remove all infeasibility by removing just the single constraint C, which destroys both IISs simultaneously and leaves a maximum cardinality feasible set of size *m* – 1. The single removed constraint C contributes to infeasibility in both of the IISs in the model, and hence is a better focus for the initial diagnostic effort.

Nothing in the previous paragraph restricts this concept to sets of linear constraints. However the current state of the art is indeed limited to methods for linear systems. For this reason we restrict our attention in this chapter mainly to linear constraints. There is wide scope for extending the methods to other forms of optimization problems.

## Keywords

Linear Constraint Linear Inequality List Length Minimum Cardinality Elastic Variable## Preview

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