Remove VariableBasisStatus and clarify when NONBASIC_AT_* should be used

[mlubin]

Ref #678 (comment)
CC @EdvinAblad

[codecov-io]

Codecov Report

Merging #679 into master will not change coverage.
The diff coverage is n/a.

@@           Coverage Diff           @@
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  Coverage   94.96%   94.96%           
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  Files          50       50           
  Lines        5347     5347           
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  Hits         5078     5078           
  Misses        269      269

Impacted Files	Coverage Δ
src/attributes.jl	95.71% <ø> (ø)	⬆️

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[mlubin]

I'm looking at this again and I'm questioning why we have a VariableBasisStatus() attribute. Shouldn't this be the BasisStatus() on the single-variable constraints instead? As it is currently, this is the only place in MOI where "variable bounds" need to be discussed as a concept, which seems inconsistent.

[EdvinAblad]

Consistency is important, however, I think that the most common use case is for a user to query whether a variable is basic or non-basic, not if a constraint is...

Is it possible (and sane) to have both?
E.g., calling MOI.get(m, MOI.BasisStatus(), var_index), is basically an indirect call to MOI.get(m, MOI.BasisStatus(), var_constr). (where var_constr is the applicable one)

Also I'm not quite sure how these single variable constraints works if they are called multiple times, e.g. x => 0, x >= 3, x =>2. In JuMP they seems to be available from set_lower_bound, in LQOI adding a single-variable bound also (seems to) replace the previous ones. Is it possible for multiple variable bounds to be present? If so, the basis status of the dominated constraints should always return MOI.BASIC (where should this domination check be done, LQOI?).

[blegat]

Is it possible for multiple variable bounds to be present?

No, see #570 The LowerBoundRef gives in JuMP gives you the reference of the last bound set

[mlubin]

I updated the PR to remove VariableBasisStatus. It's not only inconsistent with how we report dual solutions, it's also redundant because you can ask for the ConstraintBasisStatus of a SingleVariable constraint anyway. Given that there's a direct mapping between the two, we shouldn't force solver interfaces to implement both. Convenience functions can be added at the JuMP level like we do for lower and upper bounds.

@odow @joaquimg