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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
feasibility 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 infeasibility.
- Use
feasRelaxto 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.
- First step is to tighten the
FeasibilityTol to
- Another source of seemlingly contradictory results is due to numerical issues of the model and will be discussed in the following subsections.
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