Gradient Sensitivity

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

from __future__ import annotations

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()

<RootLogger root (INFO)>

Description

The GradientSensitivity post-processor builds histograms of derivatives of the objective and the constraints.

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,
)

The differentiation method used by default is "user", which means that the gradient will be evaluated from the Jacobian defined in each discipline. However, some disciplines may not provide one, in that case, the gradient may be approximated with the techniques "finite_differences" or "complex_step" with the method set_differentiation_method(). The following line is shown as an example, it has no effect because it does not change the default method.

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})

    INFO - 14:43:29:
    INFO - 14:43:29: *** Start MDOScenario execution ***
    INFO - 14:43:29: MDOScenario
    INFO - 14:43:29:    Disciplines: SobieskiAerodynamics SobieskiMission SobieskiPropulsion SobieskiStructure
    INFO - 14:43:29:    MDO formulation: MDF
    INFO - 14:43:30: Optimization problem:
    INFO - 14:43:30:    minimize -y_4(x_shared, x_1, x_2, x_3)
    INFO - 14:43:30:    with respect to x_1, x_2, x_3, x_shared
    INFO - 14:43:30:    subject to constraints:
    INFO - 14:43:30:       g_1(x_shared, x_1, x_2, x_3) <= 0.0
    INFO - 14:43:30:       g_2(x_shared, x_1, x_2, x_3) <= 0.0
    INFO - 14:43:30:       g_3(x_shared, x_1, x_2, x_3) <= 0.0
    INFO - 14:43:30:    over the design space:
    INFO - 14:43:30:    +-------------+-------------+-------+-------------+-------+
    INFO - 14:43:30:    | name        | lower_bound | value | upper_bound | type  |
    INFO - 14:43:30:    +-------------+-------------+-------+-------------+-------+
    INFO - 14:43:30:    | x_shared[0] |     0.01    |  0.05 |     0.09    | float |
    INFO - 14:43:30:    | x_shared[1] |    30000    | 45000 |    60000    | float |
    INFO - 14:43:30:    | x_shared[2] |     1.4     |  1.6  |     1.8     | float |
    INFO - 14:43:30:    | x_shared[3] |     2.5     |  5.5  |     8.5     | float |
    INFO - 14:43:30:    | x_shared[4] |      40     |   55  |      70     | float |
    INFO - 14:43:30:    | x_shared[5] |     500     |  1000 |     1500    | float |
    INFO - 14:43:30:    | x_1[0]      |     0.1     |  0.25 |     0.4     | float |
    INFO - 14:43:30:    | x_1[1]      |     0.75    |   1   |     1.25    | float |
    INFO - 14:43:30:    | x_2         |     0.75    |   1   |     1.25    | float |
    INFO - 14:43:30:    | x_3         |     0.1     |  0.5  |      1      | float |
    INFO - 14:43:30:    +-------------+-------------+-------+-------------+-------+
    INFO - 14:43:30: Solving optimization problem with algorithm SLSQP:
    INFO - 14:43:30: ...   0%|          | 0/10 [00:00<?, ?it]
    INFO - 14:43:30: ...  20%|██        | 2/10 [00:00<00:00, 40.63 it/sec, obj=-2.12e+3]
 WARNING - 14:43:30: MDAJacobi has reached its maximum number of iterations but the normed residual 1.4486313079508508e-06 is still above the tolerance 1e-06.
    INFO - 14:43:30: ...  30%|███       | 3/10 [00:00<00:00, 23.56 it/sec, obj=-3.75e+3]
    INFO - 14:43:30: ...  40%|████      | 4/10 [00:00<00:00, 17.21 it/sec, obj=-4.01e+3]
 WARNING - 14:43:30: MDAJacobi has reached its maximum number of iterations but the normed residual 2.928004141058104e-06 is still above the tolerance 1e-06.
    INFO - 14:43:30: ...  50%|█████     | 5/10 [00:00<00:00, 13.11 it/sec, obj=-4.49e+3]
    INFO - 14:43:30: ...  60%|██████    | 6/10 [00:00<00:00, 11.11 it/sec, obj=-3.4e+3]
    INFO - 14:43:31: ...  80%|████████  | 8/10 [00:01<00:00,  9.46 it/sec, obj=-4.76e+3]
    INFO - 14:43:31: ... 100%|██████████| 10/10 [00:01<00:00,  8.23 it/sec, obj=-4.56e+3]
    INFO - 14:43:31: ... 100%|██████████| 10/10 [00:01<00:00,  8.21 it/sec, obj=-4.56e+3]
    INFO - 14:43:31: Optimization result:
    INFO - 14:43:31:    Optimizer info:
    INFO - 14:43:31:       Status: None
    INFO - 14:43:31:       Message: Maximum number of iterations reached. GEMSEO Stopped the driver
    INFO - 14:43:31:       Number of calls to the objective function by the optimizer: 12
    INFO - 14:43:31:    Solution:
    INFO - 14:43:31:       The solution is feasible.
    INFO - 14:43:31:       Objective: -3749.8868975554387
    INFO - 14:43:31:       Standardized constraints:
    INFO - 14:43:31:          g_1 = [-0.01671296 -0.03238836 -0.04350867 -0.05123129 -0.05681738 -0.13780658
    INFO - 14:43:31:  -0.10219342]
    INFO - 14:43:31:          g_2 = -0.0004062839430756249
    INFO - 14:43:31:          g_3 = [-0.66482546 -0.33517454 -0.11023156 -0.183255  ]
    INFO - 14:43:31:       Design space:
    INFO - 14:43:31:       +-------------+-------------+---------------------+-------------+-------+
    INFO - 14:43:31:       | name        | lower_bound |        value        | upper_bound | type  |
    INFO - 14:43:31:       +-------------+-------------+---------------------+-------------+-------+
    INFO - 14:43:31:       | x_shared[0] |     0.01    | 0.05989842901423112 |     0.09    | float |
    INFO - 14:43:31:       | x_shared[1] |    30000    |  59853.73840058666  |    60000    | float |
    INFO - 14:43:31:       | x_shared[2] |     1.4     |         1.4         |     1.8     | float |
    INFO - 14:43:31:       | x_shared[3] |     2.5     |  2.527371250092273  |     8.5     | float |
    INFO - 14:43:31:       | x_shared[4] |      40     |  69.86825198198687  |      70     | float |
    INFO - 14:43:31:       | x_shared[5] |     500     |  1495.734648986894  |     1500    | float |
    INFO - 14:43:31:       | x_1[0]      |     0.1     |         0.4         |     0.4     | float |
    INFO - 14:43:31:       | x_1[1]      |     0.75    |  0.7521124139939552 |     1.25    | float |
    INFO - 14:43:31:       | x_2         |     0.75    |  0.7520888531444992 |     1.25    | float |
    INFO - 14:43:31:       | x_3         |     0.1     |  0.1398000762238233 |      1      | float |
    INFO - 14:43:31:       +-------------+-------------+---------------------+-------------+-------+
    INFO - 14:43:31: *** End MDOScenario execution (time: 0:00:01.233712) ***

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

Post-process scenario

Lastly, we post-process the scenario by means of the GradientSensitivity post-processor which builds histograms of derivatives of objective and constraints. The sensitivities shown in the plot are calculated with the gradient at the optimum or the least-non feasible point when the result is not feasible. One may choose any other iteration instead.

Note

In some cases, the iteration that is being used to compute the sensitivities corresponds to a point for which the algorithm did not request the evaluation of the gradients, and a ValueError is raised. A way to avoid this issue is to set the option compute_missing_gradients of GradientSensitivity to True, this way GEMSEO will compute the gradients for the requested iteration if they are not available.

Warning

Please note that this extra computation may be expensive depending on the OptimizationProblem defined by the user. Additionally, keep in mind that GEMSEO cannot compute missing gradients for an OptimizationProblem that was imported from an HDF5 file.

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(
    "GradientSensitivity", save=False, show=False, compute_missing_gradients=True
)
# Workaround for HTML rendering, instead of ``show=True``
plt.show()

 WARNING - 14:43:31: MDAJacobi has reached its maximum number of iterations but the normed residual 1.4486313079508508e-06 is still above the tolerance 1e-06.
/home/docs/checkouts/readthedocs.org/user_builds/gemseo/envs/4.1.0/lib/python3.9/site-packages/gemseo/post/gradient_sensitivity.py:201: UserWarning: FixedFormatter should only be used together with FixedLocator
  axe.set_xticklabels(design_names, fontsize=font_size, rotation=rotation)