Add VariableBasisStatus

[odow]

VariableBasisStatus was removed here: #679

However, we over-looked an edge case: without it, we can't tell if variables with free bounds with a VariablePrimal of 0.0 are in the (degenerate) basis or not.

This became a problem here: jump-dev/JuMP.jl#2560, jump-dev/JuMP.jl#2562

Given the redundancy with ConstraintBasisStatus on variable bounds, perhaps we just need

"""
    MOI.VariableInBasis

A variable attribute which indicates `true` or `false` if some variable is in 
the current optimal solution basis.
"""
struct MOI.VariableInBasis <: MOI.AbstractVariableAttribute end

cc @geoffleyland

[joaquimg]

Maybe a NoBoundRef or FreeBoundRef. Together with a Real (scalar set).

Or even simpler, a FreeVariableBasisStatus.
Returns error or nothing for bounded variables.

[odow]

I guess we could go:

function FreeVariableRef(x::VariableRef)
    # Check lower_bound, upper_bound, is_fix, etc.
    return MOI.ConstraintIndex{MOI.SingleVariable,MOI.Reals}(x.index.value)
end

and define something along the likes of

function MOI.get(
    model::Optimizer,
    ::MOI.ConstraintBasisStatus,
    c::MOI.ConstraintIndex{MOI.SingleVariable,MOI.Reals},
)
    return in_basis(MOI.VariableIndex(c.value)) ? MOI.BASIC : MOI.SUPER_BASIC
end

But that's a pretty unusual interpretation.

[mlubin]

Can you point to solver documentation on non-basic free variables? Which solvers have this concept?

[joaquimg]

Here is Xpress:

https://www.fico.com/fico-xpress-optimization/docs/latest/solver/optimizer/HTML/XPRSgetbasis.html

[odow]

GLPK has

CPLEX has

[mlubin]

Maybe we should stop fighting against the solver APIs by reviving VariableBasisStatus, and saying that SingleVariable constraints don't have a ConstraintBasisStatus.