MDF-based MDO on the Sobieski SSBJ test case

from __future__ import annotations

from gemseo.api import configure_logger
from gemseo.api import create_discipline
from gemseo.api import create_scenario
from gemseo.api import generate_n2_plot
from gemseo.problems.sobieski.core.problem import SobieskiProblem

configure_logger()
<RootLogger root (INFO)>

Instantiate the disciplines

First, we instantiate the four disciplines of the use case: SobieskiPropulsion, SobieskiAerodynamics, SobieskiMission and SobieskiStructure.

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

We can quickly access the most relevant information of any discipline (name, inputs, and outputs) with Python’s print() function. Moreover, we can get the default input values of a discipline with the attribute MDODiscipline.default_inputs

for discipline in disciplines:
    print(discipline)
    print(f"Default inputs: {discipline.default_inputs}")
SobieskiPropulsion
Default inputs: {'y_23': array([12562.01206488]), 'x_3': array([0.5]), 'x_shared': array([5.0e-02, 4.5e+04, 1.6e+00, 5.5e+00, 5.5e+01, 1.0e+03]), 'c_3': array([4360.])}
SobieskiAerodynamics
Default inputs: {'x_2': array([1.]), 'y_32': array([0.50279625]), 'x_shared': array([5.0e-02, 4.5e+04, 1.6e+00, 5.5e+00, 5.5e+01, 1.0e+03]), 'y_12': array([5.06069742e+04, 9.50000000e-01]), 'c_4': array([0.01375])}
SobieskiMission
Default inputs: {'y_14': array([50606.9741711 ,  7306.20262124]), 'x_shared': array([5.0e-02, 4.5e+04, 1.6e+00, 5.5e+00, 5.5e+01, 1.0e+03]), 'y_24': array([4.15006276]), 'y_34': array([1.10754577])}
SobieskiStructure
Default inputs: {'y_21': array([50606.9741711]), 'y_31': array([6354.32430691]), 'x_1': array([0.25, 1.  ]), 'x_shared': array([5.0e-02, 4.5e+04, 1.6e+00, 5.5e+00, 5.5e+01, 1.0e+03]), 'c_0': array([2000.]), 'c_1': array([25000.]), 'c_2': array([6.])}

You may also be interested in plotting the couplings of your disciplines. A quick way of getting this information is the API function generate_n2_plot(). A much more detailed explanation of coupling visualization is available here.

generate_n2_plot(disciplines, save=False, show=True)
plot sobieski mdf example

Build, execute and post-process the scenario

Then, we build the scenario which links the disciplines with the formulation and the optimization algorithm. Here, we use the MDF formulation. We tell the scenario to minimize -y_4 instead of minimizing y_4 (range), which is the default option.

Instantiate the scenario

During the instantiation of the scenario, we provide some options for the MDF formulations:

formulation_options = {
    "tolerance": 1e-10,
    "max_mda_iter": 50,
    "warm_start": True,
    "use_lu_fact": True,
    "linear_solver_tolerance": 1e-15,
}
  • 'warm_start: warm starts MDA,

  • 'warm_start: optimize the adjoints resolution by storing the Jacobian matrix LU factorization for the multiple RHS (objective + constraints). This saves CPU time if you can pay for the memory and have the full Jacobians available, not just matrix vector products.

  • 'linear_solver_tolerance': set the linear solver tolerance, idem we need full convergence

design_space = SobieskiProblem().design_space
print(design_space)
scenario = create_scenario(
    disciplines,
    "MDF",
    objective_name="y_4",
    design_space=design_space,
    maximize_objective=True,
    **formulation_options,
)
Design space:
+-------------+-------------+--------------------+-------------+-------+
| name        | lower_bound |       value        | upper_bound | type  |
+-------------+-------------+--------------------+-------------+-------+
| x_shared[0] |     0.01    |        0.05        |     0.09    | float |
| x_shared[1] |    30000    |       45000        |    60000    | float |
| x_shared[2] |     1.4     |        1.6         |     1.8     | float |
| x_shared[3] |     2.5     |        5.5         |     8.5     | float |
| x_shared[4] |      40     |         55         |      70     | float |
| x_shared[5] |     500     |        1000        |     1500    | float |
| x_1[0]      |     0.1     |        0.25        |     0.4     | float |
| x_1[1]      |     0.75    |         1          |     1.25    | float |
| x_2         |     0.75    |         1          |     1.25    | float |
| x_3         |     0.1     |        0.5         |      1      | float |
| y_14[0]     |    24850    |   50606.9741711    |    77100    | float |
| y_14[1]     |    -7700    |   7306.20262124    |    45000    | float |
| y_32        |    0.235    | 0.5027962499999999 |    0.795    | float |
| y_31        |     2960    |   6354.32430691    |    10185    | float |
| y_24        |     0.44    |     4.15006276     |    11.13    | float |
| y_34        |     0.44    |     1.10754577     |     1.98    | float |
| y_23        |     3365    |   12194.2671934    |    26400    | float |
| y_21        |    24850    |   50606.9741711    |    77250    | float |
| y_12[0]     |    24850    |     50606.9742     |    77250    | float |
| y_12[1]     |     0.45    |        0.95        |     1.5     | float |
+-------------+-------------+--------------------+-------------+-------+

Set the design constraints

for c_name in ["g_1", "g_2", "g_3"]:
    scenario.add_constraint(c_name, "ineq")

XDSMIZE the scenario

Generate the XDSM file on the fly, setting print_statuses=True will print the status in the console html_output (default True), will generate a self-contained HTML file, that can be automatically open using open_browser=True

scenario.xdsmize()
INFO - 11:19:21: Generating HTML XDSM file in : xdsm.html

Define the algorithm inputs

We set the maximum number of iterations, the optimizer and the optimizer options. Algorithm specific options are passed there. Use get_algorithm_options_schema() API function for more information or read the documentation.

Here ftol_rel option is a stop criteria based on the relative difference in the objective between two iterates ineq_tolerance the tolerance determination of the optimum; this is specific to the GEMSEO wrapping and not in the solver.

algo_options = {
    "ftol_rel": 1e-10,
    "ineq_tolerance": 2e-3,
    "normalize_design_space": True,
}
scn_inputs = {"max_iter": 10, "algo": "SLSQP", "algo_options": algo_options}

See also

We can also generate a backup file for the optimization, as well as plots on the fly of the optimization history if option generate_opt_plot is True. This slows down a lot the process, here since SSBJ is very light

scenario.set_optimization_history_backup(file_path="mdf_backup.h5",
                                         each_new_iter=True,
                                         each_store=False, erase=True,
                                         pre_load=False,
                                         generate_opt_plot=True)

Execute the scenario

scenario.execute(scn_inputs)
    INFO - 11:19:21:
    INFO - 11:19:21: *** Start MDOScenario execution ***
    INFO - 11:19:21: MDOScenario
    INFO - 11:19:21:    Disciplines: SobieskiAerodynamics SobieskiMission SobieskiPropulsion SobieskiStructure
    INFO - 11:19:21:    MDO formulation: MDF
    INFO - 11:19:21: Optimization problem:
    INFO - 11:19:21:    minimize -y_4(x_shared, x_1, x_2, x_3)
    INFO - 11:19:21:    with respect to x_1, x_2, x_3, x_shared
    INFO - 11:19:21:    subject to constraints:
    INFO - 11:19:21:       g_1(x_shared, x_1, x_2, x_3) <= 0.0
    INFO - 11:19:21:       g_2(x_shared, x_1, x_2, x_3) <= 0.0
    INFO - 11:19:21:       g_3(x_shared, x_1, x_2, x_3) <= 0.0
    INFO - 11:19:21:    over the design space:
    INFO - 11:19:21:    +-------------+-------------+-------+-------------+-------+
    INFO - 11:19:21:    | name        | lower_bound | value | upper_bound | type  |
    INFO - 11:19:21:    +-------------+-------------+-------+-------------+-------+
    INFO - 11:19:21:    | x_shared[0] |     0.01    |  0.05 |     0.09    | float |
    INFO - 11:19:21:    | x_shared[1] |    30000    | 45000 |    60000    | float |
    INFO - 11:19:21:    | x_shared[2] |     1.4     |  1.6  |     1.8     | float |
    INFO - 11:19:21:    | x_shared[3] |     2.5     |  5.5  |     8.5     | float |
    INFO - 11:19:21:    | x_shared[4] |      40     |   55  |      70     | float |
    INFO - 11:19:21:    | x_shared[5] |     500     |  1000 |     1500    | float |
    INFO - 11:19:21:    | x_1[0]      |     0.1     |  0.25 |     0.4     | float |
    INFO - 11:19:21:    | x_1[1]      |     0.75    |   1   |     1.25    | float |
    INFO - 11:19:21:    | x_2         |     0.75    |   1   |     1.25    | float |
    INFO - 11:19:21:    | x_3         |     0.1     |  0.5  |      1      | float |
    INFO - 11:19:21:    +-------------+-------------+-------+-------------+-------+
    INFO - 11:19:21: Solving optimization problem with algorithm SLSQP:
    INFO - 11:19:21: ...   0%|          | 0/10 [00:00<?, ?it]
    INFO - 11:19:21: ...  10%|█         | 1/10 [00:00<00:00, 99.50 it/sec, obj=-536]
    INFO - 11:19:22: ...  30%|███       | 3/10 [00:00<00:00, 27.42 it/sec, obj=-3.64e+3]
    INFO - 11:19:22: ...  40%|████      | 4/10 [00:00<00:00, 21.49 it/sec, obj=-4.01e+3]
    INFO - 11:19:22: ...  50%|█████     | 5/10 [00:00<00:00, 16.38 it/sec, obj=-4.51e+3]
 WARNING - 11:19:22: Optimization found no feasible point !  The least infeasible point is selected.
    INFO - 11:19:22: ...  50%|█████     | 5/10 [00:00<00:00, 15.81 it/sec, obj=-4.51e+3]
    INFO - 11:19:22: Optimization result:
    INFO - 11:19:22:    Optimizer info:
    INFO - 11:19:22:       Status: 8
    INFO - 11:19:22:       Message: Positive directional derivative for linesearch
    INFO - 11:19:22:       Number of calls to the objective function by the optimizer: 6
    INFO - 11:19:22:    Solution:
 WARNING - 11:19:22:       The solution is not feasible.
    INFO - 11:19:22:       Objective: -3643.2646614710907
    INFO - 11:19:22:       Standardized constraints:
    INFO - 11:19:22:          g_1 = [-0.02648406 -0.03933265 -0.04887821 -0.05560436 -0.06050463 -0.13630937
    INFO - 11:19:22:  -0.10369063]
    INFO - 11:19:22:          g_2 = 0.002396186936539646
    INFO - 11:19:22:          g_3 = [-0.50236422 -0.49763578 -0.23179683 -0.18266046]
    INFO - 11:19:22:       Design space:
    INFO - 11:19:22:       +-------------+-------------+---------------------+-------------+-------+
    INFO - 11:19:22:       | name        | lower_bound |        value        | upper_bound | type  |
    INFO - 11:19:22:       +-------------+-------------+---------------------+-------------+-------+
    INFO - 11:19:22:       | x_shared[0] |     0.01    | 0.06059904673413494 |     0.09    | float |
    INFO - 11:19:22:       | x_shared[1] |    30000    |        60000        |    60000    | float |
    INFO - 11:19:22:       | x_shared[2] |     1.4     |   1.40455692827199  |     1.8     | float |
    INFO - 11:19:22:       | x_shared[3] |     2.5     |         2.5         |     8.5     | float |
    INFO - 11:19:22:       | x_shared[4] |      40     |          70         |      70     | float |
    INFO - 11:19:22:       | x_shared[5] |     500     |         1500        |     1500    | float |
    INFO - 11:19:22:       | x_1[0]      |     0.1     |  0.3947569275200153 |     0.4     | float |
    INFO - 11:19:22:       | x_1[1]      |     0.75    |         0.75        |     1.25    | float |
    INFO - 11:19:22:       | x_2         |     0.75    |         0.75        |     1.25    | float |
    INFO - 11:19:22:       | x_3         |     0.1     |  0.1205118700740507 |      1      | float |
    INFO - 11:19:22:       +-------------+-------------+---------------------+-------------+-------+
    INFO - 11:19:22: *** End MDOScenario execution (time: 0:00:00.648136) ***

{'max_iter': 10, 'algo': 'SLSQP', 'algo_options': {'ftol_rel': 1e-10, 'ineq_tolerance': 0.002, 'normalize_design_space': True}}

Save the optimization history

We can save the whole optimization problem and its history for further post processing:

scenario.save_optimization_history("mdf_history.h5", file_format="hdf5")
INFO - 11:19:22: Export optimization problem to file: mdf_history.h5

We can also save only calls to functions and design variables history:

scenario.save_optimization_history("mdf_history.xml", file_format="ggobi")
INFO - 11:19:22: Export to ggobi for functions: ['-y_4', 'Iter', 'g_1', 'g_2', 'g_3']
INFO - 11:19:22: Export to ggobi file: mdf_history.xml

Post-process the results

Plot the optimization history view

scenario.post_process("OptHistoryView", save=False, show=True)
  • Evolution of the optimization variables
  • Evolution of the objective value
  • Distance to the optimum
  • Hessian diagonal approximation
  • Evolution of the inequality constraints
 WARNING - 11:19:22: Optimization found no feasible point !  The least infeasible point is selected.

<gemseo.post.opt_history_view.OptHistoryView object at 0x7f3d0b6b78b0>

Plot the basic history view

scenario.post_process(
    "BasicHistory", variable_names=["x_shared"], save=False, show=True
)
History plot
<gemseo.post.basic_history.BasicHistory object at 0x7f3d0b92e250>

Plot the constraints and objective history

scenario.post_process("ObjConstrHist", save=False, show=True)
Evolution of the objective value and maximal constraint
<gemseo.post.obj_constr_hist.ObjConstrHist object at 0x7f3d22e93e80>

Plot the constraints history

scenario.post_process(
    "ConstraintsHistory",
    constraint_names=["g_1", "g_2", "g_3"],
    save=False,
    show=True,
)
Evolution of the constraints w.r.t. iterations, g_1 (0), g_1 (1), g_1 (2), g_1 (3), g_1 (4), g_1 (5), g_1 (6), g_2, g_3 (0), g_3 (1), g_3 (2), g_3 (3)
<gemseo.post.constraints_history.ConstraintsHistory object at 0x7f3d2287a850>

Plot the constraints history using a radar chart

scenario.post_process(
    "RadarChart",
    constraint_names=["g_1", "g_2", "g_3"],
    save=False,
    show=True,
)
Constraints at iteration 2 (optimum)
<gemseo.post.radar_chart.RadarChart object at 0x7f3d20739490>

Plot the quadratic approximation of the objective

scenario.post_process("QuadApprox", function="-y_4", save=False, show=True)
  • Hessian matrix SR1 approximation of -y_4
  • plot sobieski mdf example
<gemseo.post.quad_approx.QuadApprox object at 0x7f3d20335be0>

Plot the functions using a SOM

scenario.post_process("SOM", save=False, show=True)
Self Organizing Maps of the design space, -y_4, g_1_0, g_1_1, g_1_2, g_1_3, g_1_4, g_1_5, g_1_6, g_2, g_3_0, g_3_1, g_3_2, g_3_3
    INFO - 11:19:26: Building Self Organizing Map from optimization history:
    INFO - 11:19:26:     Number of neurons in x direction = 4
    INFO - 11:19:26:     Number of neurons in y direction = 4

<gemseo.post.som.SOM object at 0x7f3d20335c70>

Plot the scatter matrix of variables of interest

scenario.post_process(
    "ScatterPlotMatrix",
    variable_names=["-y_4", "g_1"],
    save=False,
    show=True,
    fig_size=(14, 14),
)
plot sobieski mdf example
<gemseo.post.scatter_mat.ScatterPlotMatrix object at 0x7f3d20b50dc0>

Plot the variables using the parallel coordinates

scenario.post_process("ParallelCoordinates", save=False, show=True)
  • Design variables history colored by '-y_4' value
  • Objective function and constraints history colored by '-y_4' value.
<gemseo.post.para_coord.ParallelCoordinates object at 0x7f3d2113faf0>

Plot the robustness of the solution

scenario.post_process("Robustness", save=True, show=True)
<gemseo.post.robustness.Robustness object at 0x7f3d0ab44430>

Plot the influence of the design variables

scenario.post_process("VariableInfluence", fig_size=(14, 14), save=False, show=True)
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
 WARNING - 11:19:30: Optimization found no feasible point !  The least infeasible point is selected.
    INFO - 11:19:31: VariableInfluence for function -y_4
    INFO - 11:19:31: Most influential variables indices to explain % of the function variation: 99
    INFO - 11:19:31: [1 4 3 2 5 9 7 8 0]
/home/docs/checkouts/readthedocs.org/user_builds/gemseo/envs/develop/lib/python3.9/site-packages/gemseo/post/variable_influence.py:230: UserWarning: FixedFormatter should only be used together with FixedLocator
  axe.set_xticklabels(x_labels, fontsize=font_size, rotation=rotation)
    INFO - 11:19:31: VariableInfluence for function g_1_0
    INFO - 11:19:31: Most influential variables indices to explain % of the function variation: 99
    INFO - 11:19:31: [0 7 3 5 6]
    INFO - 11:19:31: VariableInfluence for function g_1_1
    INFO - 11:19:31: Most influential variables indices to explain % of the function variation: 99
    INFO - 11:19:31: [0 7 3 5 6]
    INFO - 11:19:31: VariableInfluence for function g_1_2
    INFO - 11:19:31: Most influential variables indices to explain % of the function variation: 99
    INFO - 11:19:31: [7 0 3 5 6]
    INFO - 11:19:31: VariableInfluence for function g_1_3
    INFO - 11:19:31: Most influential variables indices to explain % of the function variation: 99
    INFO - 11:19:31: [7 0 3 5 6]
    INFO - 11:19:31: VariableInfluence for function g_1_4
    INFO - 11:19:31: Most influential variables indices to explain % of the function variation: 99
    INFO - 11:19:31: [7 0 3 5 6]
    INFO - 11:19:31: VariableInfluence for function g_1_5
    INFO - 11:19:31: Most influential variables indices to explain % of the function variation: 99
    INFO - 11:19:31: [3 7 5 6]
    INFO - 11:19:31: VariableInfluence for function g_1_6
    INFO - 11:19:31: Most influential variables indices to explain % of the function variation: 99
    INFO - 11:19:31: [3 7 5 6]
    INFO - 11:19:31: VariableInfluence for function g_2
    INFO - 11:19:31: Most influential variables indices to explain % of the function variation: 99
    INFO - 11:19:31: [0]
    INFO - 11:19:31: VariableInfluence for function g_3_0
    INFO - 11:19:31: Most influential variables indices to explain % of the function variation: 99
    INFO - 11:19:31: [9 1 5 2 4 0 8]
    INFO - 11:19:31: VariableInfluence for function g_3_1
    INFO - 11:19:31: Most influential variables indices to explain % of the function variation: 99
    INFO - 11:19:31: [9 1 5 2 4 0 8]
    INFO - 11:19:31: VariableInfluence for function g_3_2
    INFO - 11:19:31: Most influential variables indices to explain % of the function variation: 99
    INFO - 11:19:31: [1 9 2]
    INFO - 11:19:31: VariableInfluence for function g_3_3
    INFO - 11:19:31: Most influential variables indices to explain % of the function variation: 99
    INFO - 11:19:31: [9 1 2]

<gemseo.post.variable_influence.VariableInfluence object at 0x7f3d0aa73d00>

Total running time of the script: ( 0 minutes 10.903 seconds)

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