MDF-based MDO on the Sobieski SSBJ test case

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

configure_logger()

Out:

<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",
    ]
)

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
scenario = create_scenario(
    disciplines,
    "MDF",
    objective_name="y_4",
    design_space=design_space,
    maximize_objective=True,
    **formulation_options,
)

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(html_output=True, print_statuses=False, open_browser=False)

Out:

INFO - 10:05:51: 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 generates 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)

Out:

    INFO - 10:05:51:
    INFO - 10:05:51: *** Start MDOScenario execution ***
    INFO - 10:05:51: MDOScenario
    INFO - 10:05:51:    Disciplines: SobieskiPropulsion SobieskiAerodynamics SobieskiMission SobieskiStructure
    INFO - 10:05:51:    MDO formulation: MDF
    INFO - 10:05:51: Optimization problem:
    INFO - 10:05:51:    minimize -y_4(x_shared, x_1, x_2, x_3)
    INFO - 10:05:51:    with respect to x_1, x_2, x_3, x_shared
    INFO - 10:05:51:    subject to constraints:
    INFO - 10:05:51:       g_1(x_shared, x_1, x_2, x_3) <= 0.0
    INFO - 10:05:51:       g_2(x_shared, x_1, x_2, x_3) <= 0.0
    INFO - 10:05:51:       g_3(x_shared, x_1, x_2, x_3) <= 0.0
    INFO - 10:05:51:    over the design space:
    INFO - 10:05:51:    +----------+-------------+-------+-------------+-------+
    INFO - 10:05:51:    | name     | lower_bound | value | upper_bound | type  |
    INFO - 10:05:51:    +----------+-------------+-------+-------------+-------+
    INFO - 10:05:51:    | x_shared |     0.01    |  0.05 |     0.09    | float |
    INFO - 10:05:51:    | x_shared |    30000    | 45000 |    60000    | float |
    INFO - 10:05:51:    | x_shared |     1.4     |  1.6  |     1.8     | float |
    INFO - 10:05:51:    | x_shared |     2.5     |  5.5  |     8.5     | float |
    INFO - 10:05:51:    | x_shared |      40     |   55  |      70     | float |
    INFO - 10:05:51:    | x_shared |     500     |  1000 |     1500    | float |
    INFO - 10:05:51:    | x_1      |     0.1     |  0.25 |     0.4     | float |
    INFO - 10:05:51:    | x_1      |     0.75    |   1   |     1.25    | float |
    INFO - 10:05:51:    | x_2      |     0.75    |   1   |     1.25    | float |
    INFO - 10:05:51:    | x_3      |     0.1     |  0.5  |      1      | float |
    INFO - 10:05:51:    +----------+-------------+-------+-------------+-------+
    INFO - 10:05:51: Solving optimization problem with algorithm SLSQP:
    INFO - 10:05:51: ...   0%|          | 0/10 [00:00<?, ?it]
    INFO - 10:05:51: ...  10%|█         | 1/10 [00:00<00:00, 99.52 it/sec]
    INFO - 10:05:51: ...  30%|███       | 3/10 [00:00<00:00, 29.47 it/sec, obj=-3.76e+3]
    INFO - 10:05:51: ...  50%|█████     | 5/10 [00:00<00:00, 20.02 it/sec, obj=-3.96e+3]
    INFO - 10:05:51: ...  70%|███████   | 7/10 [00:00<00:00, 12.20 it/sec, obj=-4.51e+3]
    INFO - 10:05:51: ...  80%|████████  | 8/10 [00:00<00:00, 10.67 it/sec, obj=-4.52e+3]
    INFO - 10:05:52: ...  90%|█████████ | 9/10 [00:01<00:00,  9.10 it/sec, obj=-4.53e+3]
    INFO - 10:05:52: ... 100%|██████████| 10/10 [00:01<00:00,  8.01 it/sec, obj=-4.52e+3]
    INFO - 10:05:52: ... 100%|██████████| 10/10 [00:01<00:00,  7.99 it/sec, obj=-4.52e+3]
    INFO - 10:05:52: Optimization result:
    INFO - 10:05:52:    Optimizer info:
    INFO - 10:05:52:       Status: None
    INFO - 10:05:52:       Message: Maximum number of iterations reached. GEMSEO Stopped the driver
    INFO - 10:05:52:       Number of calls to the objective function by the optimizer: 12
    INFO - 10:05:52:    Solution:
    INFO - 10:05:52:       The solution is feasible.
    INFO - 10:05:52:       Objective: -3963.5131848833107
    INFO - 10:05:52:       Standardized constraints:
    INFO - 10:05:52:          g_1 = [-0.01809992 -0.03337438 -0.0442712  -0.05185236 -0.05734108 -0.13720865
    INFO - 10:05:52:  -0.10279135]
    INFO - 10:05:52:          g_2 = 1.6156768581687686e-05
    INFO - 10:05:52:          g_3 = [-7.67159033e-01 -2.32840967e-01 -2.38127604e-06 -1.83255000e-01]
    INFO - 10:05:52:       Design space:
    INFO - 10:05:52:       +----------+-------------+--------------------+-------------+-------+
    INFO - 10:05:52:       | name     | lower_bound |       value        | upper_bound | type  |
    INFO - 10:05:52:       +----------+-------------+--------------------+-------------+-------+
    INFO - 10:05:52:       | x_shared |     0.01    | 0.0600040391921454 |     0.09    | float |
    INFO - 10:05:52:       | x_shared |    30000    |       60000        |    60000    | float |
    INFO - 10:05:52:       | x_shared |     1.4     |        1.4         |     1.8     | float |
    INFO - 10:05:52:       | x_shared |     2.5     |        2.5         |     8.5     | float |
    INFO - 10:05:52:       | x_shared |      40     |         70         |      70     | float |
    INFO - 10:05:52:       | x_shared |     500     |        1500        |     1500    | float |
    INFO - 10:05:52:       | x_1      |     0.1     |        0.4         |     0.4     | float |
    INFO - 10:05:52:       | x_1      |     0.75    |        0.75        |     1.25    | float |
    INFO - 10:05:52:       | x_2      |     0.75    |        0.75        |     1.25    | float |
    INFO - 10:05:52:       | x_3      |     0.1     | 0.1562443735844188 |      1      | float |
    INFO - 10:05:52:       +----------+-------------+--------------------+-------------+-------+
    INFO - 10:05:52: *** End MDOScenario execution (time: 0:00:01.264564) ***

{'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")

Out:

INFO - 10:05:52: 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")

Out:

INFO - 10:05:52: Export to ggobi for functions: ['-y_4', 'Iter', 'g_1', 'g_2', 'g_3']
INFO - 10:05:52: Export to ggobi file: mdf_history.xml

Post-process the results

Plot the optimization history view

scenario.post_process("OptHistoryView", save=False, show=False)
  • Evolution of the optimization variables
  • Evolution of the objective value
  • Distance to the optimum
  • Hessian diagonal approximation
  • Evolution of the inequality constraints

Out:

<gemseo.post.opt_history_view.OptHistoryView object at 0x7fdbe1775ca0>

Plot the basic history view

scenario.post_process(
    "BasicHistory", variable_names=["x_shared"], save=False, show=False
)
History plot

Out:

<gemseo.post.basic_history.BasicHistory object at 0x7fdbe1d67d00>

Plot the constraints and objective history

scenario.post_process("ObjConstrHist", save=False, show=False)
Evolution of the objective value and maximal constraint

Out:

<gemseo.post.obj_constr_hist.ObjConstrHist object at 0x7fdbe0f0fe50>

Plot the constraints history

scenario.post_process(
    "ConstraintsHistory", save=False, show=False, constraint_names=["g_1", "g_2", "g_3"]
)
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)

Out:

<gemseo.post.constraints_history.ConstraintsHistory object at 0x7fdbf81328e0>

Plot the constraints history using a radar chart

scenario.post_process(
    "RadarChart", save=False, show=False, constraint_names=["g_1", "g_2", "g_3"]
)
Constraints at iteration 4 (optimum)

Out:

<gemseo.post.radar_chart.RadarChart object at 0x7fdbfa5ae670>

Plot the quadratic approximation of the objective

scenario.post_process("QuadApprox", function="-y_4", save=False, show=False)
  • Hessian matrix SR1 approximation of -y_4
  • plot sobieski mdf example

Out:

<gemseo.post.quad_approx.QuadApprox object at 0x7fdbf8064250>

Plot the functions using a SOM

scenario.post_process("SOM", save=False, show=False)
Self Organizing Maps of the design space, -y_4, g_1_6, g_2, g_3_3

Out:

    INFO - 10:05:55: Building Self Organizing Map from optimization history:
    INFO - 10:05:55:     Number of neurons in x direction = 4
    INFO - 10:05:55:     Number of neurons in y direction = 4

<gemseo.post.som.SOM object at 0x7fdbf84a9910>

Plot the scatter matrix of variables of interest

scenario.post_process(
    "ScatterPlotMatrix",
    save=False,
    show=False,
    variable_names=["-y_4", "g_1"],
    fig_size=(14, 14),
)
plot sobieski mdf example

Out:

<gemseo.post.scatter_mat.ScatterPlotMatrix object at 0x7fdbfa3bdfd0>

Plot the variables using the parallel coordinates

scenario.post_process("ParallelCoordinates", save=False, show=False)
  • Design variables history colored by '-y_4' value
  • Objective function and constraints history colored by '-y_4' value.

Out:

<gemseo.post.para_coord.ParallelCoordinates object at 0x7fdbe05ce460>

Plot the robustness of the solution

scenario.post_process("Robustness", save=False, show=False)
Boxplot of the optimization functions with normalized stddev 0.01

Out:

<gemseo.post.robustness.Robustness object at 0x7fdbe037fbe0>

Plot the influence of the design variables

scenario.post_process("VariableInfluence", save=False, show=False, fig_size=(14, 14))
# 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:

    INFO - 10:06:00: VariableInfluence for function -y_4
    INFO - 10:06:00: Most influential variables indices to explain % of the function variation: 99
    INFO - 10:06:00: [1 4 3 2 5 9 7 8 0]
/home/docs/checkouts/readthedocs.org/user_builds/gemseo/envs/4.0.1/lib/python3.9/site-packages/gemseo/post/variable_influence.py:234: UserWarning: FixedFormatter should only be used together with FixedLocator
  axe.set_xticklabels(x_labels, fontsize=font_size, rotation=rotation)
    INFO - 10:06:00: VariableInfluence for function g_1_0
    INFO - 10:06:00: Most influential variables indices to explain % of the function variation: 99
    INFO - 10:06:00: [0 7 3 5 6]
    INFO - 10:06:00: VariableInfluence for function g_1_1
    INFO - 10:06:00: Most influential variables indices to explain % of the function variation: 99
    INFO - 10:06:00: [7 0 3 5 6]
    INFO - 10:06:00: VariableInfluence for function g_1_2
    INFO - 10:06:00: Most influential variables indices to explain % of the function variation: 99
    INFO - 10:06:00: [7 0 3 5 6]
    INFO - 10:06:00: VariableInfluence for function g_1_3
    INFO - 10:06:00: Most influential variables indices to explain % of the function variation: 99
    INFO - 10:06:00: [7 0 3 5 6]
    INFO - 10:06:00: VariableInfluence for function g_1_4
    INFO - 10:06:00: Most influential variables indices to explain % of the function variation: 99
    INFO - 10:06:00: [7 0 3 5 6]
    INFO - 10:06:00: VariableInfluence for function g_1_5
    INFO - 10:06:00: Most influential variables indices to explain % of the function variation: 99
    INFO - 10:06:00: [3 7 5 6]
    INFO - 10:06:00: VariableInfluence for function g_1_6
    INFO - 10:06:00: Most influential variables indices to explain % of the function variation: 99
    INFO - 10:06:00: [3 7 5 6]
    INFO - 10:06:00: VariableInfluence for function g_2
    INFO - 10:06:00: Most influential variables indices to explain % of the function variation: 99
    INFO - 10:06:00: [0]
    INFO - 10:06:00: VariableInfluence for function g_3_0
    INFO - 10:06:00: Most influential variables indices to explain % of the function variation: 99
    INFO - 10:06:00: [1 9 5 2 4 0 8]
    INFO - 10:06:00: VariableInfluence for function g_3_1
    INFO - 10:06:00: Most influential variables indices to explain % of the function variation: 99
    INFO - 10:06:00: [1 9 5 2 4 0 8]
    INFO - 10:06:00: VariableInfluence for function g_3_2
    INFO - 10:06:00: Most influential variables indices to explain % of the function variation: 99
    INFO - 10:06:00: [1 9 2]
    INFO - 10:06:00: VariableInfluence for function g_3_3
    INFO - 10:06:00: Most influential variables indices to explain % of the function variation: 99
    INFO - 10:06:00: [9 1 2]

Total running time of the script: ( 0 minutes 10.142 seconds)

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