Variables influence

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

from __future__ import division, unicode_literals

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.

from gemseo.api import configure_logger, create_discipline, create_scenario
from gemseo.problems.sobieski.core import SobieskiProblem

configure_logger()

Out:

<RootLogger root (INFO)>

Create disciplines

Then, we instantiate the disciplines of the 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().read_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 - 09:25:50:
    INFO - 09:25:50: *** Start MDO Scenario execution ***
    INFO - 09:25:50: MDOScenario
    INFO - 09:25:50:    Disciplines: SobieskiPropulsion SobieskiAerodynamics SobieskiStructure SobieskiMission
    INFO - 09:25:50:    MDOFormulation: MDF
    INFO - 09:25:50:    Algorithm: SLSQP
    INFO - 09:25:50: Optimization problem:
    INFO - 09:25:50:    Minimize: -y_4(x_shared, x_1, x_2, x_3)
    INFO - 09:25:50:    With respect to: x_shared, x_1, x_2, x_3
    INFO - 09:25:50:    Subject to constraints:
    INFO - 09:25:50:       g_1(x_shared, x_1, x_2, x_3) <= 0.0
    INFO - 09:25:50:       g_2(x_shared, x_1, x_2, x_3) <= 0.0
    INFO - 09:25:50:       g_3(x_shared, x_1, x_2, x_3) <= 0.0
    INFO - 09:25:50: Design Space:
    INFO - 09:25:50: +----------+-------------+-------+-------------+-------+
    INFO - 09:25:50: | name     | lower_bound | value | upper_bound | type  |
    INFO - 09:25:50: +----------+-------------+-------+-------------+-------+
    INFO - 09:25:50: | x_shared |     0.01    |  0.05 |     0.09    | float |
    INFO - 09:25:50: | x_shared |    30000    | 45000 |    60000    | float |
    INFO - 09:25:50: | x_shared |     1.4     |  1.6  |     1.8     | float |
    INFO - 09:25:50: | x_shared |     2.5     |  5.5  |     8.5     | float |
    INFO - 09:25:50: | x_shared |      40     |   55  |      70     | float |
    INFO - 09:25:50: | x_shared |     500     |  1000 |     1500    | float |
    INFO - 09:25:50: | x_1      |     0.1     |  0.25 |     0.4     | float |
    INFO - 09:25:50: | x_1      |     0.75    |   1   |     1.25    | float |
    INFO - 09:25:50: | x_2      |     0.75    |   1   |     1.25    | float |
    INFO - 09:25:50: | x_3      |     0.1     |  0.5  |      1      | float |
    INFO - 09:25:50: +----------+-------------+-------+-------------+-------+
    INFO - 09:25:50: Optimization:   0%|          | 0/10 [00:00<?, ?it]
    INFO - 09:25:50: Optimization:  20%|██        | 2/10 [00:00<00:00, 67.11 it/sec, obj=536]
    INFO - 09:25:50: Optimization:  40%|████      | 4/10 [00:00<00:00, 21.57 it/sec, obj=3.8e+3]
 WARNING - 09:25:51: Optimization found no feasible point !  The least infeasible point is selected.
    INFO - 09:25:51: Optimization:  40%|████      | 4/10 [00:00<00:00, 16.27 it/sec, obj=3.96e+3]
    INFO - 09:25:51: Optimization result:
    INFO - 09:25:51: Objective value = 3795.0851933441872
    INFO - 09:25:51: The result is not feasible.
    INFO - 09:25:51: Status: 8
    INFO - 09:25:51: Optimizer message: Positive directional derivative for linesearch
    INFO - 09:25:51: Number of calls to the objective function by the optimizer: 5
    INFO - 09:25:51: Constraints values w.r.t. 0:
    INFO - 09:25:51:    g_1 = [-0.01940553 -0.03430815 -0.04499528 -0.05244303 -0.05783964 -0.13706197
    INFO - 09:25:51:  -0.10293803]
    INFO - 09:25:51:    g_2 = 0.0003917260521535404
    INFO - 09:25:51:    g_3 = [-0.6301543  -0.3698457  -0.14096439 -0.18315803]
    INFO - 09:25:51: Design Space:
    INFO - 09:25:51: +----------+-------------+---------------------+-------------+-------+
    INFO - 09:25:51: | name     | lower_bound |        value        | upper_bound | type  |
    INFO - 09:25:51: +----------+-------------+---------------------+-------------+-------+
    INFO - 09:25:51: | x_shared |     0.01    | 0.06009793151303839 |     0.09    | float |
    INFO - 09:25:51: | x_shared |    30000    |        60000        |    60000    | float |
    INFO - 09:25:51: | x_shared |     1.4     |  1.400744940049757  |     1.8     | float |
    INFO - 09:25:51: | x_shared |     2.5     |         2.5         |     8.5     | float |
    INFO - 09:25:51: | x_shared |      40     |          70         |      70     | float |
    INFO - 09:25:51: | x_shared |     500     |         1500        |     1500    | float |
    INFO - 09:25:51: | x_1      |     0.1     |  0.3991428961174674 |     0.4     | float |
    INFO - 09:25:51: | x_1      |     0.75    |         0.75        |     1.25    | float |
    INFO - 09:25:51: | x_2      |     0.75    |         0.75        |     1.25    | float |
    INFO - 09:25:51: | x_3      |     0.1     |  0.1343078243802689 |      1      | float |
    INFO - 09:25:51: +----------+-------------+---------------------+-------------+-------+
    INFO - 09:25:51: *** MDO Scenario run terminated in 0:00:00.631379 ***

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

Post-process scenario

Lastly, we post-process the scenario by means of the BasicHistory plot which performs first order variable influence analysis by computing df/dxi * (xi* - xi0) where xi0 is the initial value of the variable and xi* is the optimal value of the variable plots any of the constraint or objective functions w.r.t. optimization iterations or sampling snapshots.

scenario.post_process(
    "VariableInfluence", save=False, show=False, figsize_x=15, figsize_y=12
)
# Workaround for HTML rendering, instead of ``show=True``
plt.show()
Partial variation of the functions wrt design variables, 9 variables required to explain 99% of -y_4 variations, 5 variables required to explain 99% of g_1_0 variations, 5 variables required to explain 99% of g_1_1 variations, 5 variables required to explain 99% of g_1_2 variations, 5 variables required to explain 99% of g_1_3 variations, 5 variables required to explain 99% of g_1_4 variations, 4 variables required to explain 99% of g_1_5 variations, 4 variables required to explain 99% of g_1_6 variations, 1 variables required to explain 99% of g_2 variations, 7 variables required to explain 99% of g_3_0 variations, 7 variables required to explain 99% of g_3_1 variations, 3 variables required to explain 99% of g_3_2 variations, 3 variables required to explain 99% of g_3_3 variations

Out:

 WARNING - 09:25:51: Optimization found no feasible point !  The least infeasible point is selected.
    INFO - 09:25:51: VariableInfluence for function -y_4
    INFO - 09:25:51: Most influent variables indices to explain % of the function variation : 99
    INFO - 09:25:51: [1 4 3 2 5 9 7 8 0]
/home/docs/checkouts/readthedocs.org/user_builds/gemseo/conda/3.1.0/lib/python3.8/site-packages/gemseo/post/variable_influence.py:231: UserWarning: FixedFormatter should only be used together with FixedLocator
  axe.set_xticklabels(x_labels, fontsize=12, rotation=90)
    INFO - 09:25:51: VariableInfluence for function g_1_0
    INFO - 09:25:51: Most influent variables indices to explain % of the function variation : 99
    INFO - 09:25:51: [0 7 3 5 6]
    INFO - 09:25:51: VariableInfluence for function g_1_1
    INFO - 09:25:51: Most influent variables indices to explain % of the function variation : 99
    INFO - 09:25:51: [7 0 3 5 6]
    INFO - 09:25:51: VariableInfluence for function g_1_2
    INFO - 09:25:51: Most influent variables indices to explain % of the function variation : 99
    INFO - 09:25:51: [7 0 3 5 6]
    INFO - 09:25:51: VariableInfluence for function g_1_3
    INFO - 09:25:51: Most influent variables indices to explain % of the function variation : 99
    INFO - 09:25:51: [7 0 3 5 6]
    INFO - 09:25:51: VariableInfluence for function g_1_4
    INFO - 09:25:51: Most influent variables indices to explain % of the function variation : 99
    INFO - 09:25:51: [7 0 3 5 6]
    INFO - 09:25:51: VariableInfluence for function g_1_5
    INFO - 09:25:51: Most influent variables indices to explain % of the function variation : 99
    INFO - 09:25:51: [3 7 5 6]
    INFO - 09:25:51: VariableInfluence for function g_1_6
    INFO - 09:25:51: Most influent variables indices to explain % of the function variation : 99
    INFO - 09:25:51: [3 7 5 6]
    INFO - 09:25:51: VariableInfluence for function g_2
    INFO - 09:25:51: Most influent variables indices to explain % of the function variation : 99
    INFO - 09:25:51: [0]
    INFO - 09:25:52: VariableInfluence for function g_3_0
    INFO - 09:25:52: Most influent variables indices to explain % of the function variation : 99
    INFO - 09:25:52: [9 1 5 2 4 0 8]
    INFO - 09:25:52: VariableInfluence for function g_3_1
    INFO - 09:25:52: Most influent variables indices to explain % of the function variation : 99
    INFO - 09:25:52: [9 1 5 2 4 0 8]
    INFO - 09:25:52: VariableInfluence for function g_3_2
    INFO - 09:25:52: Most influent variables indices to explain % of the function variation : 99
    INFO - 09:25:52: [1 9 2]
    INFO - 09:25:52: VariableInfluence for function g_3_3
    INFO - 09:25:52: Most influent variables indices to explain % of the function variation : 99
    INFO - 09:25:52: [9 1 2]

Total running time of the script: ( 0 minutes 2.449 seconds)

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