Princeton Handbook of Test Problems: Test 9.3.4
This example is from the book Princeton Handbook of Test Problems in Local and Global Optimization Dempe, Chapter 9.3.4 -parg 223 url
Here, only the second level is described
Model of the problem First level
\[\min 2x_1+2x_2-3y_1-3y_2-60,\\ \notag s.t.\\ x_1 + x_2 + y_1 -2y_2 -40\leq 0,\\ 0 \leq x_i \leq 50, \forall i \in I,\\ -10 \leq y_j \leq 20, \forall j \in J,\\\]
Second level
\[\min (-x_1 + y_1 + 40)^2 + (-x_2 + y_2 + 20)^2,\\ \notag s.t.\\ - x_i + 2y_j <= -10,\forall (i,j) \in \{(i,j)|i\in I, j\in J, i=j\},\\ -10 \leq y_j \leq 20, \forall j \in J.\\\]
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: IpoptFirst we need to create all of the variables in the upper and lower problems:
Upper level variables
@variable(Upper(model), x[i = 1:2], start = 0)2-element Vector{BilevelVariableRef}:
x[1]
x[2]Lower level variables
@variable(Lower(model), y[i = 1:2], start = -10)2-element Vector{BilevelVariableRef}:
y[1]
y[2]Then we can add the objective and constraints of the upper problem:
Upper level objecive function
@objective(Upper(model), Min, 2x[1] + 2x[2] - 3y[1] - 3y[2] - 60)$ 2 x{1} + 2 x{2} - 3 y{1} - 3 y{2} - 60 $
Upper level constraints
@constraint(Upper(model), x[1] + x[2] + y[1] - 2y[2] - 40 <= 0)
@constraint(Upper(model), [i = 1:2], x[i] >= 0)
@constraint(Upper(model), [i = 1:2], x[i] <= 50)
@constraint(Upper(model), [i = 1:2], y[i] >= -10)
@constraint(Upper(model), [i = 1:2], y[i] <= 20)2-element Vector{ConstraintRef{BilevelModel, Int64, ScalarShape}}:
y[1] ≤ 20
y[2] ≤ 20Followed by the objective and constraints of the lower problem:
Lower objective function
@objective(Lower(model), Min, (-x[1] + y[1] + 40)^2 + (-x[2] + y[2] + 20)^2)$ x{1}^2 - 2 y{1}\times x{1} + y{1}^2 + x{2}^2 - 2 y{2}\times x{2} + y{2}^2 - 80 x{1} + 80 y{1} - 40 x{2} + 40 y{2} + 2000 $
Lower constraints
@constraint(Lower(model), [i = 1:2], -x[i] + 2y[i] <= -10)
@constraint(Lower(model), [i = 1:2], y[i] >= -10)
@constraint(Lower(model), [i = 1:2], y[i] <= 20)2-element Vector{ConstraintRef{BilevelModel, Int64, ScalarShape}}:
y[1] ≤ 20
y[2] ≤ 20Now we can solve the problem and verify the solution again that reported by the book
optimize!(model)This is Ipopt version 3.14.4, running with linear solver MUMPS 5.4.1.
Number of nonzeros in equality constraint Jacobian...: 10
Number of nonzeros in inequality constraint Jacobian.: 42
Number of nonzeros in Lagrangian Hessian.............: 8
Total number of variables............................: 10
variables with only lower bounds: 2
variables with lower and upper bounds: 0
variables with only upper bounds: 4
Total number of equality constraints.................: 2
Total number of inequality constraints...............: 21
inequality constraints with only lower bounds: 6
inequality constraints with lower and upper bounds: 0
inequality constraints with only upper bounds: 15
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
0 0.0000000e+00 6.00e+01 9.57e-01 -1.0 0.00e+00 - 0.00e+00 0.00e+00 0
1 4.4814823e-01 5.98e+01 1.28e+00 -1.0 5.21e+01 - 2.06e-03 4.18e-03h 1
2 1.6046697e+00 5.89e+01 2.12e+00 -1.0 4.35e+01 - 1.53e-03 1.42e-02h 1
3 1.6995414e+01 4.70e+01 1.06e+01 -1.0 3.84e+01 - 8.81e-03 2.02e-01h 1
4 1.9250234e+01 4.52e+01 3.77e+01 -1.0 3.00e+01 - 1.34e-01 3.79e-02h 1
5 3.5204890e+01 2.81e+01 6.90e+01 -1.0 2.11e+01 - 3.21e-02 3.78e-01h 1
6 3.7687463e+01 2.21e+01 1.93e+02 -1.0 1.84e+01 0.0 1.54e-02 2.14e-01h 1
7 3.8166798e+01 2.11e+01 8.95e+01 -1.0 1.36e+01 0.4 5.36e-01 4.66e-02h 1
8 3.4396171e+01 1.82e+01 3.30e+02 -1.0 5.43e+01 - 4.41e-01 1.37e-01f 1
9 3.3236150e+01 1.66e+01 2.79e+02 -1.0 2.52e+01 - 5.77e-02 8.71e-02f 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
10 3.0766710e+01 1.60e+01 4.15e+02 -1.0 4.61e+01 - 3.12e-01 3.79e-02f 1
11 1.0078764e+01 1.11e+01 3.77e+02 -1.0 8.40e+01 - 5.03e-01 3.05e-01f 1
12 2.4302311e-01 9.16e+00 4.41e+02 -1.0 6.72e+01 - 2.05e-02 1.75e-01f 1
13 1.1671615e-01 4.12e+00 3.93e+02 -1.0 9.35e+00 - 1.00e+00 5.50e-01h 1
14 1.7254023e-01 1.32e+00 5.89e+02 -1.0 4.03e+00 - 1.00e+00 6.79e-01h 1
15 1.8014838e-01 7.75e-01 3.02e+03 -1.0 1.30e+00 - 1.00e+00 4.14e-01h 1
16 1.9235506e-01 2.03e-01 2.82e+03 -1.0 7.56e-01 - 1.00e+00 7.38e-01h 1
17 1.9390169e-01 1.15e-01 1.85e+04 -1.0 1.99e-01 - 1.00e+00 4.33e-01h 1
18 1.9562312e-01 3.19e-02 1.96e+04 -1.0 1.12e-01 - 1.00e+00 7.22e-01h 1
19 1.9604520e-01 1.79e-02 1.16e+05 -1.0 3.12e-02 - 1.00e+00 4.38e-01h 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
20 1.9619242e-01 4.98e-03 1.23e+05 -1.0 1.75e-02 - 1.00e+00 7.22e-01h 1
21 1.9632743e-01 3.88e-03 9.52e+05 -1.0 4.75e-03 - 1.00e+00 2.21e-01f 2
22 1.9565901e-01 1.07e-03 6.21e+05 -1.0 4.19e-03 - 1.00e+00 7.24e-01h 1
23 1.9573024e-01 1.02e-03 4.02e+06 -1.0 8.16e-04 - 1.00e+00 4.36e-02f 5
24 1.9569163e-01 1.11e-04 7.43e+05 -1.0 7.92e-04 - 1.00e+00 8.91e-01h 1
25 1.9645947e-01 8.32e-05 5.40e+06 -1.0 2.01e-03 - 1.00e+00 2.51e-01f 2
26 1.9874099e-01 1.42e-14 6.83e+02 -1.0 1.51e-03 - 1.00e+00 1.00e+00h 1
27 6.8772523e-03 1.42e-14 8.22e+04 -2.5 9.71e-02 - 1.00e+00 1.00e+00f 1
28 5.6568726e-03 7.11e-15 3.16e+00 -2.5 1.19e-03 - 1.00e+00 1.00e+00f 1
29 3.0104856e-04 1.42e-14 6.46e+01 -3.8 2.68e-03 - 1.00e+00 1.00e+00f 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
30 3.0100656e-04 0.00e+00 7.46e-07 -3.8 2.57e-07 - 1.00e+00 1.00e+00h 1
31 1.6349148e-07 1.42e-14 5.98e+00 -8.6 1.50e-04 - 1.00e+00 1.00e+00f 1
32 -3.6260538e-08 2.58e-14 5.60e-03 -8.6 2.01e-06 - 1.00e+00 9.89e-01h 1
33 -3.2290178e-08 1.42e-14 1.03e-12 -8.6 3.05e-08 - 1.00e+00 1.00e+00f 1
Number of Iterations....: 33
(scaled) (unscaled)
Objective...............: -3.2290177642835260e-08 -3.2290177642835260e-08
Dual infeasibility......: 1.0289546992225951e-12 1.0289546992225951e-12
Constraint violation....: 1.4210854715202004e-14 1.4210854715202004e-14
Variable bound violation: 4.8166666662694811e-09 4.8166666662694811e-09
Complementarity.........: 2.5157965850181528e-09 2.5157965850181528e-09
Overall NLP error.......: 2.5157965850181528e-09 2.5157965850181528e-09
Number of objective function evaluations = 40
Number of objective gradient evaluations = 34
Number of equality constraint evaluations = 40
Number of inequality constraint evaluations = 40
Number of equality constraint Jacobian evaluations = 34
Number of inequality constraint Jacobian evaluations = 34
Number of Lagrangian Hessian evaluations = 33
Total seconds in IPOPT = 0.014
EXIT: Optimal Solution Found.primal_status(model)FEASIBLE_POINT::ResultStatusCode = 1
termination_status(model)LOCALLY_SOLVED::TerminationStatusCode = 4
objective_value(model)-3.229017764283526e-8
value.(x)2-element Vector{Float64}:
-8.747048220055642e-9
-8.747048220161989e-9value.(y)2-element Vector{Float64}:
-10.000000000634337
-10.000000000265Like any other optimization problem, there is a chance in bilevel
optimization to find multiple solutions with the same optimal value; based on the inherent stochasticity of the algorithm and random seed, we are expecting two optimal solutions for this problem.
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