[SCIP] The rule of the slack variable in an indicator constraint

[Abbas Omidi]

Dear Mathieu,

Thank you so much for your reply and also the provided information. I
actually saw the link corresponds to the indicator constraints, but I am
not sure to fully understand how this separation works on the slack
variable. Also, as the constraint that should be used in an indicator
constraint is of the form (Ax <= b), the slack variable might be a
violation on the constraint. Now, I have some questions and I was wondering
if you can give some points to that:

> This constraint is equivalent to a linear constraint ax−s≤b and an SOS1
> constraint on y and s (at most one should be nonzero)


1) What is it meaning of SOS1 on "y" and "s"? I mean, AFAIK, the SOS1 is
the form (\sum_{i} x_{i} = 1), how we can interpret the relation between
"y" and "s" as a SOS1?

In the implementation, we assume that bounds on the original variables x cannot
> be influenced by the indicator constraint. If it should be possible to
> relax these constraints as well, then these constraints have to be added as
> indicator constraints.


2) Does it mean we cannot use the bounded variables as an input into the
indicator constraint? If so, How can we deal with such a case?
3) Unfortunately, I cannot understand of how the SCIP can deal with the
slack variable in the separation via the alternative polyhedron. It seems
just to be as the SOS1. Would you please, explain a bit more only about the
slack variable?


All the best
regards
Abbas

On Mon, Aug 21, 2023 at 10:25 AM Mathieu Besançon <
mathieu.besancon at gmail.com> wrote:

> Dear Abbas,
>
> The top-level description of the indicator constraint handler of SCIP
> should be a good start:
> https://www.scipopt.org/doc/html/cons__indicator_8c.php
>
> There are three ways indicator constraints are handled, separation,
> preprocessing, and a heuristic. Separation is based on this paper in
> particular: https://doi.org/10.1137/050645828
>
> Performance with a big M approach will typically differ. In my experience,
> if you have a good bound M, a big M approach won't perform too bad, but may
> be numerically more unstable.
>
> SCIP only supports
> *LHS - slack <= RHS*
> At least for now.
>
> Hope this helps,
> Mathieu
>
> On Mon, Aug 21, 2023 at 8:34 AM Abbas Omidi <abb.omidi at gmail.com> wrote:
>
>> Dear support team,
>>
>> In the SCIP the indicator constraint can be written as indcons(expression,
>> binary_var). Then it is interpreted as follows: (based on ".cip" format)
>>
>> *LHS - slack (<=/=/>=) RHS*
>> *If: (binary_var = 1) -> (slack = 0)*
>>
>> As far as I know, CPLEX and maybe Gurobi have used binary variables and
>> whose linking to linearize the logical constraints. I am actually not aware
>> of their internal mechanism. Now, I would like to know how the SCIP can
>> deal with indicator constraint internally, and also the rule of the slack
>> variable in this form, and if is it necessary to use that instead of
>> using e.g. a *BigM* approach.
>>
>>
>> Best regards
>> Abbas
>> _______________________________________________
>> Scip mailing list
>> Scip at zib.de
>> https://listserv.zib.de/mailman/listinfo/scip
>>
>
>
> --
> Mathieu Besançon
>
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