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.add_constraint(constraint, "ineq")
scenario.execute({"algo": "SLSQP", "max_iter": 10})

Out:

    INFO - 10:02:56:
    INFO - 10:02:56: *** Start MDOScenario execution ***
    INFO - 10:02:56: MDOScenario
    INFO - 10:02:56:    Disciplines: SobieskiPropulsion SobieskiAerodynamics SobieskiStructure SobieskiMission
    INFO - 10:02:56:    MDO formulation: MDF
    INFO - 10:02:56: Optimization problem:
    INFO - 10:02:56:    minimize -y_4(x_shared, x_1, x_2, x_3)
    INFO - 10:02:56:    with respect to x_1, x_2, x_3, x_shared
    INFO - 10:02:56:    subject to constraints:
    INFO - 10:02:56:       g_1(x_shared, x_1, x_2, x_3) <= 0.0
    INFO - 10:02:56:       g_2(x_shared, x_1, x_2, x_3) <= 0.0
    INFO - 10:02:56:       g_3(x_shared, x_1, x_2, x_3) <= 0.0
    INFO - 10:02:56:    over the design space:
    INFO - 10:02:56:    +----------+-------------+-------+-------------+-------+
    INFO - 10:02:56:    | name     | lower_bound | value | upper_bound | type  |
    INFO - 10:02:56:    +----------+-------------+-------+-------------+-------+
    INFO - 10:02:56:    | x_shared |     0.01    |  0.05 |     0.09    | float |
    INFO - 10:02:56:    | x_shared |    30000    | 45000 |    60000    | float |
    INFO - 10:02:56:    | x_shared |     1.4     |  1.6  |     1.8     | float |
    INFO - 10:02:56:    | x_shared |     2.5     |  5.5  |     8.5     | float |
    INFO - 10:02:56:    | x_shared |      40     |   55  |      70     | float |
    INFO - 10:02:56:    | x_shared |     500     |  1000 |     1500    | float |
    INFO - 10:02:56:    | x_1      |     0.1     |  0.25 |     0.4     | float |
    INFO - 10:02:56:    | x_1      |     0.75    |   1   |     1.25    | float |
    INFO - 10:02:56:    | x_2      |     0.75    |   1   |     1.25    | float |
    INFO - 10:02:56:    | x_3      |     0.1     |  0.5  |      1      | float |
    INFO - 10:02:56:    +----------+-------------+-------+-------------+-------+
    INFO - 10:02:56: Solving optimization problem with algorithm SLSQP:
    INFO - 10:02:56: ...   0%|          | 0/10 [00:00<?, ?it]
    INFO - 10:02:56: ...  20%|██        | 2/10 [00:00<00:00, 41.03 it/sec, obj=-2.12e+3]
    INFO - 10:02:56: ...  30%|███       | 3/10 [00:00<00:00, 24.64 it/sec, obj=-3.15e+3]
    INFO - 10:02:56: ...  40%|████      | 4/10 [00:00<00:00, 17.58 it/sec, obj=-3.96e+3]
    INFO - 10:02:56: ...  50%|█████     | 5/10 [00:00<00:00, 13.65 it/sec, obj=-3.98e+3]
    INFO - 10:02:57: ...  50%|█████     | 5/10 [00:00<00:00, 12.27 it/sec, obj=-3.98e+3]
    INFO - 10:02:57: Optimization result:
    INFO - 10:02:57:    Optimizer info:
    INFO - 10:02:57:       Status: 8
    INFO - 10:02:57:       Message: Positive directional derivative for linesearch
    INFO - 10:02:57:       Number of calls to the objective function by the optimizer: 6
    INFO - 10:02:57:    Solution:
    INFO - 10:02:57:       The solution is feasible.
    INFO - 10:02:57:       Objective: -3960.1367790933214
    INFO - 10:02:57:       Standardized constraints:
    INFO - 10:02:57:          g_1 = [-0.01805983 -0.03334555 -0.04424879 -0.05183405 -0.05732561 -0.13720865
    INFO - 10:02:57:  -0.10279135]
    INFO - 10:02:57:          g_2 = 2.9360600315442298e-06
    INFO - 10:02:57:          g_3 = [-0.76310174 -0.23689826 -0.00553375 -0.183255  ]
    INFO - 10:02:57:       Design space:
    INFO - 10:02:57:       +----------+-------------+---------------------+-------------+-------+
    INFO - 10:02:57:       | name     | lower_bound |        value        | upper_bound | type  |
    INFO - 10:02:57:       +----------+-------------+---------------------+-------------+-------+
    INFO - 10:02:57:       | x_shared |     0.01    | 0.06000073401500788 |     0.09    | float |
    INFO - 10:02:57:       | x_shared |    30000    |        60000        |    60000    | float |
    INFO - 10:02:57:       | x_shared |     1.4     |         1.4         |     1.8     | float |
    INFO - 10:02:57:       | x_shared |     2.5     |         2.5         |     8.5     | float |
    INFO - 10:02:57:       | x_shared |      40     |          70         |      70     | float |
    INFO - 10:02:57:       | x_shared |     500     |         1500        |     1500    | float |
    INFO - 10:02:57:       | x_1      |     0.1     |         0.4         |     0.4     | float |
    INFO - 10:02:57:       | x_1      |     0.75    |         0.75        |     1.25    | float |
    INFO - 10:02:57:       | x_2      |     0.75    |         0.75        |     1.25    | float |
    INFO - 10:02:57:       | x_3      |     0.1     |  0.1553801266337427 |      1      | float |
    INFO - 10:02:57:       +----------+-------------+---------------------+-------------+-------+
    INFO - 10:02:57: *** End MDOScenario execution (time: 0:00:00.828125) ***

{'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()