Dealing with big-M constraints
Big-M constraints are a regular source of instability for optimization
problems. They are so named because they typically involve a large
coefficient 
that is chosen to be larger than any reasonable value
that a continuous variable or expression may take. Here's a simple
example:
satisfies the default
integrality tolerance (IntFeasTol=
), which allows
to take a value of 
. In other words, can take a
positive value without incurring an expensive fixed charge on 
,
which subverts the intent of only allowing a non-zero value
for when the binary variable has the value of 1. You can reduce
the effect of this behavior by adjusting the IntFeasTol
parameter, but you can't avoid it entirely.
However, if the modeler has additional information that the
variable will never be larger than 
, then you could reformulate
the earlier constraint as:
would only allow for 
.
For cases when it is not possible to either rescale variable
or tighten its bounds, an SOS constraints or an indicator constraint
(of the form
) may produce more accurate
solutions, but often at the expense of additional processing time.
