Consider adding parameter set for dual parameters

[klamike]

When dualizing a parametric model, the current behavior is to add variables corresponding to primal parameters and to keep track of them in the primal_parameter_to_dual_parameter. I'd like to propose that dual models should constrain those dual parameters to the Parameter set with value matching that of the primal.

Consider

m = Model()
@variable m p ∈ Parameter(1.0)
@variable m x
@constraint m c x ≥ p
@objective m Min x
println(m)
print(dualize(m, dual_names=DualNames()))

which currently gives

Min x
Subject to
 c : -p + x ≥ 0
 p ∈ MathOptInterface.Parameter{Float64}(1.0)

Max param_p*c
Subject to
 x : c = 1
 c ≥ 0

The problem is, directly optimizing this dual allows the solver to optimize param_p. To me that seems unnatural, since we told JuMP it is a parameter, not a decision variable, in the primal. Note that DualOptimizer seems to get around this somehow (I think it has to do with not allowing DIRECT mode which ends up always bridging the parameters prior to dualization?)

Under my proposal, Dualization would give the below dual, which retains the parameter value we set in the primal.

Max param_p*c
Subject to
 x : c = 1
 c ≥ 0
 param_p ∈ MathOptInterface.Parameter{Float64}(1.0)

[odow]

@joaquimg was this intentional?

[joaquimg]

Partially,
There are 2 types of parameters.
The variables in the parameter set and the variables that we pass as a list.
The variables in the parameter set can also be in that set, no issue, I just forgot to do that.
However the one we pass a list, need to be free variabels for BilevelJuMP.