Princeton Handbook of Test Problems: Test 9.3.2

This example is from the book Princeton Handbook of Test Problems in Local and Global Optimization, Floudas et al., Chapter 9.3.2, Page 221, url.

Model of the problem First level

\[\min (x-5)^2+(2y+1)^2,\\ \notag s.t.\\ x \geq 0,\\ y \geq 0,\\\]

Second level

\[\min (y-1)^2-1.5xy,\\ \notag s.t.\\ -3x+y \leq -3,\\ x-0.5y \leq 4,\\ x+y \leq 7,\\\]

using BilevelJuMP
using Ipopt

model = BilevelModel(Ipopt.Optimizer; mode = BilevelJuMP.ProductMode(1e-9))

An Abstract JuMP Model
Feasibility problem with:
Variables: 0
Upper Constraints: 0
Lower Constraints: 0
Bilevel Model
Solution method: BilevelJuMP.ProductMode{Float64}(1.0e-9, false, 0, nothing)
Solver name: Ipopt

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

Upper level variables

@variable(Upper(model), x)

#Lower level variables
@variable(Lower(model), y)

\[ y \]

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

Upper level objecive function

@objective(Upper(model), Min, (x - 5)^2 + (2y + 1)^2)

\[ x^2 + 4 y^2 - 10 x + 4 y + 26 \]

Upper level constraints

@constraint(Upper(model), x >= 0)
@constraint(Upper(model), y >= 0)

\[ y \geq 0 \]

Followed by the objective and constraints of the lower problem:

Lower objective function

@objective(Lower(model), Min, (y - 1)^2 - 1.5 * x * y)

\[ y^2 - 1.5 y\times x - 2 y + 1 \]

Lower constraints

@constraint(Lower(model), -3x + y <= -3)
@constraint(Lower(model), x - 0.5y <= 4)
@constraint(Lower(model), x + y <= 7)

\[ x + y \leq 7 \]

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

optimize!(model)

primal_status(model)

termination_status(model)

value(x)

value(y)

-8.120572244645409e-9

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