Decomposition Techniques in Mathematical Programming: Example 7.4

This example is from the book Decomposition Techniques in Mathematical Programming Chapter 7.2, page 281, url

Model of the problem First level

\[\min -4y - x,\\ \notag s.t.\\ y + 2x \leq 8,\\\]

Second level

\[\min - y - x,\\ \notag s.t.\\ - y \leq 0,\\ y + x \leq 7,\\ - x \leq 0,\\ x \leq 4,\\\]

using BilevelJuMP
using Ipopt

model = BilevelModel(Ipopt.Optimizer; mode = BilevelJuMP.ProductMode(1e-5))

set_silent(model)

First we need to create all of the variables in the upper and lower problems:

Upper level variables

@variable(Upper(model), x, start = 1)

\[ x \]

Lower level variables

@variable(Lower(model), y, start = 6)

\[ y \]

Then we can add the objective and constraints of the upper problem:

Upper level objecive function

@objective(Upper(model), Min, 4y - x)

\[ 4 y - x \]

Upper level constraints

@constraint(Upper(model), y + 2x <= 8)

\[ 2 x + y \leq 8 \]

Followed by the objective and constraints of the lower problem:

Lower objective function

@objective(Lower(model), Min, - y -x )

\[ -y - x \]

Lower constraints

@constraint(Lower(model), - y <= 0)
@constraint(Lower(model), y + x <= 7)
@constraint(Lower(model), - x <= 0)
@constraint(Lower(model), x <= 4)

\[ x \leq 4 \]

Now we can solve the problem and verify the solution again that reported by book

optimize!(model)

Results

value(x)

1.000010089212264

value(y)

5.999979901074291

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