Example for exterior penalty applied to the Sobieski test case.

This section describes how to set up and solve the MDO problem relative to the Sobieski test case with GEMSEO applying external penalty.

See also

To start with a simpler MDO problem, and for a detailed description of how to plug a test case into GEMSEO, see Tutorial: How to solve an MDO problem.

Solving with an MDF formulation

In this example, we solve the range optimization using the following MDF formulation:


All the imports needed for the tutorials are performed here.

from __future__ import annotations

from gemseo import configure_logger
from gemseo import create_discipline
from gemseo import create_scenario
from gemseo import get_available_formulations
from gemseo.disciplines.utils import get_all_inputs
from gemseo.disciplines.utils import get_all_outputs
from gemseo.problems.sobieski.core.problem import SobieskiProblem


Step 1: Creation of MDODiscipline

To build the scenario, we first instantiate the disciplines. Here, the disciplines themselves have already been developed and interfaced with GEMSEO (see Benchmark problems).

disciplines = create_discipline(


For the disciplines that are not interfaced with GEMSEO, the GEMSEO’s gemseo eases the creation of disciplines without having to import them.

See High-level functions to use GEMSEO.

Step 2: Creation of Scenario

The scenario delegates the creation of the optimization problem to the MDO formulation.

Therefore, it needs the list of disciplines, the names of the formulation, the name of the objective function and the design space.

  • The design_space (shown below for reference) defines the unknowns of the optimization problem, and their bounds. It contains all the design variables needed by the MDF formulation. It can be imported from a text file, or created from scratch with the methods create_design_space() and add_variable(). In this case, we will retrieve it from the SobieskiProblem already defined in GEMSEO.

design_space = SobieskiProblem().design_space
x_0 = design_space.get_current_value(as_dict=True)
name      lower_bound      value      upper_bound  type
x_shared      0.01          0.05          0.09     float
x_shared    30000.0       45000.0       60000.0    float
x_shared      1.4           1.6           1.8      float
x_shared      2.5           5.5           8.5      float
x_shared      40.0          55.0          70.0     float
x_shared     500.0         1000.0        1500.0    float
x_1           0.1           0.25          0.4      float
x_1           0.75          1.0           1.25     float
x_2           0.75          1.0           1.25     float
x_3           0.1           0.5           1.0      float
y_14        24850.0    50606.9741711    77100.0    float
y_14        -7700.0    7306.20262124    45000.0    float
y_32         0.235       0.50279625      0.795     float
y_31         2960.0    6354.32430691    10185.0    float
y_24          0.44       4.15006276      11.13     float
y_34          0.44       1.10754577       1.98     float
y_23         3365.0    12194.2671934    26400.0    float
y_21        24850.0    50606.9741711    77250.0    float
y_12        24850.0      50606.9742     77250.0    float
y_12          0.45          0.95          1.5      float
  • The available MDO formulations are located in the gemseo.formulations package, see Extend GEMSEO features for extending GEMSEO with other formulations.

  • The formulation classname (here, "MDF") shall be passed to the scenario to select them.

  • The list of available formulations can be obtained by using get_available_formulations().

  • \(y\_4\) corresponds to the objective_name. This name must be one of the disciplines outputs, here the SobieskiMission discipline. The list of all outputs of the disciplines can be obtained by using get_all_outputs():


From these MDODiscipline, design space, MDO formulation name and objective function name, we build the scenario:

scenario = create_scenario(

The range function (\(y\_4\)) should be maximized. However, optimizers minimize functions by default. Which is why, when creating the scenario, the argument maximize_objective shall be set to True.

Differentiation method

We may choose the way derivatives are computed:

Function derivatives. As analytical disciplinary derivatives are available for the Sobieski test-case, they can be used instead of computing the derivatives with finite-differences or with the complex-step method. The easiest way to set a method is to let the optimizer determine it:


The default behavior of the optimizer triggers finite differences. It corresponds to:


It it also possible to differentiate functions by means of the complex step method:



Similarly to the objective function, the constraints names are a subset of the disciplines’ outputs. They can be obtained by using get_all_outputs().

The formulation has a powerful feature to automatically dispatch the constraints (\(g\_1, g\_2, g\_3\)) and plug them to the optimizers depending on the formulation. To do that, we use the method add_constraint():

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

Step 3: Apply the exterior penalty and execute the scenario

    objective_scale=10.0, scale_inequality=10.0

In this way the L-BFGS-B algorithm can be used to solve the optimization problem. Note that this algorithm is not suited for constrained optimization problems.

algo_args = {"max_iter": 10, "algo": "L-BFGS-B"}

Post-processing options

To visualize the optimization history of the constraint violation one can use the BasicHistory:

    "BasicHistory", variable_names=["g_1", "g_2", "g_3"], save=False, show=True

This solution is almost feasible. The solution can better approximate the original problem solution increasing the value

of objective_scale and scale_inequality parameters.

Step 4: Rerun the scenario with increased penalty and objective scaling.

scenario_2 = create_scenario(
for constraint in ["g_1", "g_2", "g_3"]:
    scenario_2.add_constraint(constraint, "ineq")
    objective_scale=1000.0, scale_inequality=1000.0
algo_args_2 = {"max_iter": 1000, "algo": "L-BFGS-B"}
scenario_2.post_process("BasicHistory", variable_names=["-y_4"], save=False, show=True)
    "BasicHistory", variable_names=["g_1", "g_2", "g_3"], save=False, show=True

The solution feasibility was improved but this comes with a much higher number of iterations.

Total running time of the script: ( 0 minutes 0.000 seconds)

Gallery generated by Sphinx-Gallery