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

from __future__ import annotations

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


## Import¶

The first step is to import some high-level functions and a method to get the design space.

configure_logger()

<RootLogger root (INFO)>


## Description¶

The QuadApprox post-processing performs a quadratic approximation of a given function from an optimization history and plot the results as cuts of the approximation.

## 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().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()
for constraint in ["g_1", "g_2", "g_3"]:
scenario.execute({"algo": "SLSQP", "max_iter": 10})

    INFO - 13:56:20:
INFO - 13:56:20: *** Start MDOScenario execution ***
INFO - 13:56:20: MDOScenario
INFO - 13:56:20:    Disciplines: SobieskiAerodynamics SobieskiMission SobieskiPropulsion SobieskiStructure
INFO - 13:56:20:    MDO formulation: MDF
INFO - 13:56:20: Optimization problem:
INFO - 13:56:20:    minimize -y_4(x_shared, x_1, x_2, x_3)
INFO - 13:56:20:    with respect to x_1, x_2, x_3, x_shared
INFO - 13:56:20:    subject to constraints:
INFO - 13:56:20:       g_1(x_shared, x_1, x_2, x_3) <= 0.0
INFO - 13:56:20:       g_2(x_shared, x_1, x_2, x_3) <= 0.0
INFO - 13:56:20:       g_3(x_shared, x_1, x_2, x_3) <= 0.0
INFO - 13:56:20:    over the design space:
INFO - 13:56:20:    +-------------+-------------+-------+-------------+-------+
INFO - 13:56:20:    | name        | lower_bound | value | upper_bound | type  |
INFO - 13:56:20:    +-------------+-------------+-------+-------------+-------+
INFO - 13:56:20:    | x_shared[0] |     0.01    |  0.05 |     0.09    | float |
INFO - 13:56:20:    | x_shared[1] |    30000    | 45000 |    60000    | float |
INFO - 13:56:20:    | x_shared[2] |     1.4     |  1.6  |     1.8     | float |
INFO - 13:56:20:    | x_shared[3] |     2.5     |  5.5  |     8.5     | float |
INFO - 13:56:20:    | x_shared[4] |      40     |   55  |      70     | float |
INFO - 13:56:20:    | x_shared[5] |     500     |  1000 |     1500    | float |
INFO - 13:56:20:    | x_1[0]      |     0.1     |  0.25 |     0.4     | float |
INFO - 13:56:20:    | x_1[1]      |     0.75    |   1   |     1.25    | float |
INFO - 13:56:20:    | x_2         |     0.75    |   1   |     1.25    | float |
INFO - 13:56:20:    | x_3         |     0.1     |  0.5  |      1      | float |
INFO - 13:56:20:    +-------------+-------------+-------+-------------+-------+
INFO - 13:56:20: Solving optimization problem with algorithm SLSQP:
INFO - 13:56:20: ...   0%|          | 0/10 [00:00<?, ?it]
INFO - 13:56:20: ...  10%|█         | 1/10 [00:00<00:01,  8.78 it/sec, obj=-536]
INFO - 13:56:20: ...  20%|██        | 2/10 [00:00<00:01,  6.48 it/sec, obj=-2.12e+3]
WARNING - 13:56:20: MDAJacobi has reached its maximum number of iterations but the normed residual 1.7130677857005655e-05 is still above the tolerance 1e-06.
INFO - 13:56:20: ...  30%|███       | 3/10 [00:00<00:01,  5.46 it/sec, obj=-3.75e+3]
INFO - 13:56:20: ...  40%|████      | 4/10 [00:00<00:01,  5.26 it/sec, obj=-3.96e+3]
INFO - 13:56:21: ...  50%|█████     | 5/10 [00:00<00:00,  5.14 it/sec, obj=-3.96e+3]
INFO - 13:56:21: Optimization result:
INFO - 13:56:21:    Optimizer info:
INFO - 13:56:21:       Status: 8
INFO - 13:56:21:       Message: Positive directional derivative for linesearch
INFO - 13:56:21:       Number of calls to the objective function by the optimizer: 6
INFO - 13:56:21:    Solution:
INFO - 13:56:21:       The solution is feasible.
INFO - 13:56:21:       Objective: -3963.408265187933
INFO - 13:56:21:       Standardized constraints:
INFO - 13:56:21:          g_1 = [-0.01806104 -0.03334642 -0.04424946 -0.0518346  -0.05732607 -0.13720865
INFO - 13:56:21:  -0.10279135]
INFO - 13:56:21:          g_2 = 3.333278582928756e-06
INFO - 13:56:21:          g_3 = [-7.67181773e-01 -2.32818227e-01  8.30379541e-07 -1.83255000e-01]
INFO - 13:56:21:       Design space:
INFO - 13:56:21:       +-------------+-------------+---------------------+-------------+-------+
INFO - 13:56:21:       | name        | lower_bound |        value        | upper_bound | type  |
INFO - 13:56:21:       +-------------+-------------+---------------------+-------------+-------+
INFO - 13:56:21:       | x_shared[0] |     0.01    | 0.06000083331964572 |     0.09    | float |
INFO - 13:56:21:       | x_shared[1] |    30000    |        60000        |    60000    | float |
INFO - 13:56:21:       | x_shared[2] |     1.4     |         1.4         |     1.8     | float |
INFO - 13:56:21:       | x_shared[3] |     2.5     |         2.5         |     8.5     | float |
INFO - 13:56:21:       | x_shared[4] |      40     |          70         |      70     | float |
INFO - 13:56:21:       | x_shared[5] |     500     |         1500        |     1500    | float |
INFO - 13:56:21:       | x_1[0]      |     0.1     |         0.4         |     0.4     | float |
INFO - 13:56:21:       | x_1[1]      |     0.75    |         0.75        |     1.25    | float |
INFO - 13:56:21:       | x_2         |     0.75    |         0.75        |     1.25    | float |
INFO - 13:56:21:       | x_3         |     0.1     |  0.1562448753887276 |      1      | float |
INFO - 13:56:21:       +-------------+-------------+---------------------+-------------+-------+
INFO - 13:56:21: *** End MDOScenario execution (time: 0:00:01.109474) ***

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


## Post-process scenario¶

Lastly, we post-process the scenario by means of the QuadApprox plot which performs a quadratic approximation of a given function from an optimization history and plot the results as cuts of the approximation.

Tip

Each post-processing method requires different inputs and offers a variety of customization options. Use the high-level 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.

The first plot shows an approximation of the Hessian matrix $$\frac{\partial^2 f}{\partial x_i \partial x_j}$$ based on the Symmetric Rank 1 method (SR1) [NW06]. The color map uses a symmetric logarithmic (symlog) scale. This plots the cross influence of the design variables on the objective function or constraints. For instance, on the last figure, the maximal second-order sensitivity is $$\frac{\partial^2 -y_4}{\partial^2 x_0} = 2.10^5$$, which means that the $$x_0$$ is the most influential variable. Then, the cross derivative $$\frac{\partial^2 -y_4}{\partial x_0 \partial x_2} = 5.10^4$$ is positive and relatively high compared to the previous one but the combined effects of $$x_0$$ and $$x_2$$ are non-negligible in comparison.

scenario.post_process("QuadApprox", function="-y_4", save=False, show=True)

<gemseo.post.quad_approx.QuadApprox object at 0x7f006b656610>


The second plot represents the quadratic approximation of the objective around the optimal solution : $$a_{i}(t)=0.5 (t-x^*_i)^2 \frac{\partial^2 f}{\partial x_i^2} + (t-x^*_i) \frac{\partial f}{\partial x_i} + f(x^*)$$, where $$x^*$$ is the optimal solution. This approximation highlights the sensitivity of the objective function with respect to the design variables: we notice that the design variables $$x\_1, x\_5, x\_6$$ have little influence , whereas $$x\_0, x\_2, x\_9$$ have a huge influence on the objective. This trend is also noted in the diagonal terms of the Hessian matrix $$\frac{\partial^2 f}{\partial x_i^2}$$.

Total running time of the script: (0 minutes 2.077 seconds)

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