Pareto front¶

In this example, we illustrate the use of the ParetoFront plot on the Sobieski’s SSBJ problem.

from gemseo.api import configure_logger
from gemseo.api import create_discipline
from gemseo.api import create_scenario
from gemseo.problems.sobieski.core.problem import SobieskiProblem
from matplotlib import pyplot as plt


Import¶

The first step is to import some functions from the API and a method to get the design space.

configure_logger()


Out:

<RootLogger root (INFO)>


Description¶

The ParetoFront post-processing generates a plot or a matrix of plots (if there are more than 2 objectives). It indicates in red the locally nondominated points for the current objectives, and in green the globally (all objectives) Pareto optimal points.

Create disciplines¶

At this point, we instantiate the disciplines of Sobieski’s SSBJ problem: Propulsion, Aerodynamics, Structure and Mission

disciplines = create_discipline(
[
"SobieskiPropulsion",
"SobieskiAerodynamics",
"SobieskiStructure",
"SobieskiMission",
]
)


Create design space¶

We also read the design space from the SobieskiProblem.

design_space = SobieskiProblem().design_space


Create and execute scenario¶

The next step is to build an MDO scenario in order to maximize the range, encoded ‘y_4’, with respect to the design parameters, while satisfying the inequality constraints ‘g_1’, ‘g_2’ and ‘g_3’. We can use the MDF formulation, the SLSQP optimization algorithm and a maximum number of iterations equal to 100.

scenario = create_scenario(
disciplines,
formulation="MDF",
objective_name="y_4",
maximize_objective=True,
design_space=design_space,
)
scenario.set_differentiation_method("user")
for constraint in ["g_1", "g_2", "g_3"]:
scenario.execute({"algo": "SLSQP", "max_iter": 10})


Out:

    INFO - 15:01:09:
INFO - 15:01:09: *** Start MDOScenario execution ***
INFO - 15:01:09: MDOScenario
INFO - 15:01:09:    Disciplines: SobieskiPropulsion SobieskiAerodynamics SobieskiStructure SobieskiMission
INFO - 15:01:09:    MDO formulation: MDF
INFO - 15:01:09: Optimization problem:
INFO - 15:01:09:    minimize -y_4(x_shared, x_1, x_2, x_3)
INFO - 15:01:09:    with respect to x_1, x_2, x_3, x_shared
INFO - 15:01:09:    subject to constraints:
INFO - 15:01:09:       g_1(x_shared, x_1, x_2, x_3) <= 0.0
INFO - 15:01:09:       g_2(x_shared, x_1, x_2, x_3) <= 0.0
INFO - 15:01:09:       g_3(x_shared, x_1, x_2, x_3) <= 0.0
INFO - 15:01:09:    over the design space:
INFO - 15:01:09:    +----------+-------------+-------+-------------+-------+
INFO - 15:01:09:    | name     | lower_bound | value | upper_bound | type  |
INFO - 15:01:09:    +----------+-------------+-------+-------------+-------+
INFO - 15:01:09:    | x_shared |     0.01    |  0.05 |     0.09    | float |
INFO - 15:01:09:    | x_shared |    30000    | 45000 |    60000    | float |
INFO - 15:01:09:    | x_shared |     1.4     |  1.6  |     1.8     | float |
INFO - 15:01:09:    | x_shared |     2.5     |  5.5  |     8.5     | float |
INFO - 15:01:09:    | x_shared |      40     |   55  |      70     | float |
INFO - 15:01:09:    | x_shared |     500     |  1000 |     1500    | float |
INFO - 15:01:09:    | x_1      |     0.1     |  0.25 |     0.4     | float |
INFO - 15:01:09:    | x_1      |     0.75    |   1   |     1.25    | float |
INFO - 15:01:09:    | x_2      |     0.75    |   1   |     1.25    | float |
INFO - 15:01:09:    | x_3      |     0.1     |  0.5  |      1      | float |
INFO - 15:01:09:    +----------+-------------+-------+-------------+-------+
INFO - 15:01:09: Solving optimization problem with algorithm SLSQP:
INFO - 15:01:09: ...   0%|          | 0/10 [00:00<?, ?it]
INFO - 15:01:10: ...  20%|██        | 2/10 [00:00<00:00, 41.52 it/sec, obj=-2.12e+3]
WARNING - 15:01:10: MDAJacobi has reached its maximum number of iterations but the normed residual 9.482098563532576e-06 is still above the tolerance 1e-06.
INFO - 15:01:10: ...  30%|███       | 3/10 [00:00<00:00, 23.72 it/sec, obj=-3.76e+3]
INFO - 15:01:10: ...  40%|████      | 4/10 [00:00<00:00, 17.14 it/sec, obj=-3.96e+3]
INFO - 15:01:10: ...  50%|█████     | 5/10 [00:00<00:00, 13.44 it/sec, obj=-3.96e+3]
INFO - 15:01:10: ...  70%|███████   | 7/10 [00:00<00:00, 10.97 it/sec, obj=-3.96e+3]
INFO - 15:01:10: ...  90%|█████████ | 9/10 [00:01<00:00,  9.26 it/sec, obj=-3.96e+3]
WARNING - 15:01:11: Optimization found no feasible point !  The least infeasible point is selected.
INFO - 15:01:11: ... 100%|██████████| 10/10 [00:01<00:00,  8.56 it/sec, obj=-3.96e+3]
INFO - 15:01:11: Optimization result:
INFO - 15:01:11:    Optimizer info:
INFO - 15:01:11:       Status: None
INFO - 15:01:11:       Message: Maximum number of iterations reached. GEMSEO Stopped the driver
INFO - 15:01:11:       Number of calls to the objective function by the optimizer: 12
INFO - 15:01:11:    Solution:
WARNING - 15:01:11:       The solution is not feasible.
INFO - 15:01:11:       Objective: -3963.793570013662
INFO - 15:01:11:       Standardized constraints:
INFO - 15:01:11:          g_1 = [-0.01810854 -0.03338059 -0.04427602 -0.0518563  -0.05734441 -0.13720865
INFO - 15:01:11:  -0.10279135]
INFO - 15:01:11:          g_2 = 1.9000415800052295e-05
INFO - 15:01:11:          g_3 = [-7.67474555e-01 -2.32525445e-01  4.34281075e-04 -1.83255000e-01]
INFO - 15:01:11:       Design space:
INFO - 15:01:11:       +----------+-------------+---------------------+-------------+-------+
INFO - 15:01:11:       | name     | lower_bound |        value        | upper_bound | type  |
INFO - 15:01:11:       +----------+-------------+---------------------+-------------+-------+
INFO - 15:01:11:       | x_shared |     0.01    | 0.06000475010395003 |     0.09    | float |
INFO - 15:01:11:       | x_shared |    30000    |        60000        |    60000    | float |
INFO - 15:01:11:       | x_shared |     1.4     |         1.4         |     1.8     | float |
INFO - 15:01:11:       | x_shared |     2.5     |         2.5         |     8.5     | float |
INFO - 15:01:11:       | x_shared |      40     |          70         |      70     | float |
INFO - 15:01:11:       | x_shared |     500     |         1500        |     1500    | float |
INFO - 15:01:11:       | x_1      |     0.1     |         0.4         |     0.4     | float |
INFO - 15:01:11:       | x_1      |     0.75    |         0.75        |     1.25    | float |
INFO - 15:01:11:       | x_2      |     0.75    |         0.75        |     1.25    | float |
INFO - 15:01:11:       | x_3      |     0.1     |  0.1563125997824079 |      1      | float |
INFO - 15:01:11:       +----------+-------------+---------------------+-------------+-------+
INFO - 15:01:11: *** End MDOScenario execution (time: 0:00:01.181339) ***

{'max_iter': 10, 'algo': 'SLSQP'}


Post-process scenario¶

Lastly, we post-process the scenario by means of the ParetoFront.

Tip

Each post-processing method requires different inputs and offers a variety of customization options. Use the API function get_post_processing_options_schema() to print a table with the options for any post-processing algorithm. Or refer to our dedicated page: Post-processing algorithms.

scenario.post_process("ParetoFront", objectives=["g_3", "-y_4"], save=False, show=False)
# Workaround for HTML rendering, instead of show=True
plt.show()


Total running time of the script: ( 0 minutes 1.750 seconds)

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