Altering Constraints to Achieve Feasibility

Most available methods in this category address only infeasible linear systems, though at least one method can be applied to nonlinear systems. The methods divide into two broad classes: those that only consider shifting the constraints via a change to the right hand side constant (i.e. a parallel translation of the constraint), and those that consider the much harder problem of finding the minimum change to all of the constraint coefficients, including both the constraint bodies and the right hand side constants. Oddly, none of the research addresses the issue of incorrect relationship directions. For example, the model might be rendered feasible if a ≥ relationship is changed to a ≤ relationship, or if an = is changed to a ≥, etc. The general unaddressed question is this: what is the smallest number of constraint relationships to change such that the model is made feasible?

As we will see in this chapter, there are several ways to define the “smallest adjustment” that will render a model feasible, each having different solution complexities and yielding different results. All are based on minimizing some kind of matrix norm that expresses the difference between the “corrected” version of the model and the original version.