Find dependent variables as GenericAffExpr

Hi everybody!
I’m trying to find a way to obtain the dependent variables of a nonlinear system as GenericAffExpr, expressed as a function only of the independent variables of an optimization problem.

To be more precise, I have a nonlinear problem with independent variables x. The objective function is a function of both x and other dependent variables y. In order to compute the OF, is necessary to obtain the expressions of y in function of x (in type GenericAffExpr, I suppose).
These y can be found by solving a nonlinear system of equations and to do that, I wrote a second optimization model that (in my intentions) should execute this task.

To be clearer, below is reported a minimal example of what I would like to do.

using JuMP, Ipopt

# define optimization problem with independent variables (x)
m = Model(Ipopt.Optimizer)
@variable(m, x[1:2])

# solve nonlinear system to find dependent variables (y)
p = Model(Ipopt.Optimizer)
@variable(p, y[1:2])
@NLconstraint(p, x[1] * y[1] * y[2] == 2)
@NLconstraint(p, x[2] * y[1] * y[2] == 5)
@objective(p, 0.0)

# solve original problem
@NLobjective(m, x[1] + x[2]^2 + sum(y))

However, trying to define the ‘inner’ optimization p, JuMP provides the error Variable in nonlinear expression does not belong to the corresponding model in correspondence of the first @NLconstraint, since the variables x are inserted in the problem p but belong to the problem m.

My question is: does anyone know if exist a suitable way to solve this issue? Or, does exist another strategy to reach my purpose?

Thank you in advance for your help.

I’m guessing you want to extract the value of y that you found after solving p? If so,

@NLobjective(m, x[1] + x[2]^2 + sum(value.(y)))

In a certain sense yes, but, since the variables x are present in the problem p and the problem p only optimize the variables y, the values of y should contain the terms of x. In other words, I would like that the problem p has the following outputs:

y[1] = (10 / (x[1] * x[2]))^0.5
y[2] = ((5 / 2) * x[1] / x[2])^0.5

And the type of y should be Array{GenericAffExpr{Float64,VariableRef},1}.

However, to get the value.(y) that you suggested, I have to run the inner optimization (problem p), which is the one that is giving me problems.

Ah, ok, I took another look and you’re right that wouldn’t work. You can’t mix-and-match variables and models this way. The thing to do is use a single model that contains all the necessary variables (you can add/delete constraints/variables and change the objective as necessary).

m = Model(Ipopt.Optimizer)
@variable(m, x[1:2])
@variable(m, y[1:2])
c1 = @NLconstraint(m, x[1] * y[1] * y[2] == 2)
c2 = @NLconstraint(m, x[2] * y[1] * y[2] == 5)
@objective(m, 0.0) # (note this isn't necessary)
# should you check termination_status(m) before proceeding?

# solve original problem:
# get rid of the constraints if they don't matter any more
delete!.(m, (c1, c2))
@NLobjective(m, x[1] + x[2]^2 + sum(value.(y)))

Alternatively, you could create a new model with only variables called x and set its objective based on the other model’s value for y.

Ok, thank you very much, I’ll try your suggestions!