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### Models at the edge of infeasibility

• As we saw in the introduction, seemingly contradictory results regarding the feasibility or infeasibility of a model can legitimately occur for models that are at the boundary between feasibility and infeasibility.

• A more complicated example is , . It has two bases, one where is basic and one where is basic. If is basic, we get , which is clearly infeasible. However, if is basic we get , which is feasible within tolerance. Different algorithms could lead to either of such bases and thus come to apparently contradictory feasibility results.

• Presolve reductions can also play a role. A presolve reduction, e.g. fixing a variable to a bound, implicitly forces a tolerance of 0 for that variable. When solving the reduced model, the optimizer therefore no longer has the option to "spread" a slight infeasibility of the model over these variables and produce a solution that is feasible within tolerances. This leads to seemingly contradictory feasibilty results when solving the model with presolve enabled or disabled.

• What can be done to diagnose such cases:
• First step is to tighten the FeasibilityTol to and try again. In many cases this will lead to a consistent declaration of infeasibility of the model at hand, which tells you that the model is on this boundary of infeasibiltiy.
• Use feasRelax to solve your model (again with a tight FeasibilityTol. This boundary case is identified by a non-zero relaxation value.
• Compute the IIS (again with a tight FeasibilityTol) to analyze the infeasibility.
• Another source of seemlingly contradictory results is due to numerical issues of the model and will be discussed in the following subsections.