Non Linear models
BilevelJuMP has limited support for non-linear models. The @NLconstraint and the @NLobjective macros are supported for the upper level, but not for the lower level. Moreover, these macros can only be used if the selected solver supports non-linear constraints and objectives.
The @constraint and @objective macros can be used for both levels to represent linear and quadratic constraints and objectives
Quadratic constraints and objectives
using BilevelJuMP, Ipopt
model = BilevelModel(Ipopt.Optimizer, mode = BilevelJuMP.ProductMode(1e-5))
@variable(Upper(model), x >= 2)
@variable(Lower(model), 3 <= y <= 5)\[ y \]
We can add a non-linear objective to the upper level with @NLobjective
@NLobjective(Upper(model), Min, x^2 - y)We can also add a non-linear constraint to the upper level with @NLconstraint
@NLconstraint(Upper(model), x^2 + y^2 <= 100)\[ (x ^ {2.0} + y ^ {2.0}) - 100.0 \leq 0 \]
@NLobjective is not supported in the lower level, but we can use @constraint to add a quadratic objective to the lower level.
@objective(Lower(model), Min, y^2)\[ y^2 \]
optimize!(model)This is Ipopt version 3.14.4, running with linear solver MUMPS 5.4.1.
Number of nonzeros in equality constraint Jacobian...: 3
Number of nonzeros in inequality constraint Jacobian.: 8
Number of nonzeros in Lagrangian Hessian.............: 5
Total number of variables............................: 4
variables with only lower bounds: 2
variables with lower and upper bounds: 1
variables with only upper bounds: 1
Total number of equality constraints.................: 1
Total number of inequality constraints...............: 3
inequality constraints with only lower bounds: 0
inequality constraints with lower and upper bounds: 0
inequality constraints with only upper bounds: 3
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
0 1.0603999e+00 6.04e+00 1.92e+00 -1.0 0.00e+00 - 0.00e+00 0.00e+00 0
1 1.0812189e+00 5.99e+00 1.91e+00 -1.0 1.52e+01 - 1.42e-02 7.80e-03h 1
2 1.1009516e+00 2.41e+00 2.17e+01 -1.0 6.05e+00 - 9.94e-01 5.98e-01f 1
3 1.1079890e+00 1.31e+00 8.34e+00 -1.0 2.34e+00 - 6.07e-01 4.58e-01h 1
4 1.1069894e+00 8.19e-01 4.31e+01 -1.0 1.28e+00 - 1.00e+00 3.73e-01h 1
5 1.1018085e+00 4.02e-01 6.84e+01 -1.0 8.23e-01 - 1.00e+00 5.10e-01h 1
6 1.1008006e+00 1.63e-01 1.07e+02 -1.0 3.97e-01 - 1.00e+00 5.95e-01h 1
7 1.0997554e+00 7.12e-02 2.56e+02 -1.0 1.63e-01 - 1.00e+00 5.62e-01h 1
8 1.0995471e+00 2.88e-02 5.33e+02 -1.0 7.05e-02 - 1.00e+00 5.95e-01h 1
9 1.0993558e+00 1.19e-02 1.23e+03 -1.0 2.89e-02 - 1.00e+00 5.89e-01h 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
10 1.0993204e+00 4.67e-03 2.74e+03 -1.0 1.17e-02 - 1.00e+00 6.06e-01h 1
11 1.0992867e+00 1.72e-03 5.75e+03 -1.0 4.69e-03 - 1.00e+00 6.31e-01h 1
12 1.0992814e+00 5.11e-04 9.28e+03 -1.0 1.70e-03 - 1.00e+00 7.03e-01h 1
13 1.0992784e+00 2.83e-04 2.64e+04 -1.0 5.15e-04 - 1.00e+00 4.47e-01f 2
14 1.0992757e+00 2.64e-16 2.25e+00 -1.0 2.82e-04 - 1.00e+00 1.00e+00h 1
15 1.0014454e+00 3.21e-16 5.59e-05 -3.8 9.84e-02 - 1.00e+00 1.00e+00f 1
16 1.0000011e+00 1.36e-16 8.75e-08 -5.7 8.54e-04 - 1.00e+00 1.00e+00h 1
17 9.9999884e-01 2.85e-16 3.44e-01 -8.6 1.12e-06 - 8.33e-01 1.00e+00h 1
18 9.9999826e-01 4.95e-16 6.79e-08 -8.6 3.53e-06 - 1.00e+00 1.00e+00h 1
19 9.9999826e-01 6.22e-16 3.91e-13 -8.6 7.97e-09 - 1.00e+00 1.00e+00h 1
Number of Iterations....: 19
(scaled) (unscaled)
Objective...............: 9.9999825667536113e-01 9.9999825667536113e-01
Dual infeasibility......: 3.9099478168183055e-13 3.9099478168183055e-13
Constraint violation....: 6.2161461010035515e-16 6.2161461010035515e-16
Variable bound violation: 1.9373524118293517e-08 1.9373524118293517e-08
Complementarity.........: 2.5069412948570745e-09 2.5069412948570745e-09
Overall NLP error.......: 2.5069412948570745e-09 2.5069412948570745e-09
Number of objective function evaluations = 21
Number of objective gradient evaluations = 20
Number of equality constraint evaluations = 21
Number of inequality constraint evaluations = 21
Number of equality constraint Jacobian evaluations = 20
Number of inequality constraint Jacobian evaluations = 20
Number of Lagrangian Hessian evaluations = 19
Total seconds in IPOPT = 0.008
EXIT: Optimal Solution Found.All the quadratic objectives and constraints of the upper level can also be added with the @constraint and @objective macros. Hence, we can write the quivalent model:
using BilevelJuMP, Ipopt
model = BilevelModel(Ipopt.Optimizer, mode = BilevelJuMP.ProductMode(1e-5))
@variable(Upper(model), x >= 2)
@variable(Lower(model), 3 <= y <= 5)\[ y \]
@objective(Upper(model), Min, x^2 - y)\[ x^2 - y \]
@constraint(Upper(model), x^2 + y^2 <= 100)\[ x^2 + y^2 \leq 100 \]
@objective(Lower(model), Min, y^2)\[ y^2 \]
optimize!(model)This is Ipopt version 3.14.4, running with linear solver MUMPS 5.4.1.
Number of nonzeros in equality constraint Jacobian...: 3
Number of nonzeros in inequality constraint Jacobian.: 8
Number of nonzeros in Lagrangian Hessian.............: 5
Total number of variables............................: 4
variables with only lower bounds: 2
variables with lower and upper bounds: 1
variables with only upper bounds: 1
Total number of equality constraints.................: 1
Total number of inequality constraints...............: 3
inequality constraints with only lower bounds: 0
inequality constraints with lower and upper bounds: 0
inequality constraints with only upper bounds: 3
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
0 1.0603999e+00 6.04e+00 1.92e+00 -1.0 0.00e+00 - 0.00e+00 0.00e+00 0
1 1.0812189e+00 5.99e+00 1.91e+00 -1.0 1.52e+01 - 1.42e-02 7.80e-03h 1
2 1.1009516e+00 2.41e+00 2.17e+01 -1.0 6.05e+00 - 9.94e-01 5.98e-01f 1
3 1.1079890e+00 1.31e+00 8.34e+00 -1.0 2.34e+00 - 6.07e-01 4.58e-01h 1
4 1.1069894e+00 8.19e-01 4.31e+01 -1.0 1.28e+00 - 1.00e+00 3.73e-01h 1
5 1.1018085e+00 4.02e-01 6.84e+01 -1.0 8.23e-01 - 1.00e+00 5.10e-01h 1
6 1.1008006e+00 1.63e-01 1.07e+02 -1.0 3.97e-01 - 1.00e+00 5.95e-01h 1
7 1.0997554e+00 7.12e-02 2.56e+02 -1.0 1.63e-01 - 1.00e+00 5.62e-01h 1
8 1.0995471e+00 2.88e-02 5.33e+02 -1.0 7.05e-02 - 1.00e+00 5.95e-01h 1
9 1.0993558e+00 1.19e-02 1.23e+03 -1.0 2.89e-02 - 1.00e+00 5.89e-01h 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
10 1.0993204e+00 4.67e-03 2.74e+03 -1.0 1.17e-02 - 1.00e+00 6.06e-01h 1
11 1.0992867e+00 1.72e-03 5.75e+03 -1.0 4.69e-03 - 1.00e+00 6.31e-01h 1
12 1.0992814e+00 5.11e-04 9.28e+03 -1.0 1.70e-03 - 1.00e+00 7.03e-01h 1
13 1.0992784e+00 2.83e-04 2.64e+04 -1.0 5.15e-04 - 1.00e+00 4.47e-01f 2
14 1.0992757e+00 6.25e-16 2.25e+00 -1.0 2.82e-04 - 1.00e+00 1.00e+00h 1
15 1.0014454e+00 5.67e-16 5.59e-05 -3.8 9.84e-02 - 1.00e+00 1.00e+00f 1
16 1.0000011e+00 7.52e-16 8.75e-08 -5.7 8.54e-04 - 1.00e+00 1.00e+00h 1
17 9.9999884e-01 2.85e-16 3.44e-01 -8.6 1.12e-06 - 8.33e-01 1.00e+00h 1
18 9.9999826e-01 4.95e-16 6.79e-08 -8.6 3.53e-06 - 1.00e+00 1.00e+00h 1
19 9.9999826e-01 2.67e-16 3.91e-13 -8.6 7.97e-09 - 1.00e+00 1.00e+00h 1
Number of Iterations....: 19
(scaled) (unscaled)
Objective...............: 9.9999825667536113e-01 9.9999825667536113e-01
Dual infeasibility......: 3.9084646282463454e-13 3.9084646282463454e-13
Constraint violation....: 2.6667434739938677e-16 2.6667434739938677e-16
Variable bound violation: 1.9373524118293517e-08 1.9373524118293517e-08
Complementarity.........: 2.5069412962253321e-09 2.5069412962253321e-09
Overall NLP error.......: 2.5069412962253321e-09 2.5069412962253321e-09
Number of objective function evaluations = 21
Number of objective gradient evaluations = 20
Number of equality constraint evaluations = 21
Number of inequality constraint evaluations = 21
Number of equality constraint Jacobian evaluations = 20
Number of inequality constraint Jacobian evaluations = 20
Number of Lagrangian Hessian evaluations = 19
Total seconds in IPOPT = 0.008
EXIT: Optimal Solution Found.General non-linear constraints and objectives
General non quadratic objectives and general non-linear constraints can not be added to the lower level, but they can be added to the upper level.
using BilevelJuMP, Ipopt
model = BilevelModel(Ipopt.Optimizer, mode = BilevelJuMP.ProductMode(1e-5))
@variable(Upper(model), x >= 2)
@variable(Lower(model), 3 <= y <= 5)\[ y \]
@NLobjective(Upper(model), Min, x^4 - sin(y))@NLconstraint(Upper(model), x^3 + y^3 <= 1000)\[ (x ^ {3.0} + y ^ {3.0}) - 1000.0 \leq 0 \]
@objective(Lower(model), Min, y^2)\[ y^2 \]
optimize!(model)This is Ipopt version 3.14.4, running with linear solver MUMPS 5.4.1.
Number of nonzeros in equality constraint Jacobian...: 3
Number of nonzeros in inequality constraint Jacobian.: 8
Number of nonzeros in Lagrangian Hessian.............: 6
Total number of variables............................: 4
variables with only lower bounds: 2
variables with lower and upper bounds: 1
variables with only upper bounds: 1
Total number of equality constraints.................: 1
Total number of inequality constraints...............: 3
inequality constraints with only lower bounds: 0
inequality constraints with lower and upper bounds: 0
inequality constraints with only upper bounds: 3
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
0 1.6528370e+01 6.04e+00 1.74e+01 -1.0 0.00e+00 - 0.00e+00 0.00e+00 0
1 1.6431443e+01 6.00e+00 1.70e+01 -1.0 7.75e+01 - 2.13e-02 7.36e-03h 1
2 1.5866572e+01 5.55e+00 1.38e+01 -1.0 6.03e+00 - 9.88e-01 7.42e-02f 1
3 1.5874426e+01 5.11e+00 1.28e+01 -1.0 5.48e+00 - 7.73e-02 7.98e-02h 1
4 1.5930765e+01 2.67e+00 3.99e+01 -1.0 5.12e+00 - 3.66e-01 4.77e-01f 1
5 1.5944050e+01 1.41e+00 1.72e+01 -1.0 2.65e+00 - 1.00e+00 4.73e-01h 1
6 1.5948686e+01 9.93e-01 1.78e+02 -1.0 1.41e+00 - 1.00e+00 2.94e-01h 1
7 1.5953508e+01 4.94e-01 3.39e+02 -1.0 9.90e-01 - 1.00e+00 5.03e-01h 1
8 1.5956468e+01 2.15e-01 7.73e+02 -1.0 4.93e-01 - 1.00e+00 5.65e-01h 1
9 1.5957763e+01 8.46e-02 1.51e+03 -1.0 2.14e-01 - 1.00e+00 6.06e-01h 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
10 1.5958299e+01 3.40e-02 3.86e+03 -1.0 8.45e-02 - 1.00e+00 5.98e-01h 1
11 1.5958521e+01 1.17e-02 7.08e+03 -1.0 3.39e-02 - 1.00e+00 6.55e-01h 1
12 1.5958615e+01 2.90e-03 9.56e+03 -1.0 1.17e-02 - 1.00e+00 7.52e-01h 1
13 1.5958643e+01 4.43e-16 2.32e+01 -1.0 2.90e-03 - 1.00e+00 1.00e+00h 1
14 1.5862147e+01 1.73e-16 7.82e-04 -2.5 3.62e-02 - 1.00e+00 1.00e+00f 1
15 1.5858882e+01 3.30e-17 4.63e-07 -5.7 1.17e-03 - 1.00e+00 1.00e+00h 1
16 1.5858879e+01 4.99e-16 5.78e-02 -8.6 2.00e-06 - 9.73e-01 1.00e+00h 1
17 1.5858879e+01 1.43e-16 2.02e-09 -8.6 2.75e-06 - 1.00e+00 1.00e+00h 1
Number of Iterations....: 17
(scaled) (unscaled)
Objective...............: 1.5858879327088321e+01 1.5858879327088321e+01
Dual infeasibility......: 2.0157011055619611e-09 2.0157011055619611e-09
Constraint violation....: 1.4305454742883153e-16 1.4305454742883153e-16
Variable bound violation: 2.7634276644761258e-08 2.7634276644761258e-08
Complementarity.........: 2.6660361311290849e-09 2.6660361311290849e-09
Overall NLP error.......: 2.6660361311290849e-09 2.6660361311290849e-09
Number of objective function evaluations = 18
Number of objective gradient evaluations = 18
Number of equality constraint evaluations = 18
Number of inequality constraint evaluations = 18
Number of equality constraint Jacobian evaluations = 18
Number of inequality constraint Jacobian evaluations = 18
Number of Lagrangian Hessian evaluations = 17
Total seconds in IPOPT = 0.007
EXIT: Optimal Solution Found.Conic constraints are supported in the lower level (see Conic Bilevel tutorial).
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