Note

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# Quadratic approximations¶

In this example, we illustrate the use of the `QuadApprox`

plot
on the Sobieski’s SSBJ problem.

```
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()
```

Out:

```
<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("user")
for constraint in ["g_1", "g_2", "g_3"]:
scenario.add_constraint(constraint, "ineq")
scenario.execute({"algo": "SLSQP", "max_iter": 10})
```

Out:

```
INFO - 10:03:01:
INFO - 10:03:01: *** Start MDOScenario execution ***
INFO - 10:03:01: MDOScenario
INFO - 10:03:01: Disciplines: SobieskiPropulsion SobieskiAerodynamics SobieskiStructure SobieskiMission
INFO - 10:03:01: MDO formulation: MDF
INFO - 10:03:01: Optimization problem:
INFO - 10:03:01: minimize -y_4(x_shared, x_1, x_2, x_3)
INFO - 10:03:01: with respect to x_1, x_2, x_3, x_shared
INFO - 10:03:01: subject to constraints:
INFO - 10:03:01: g_1(x_shared, x_1, x_2, x_3) <= 0.0
INFO - 10:03:01: g_2(x_shared, x_1, x_2, x_3) <= 0.0
INFO - 10:03:01: g_3(x_shared, x_1, x_2, x_3) <= 0.0
INFO - 10:03:01: over the design space:
INFO - 10:03:01: +----------+-------------+-------+-------------+-------+
INFO - 10:03:01: | name | lower_bound | value | upper_bound | type |
INFO - 10:03:01: +----------+-------------+-------+-------------+-------+
INFO - 10:03:01: | x_shared | 0.01 | 0.05 | 0.09 | float |
INFO - 10:03:01: | x_shared | 30000 | 45000 | 60000 | float |
INFO - 10:03:01: | x_shared | 1.4 | 1.6 | 1.8 | float |
INFO - 10:03:01: | x_shared | 2.5 | 5.5 | 8.5 | float |
INFO - 10:03:01: | x_shared | 40 | 55 | 70 | float |
INFO - 10:03:01: | x_shared | 500 | 1000 | 1500 | float |
INFO - 10:03:01: | x_1 | 0.1 | 0.25 | 0.4 | float |
INFO - 10:03:01: | x_1 | 0.75 | 1 | 1.25 | float |
INFO - 10:03:01: | x_2 | 0.75 | 1 | 1.25 | float |
INFO - 10:03:01: | x_3 | 0.1 | 0.5 | 1 | float |
INFO - 10:03:01: +----------+-------------+-------+-------------+-------+
INFO - 10:03:01: Solving optimization problem with algorithm SLSQP:
INFO - 10:03:01: ... 0%| | 0/10 [00:00<?, ?it]
INFO - 10:03:01: ... 20%|██ | 2/10 [00:00<00:00, 41.39 it/sec, obj=-2.12e+3]
INFO - 10:03:01: ... 30%|███ | 3/10 [00:00<00:00, 24.85 it/sec, obj=-3.15e+3]
INFO - 10:03:01: ... 40%|████ | 4/10 [00:00<00:00, 17.71 it/sec, obj=-3.96e+3]
INFO - 10:03:01: ... 50%|█████ | 5/10 [00:00<00:00, 13.77 it/sec, obj=-3.98e+3]
INFO - 10:03:01: ... 50%|█████ | 5/10 [00:00<00:00, 12.39 it/sec, obj=-3.98e+3]
INFO - 10:03:01: Optimization result:
INFO - 10:03:01: Optimizer info:
INFO - 10:03:01: Status: 8
INFO - 10:03:01: Message: Positive directional derivative for linesearch
INFO - 10:03:01: Number of calls to the objective function by the optimizer: 6
INFO - 10:03:01: Solution:
INFO - 10:03:01: The solution is feasible.
INFO - 10:03:01: Objective: -3960.1367790933214
INFO - 10:03:01: Standardized constraints:
INFO - 10:03:01: g_1 = [-0.01805983 -0.03334555 -0.04424879 -0.05183405 -0.05732561 -0.13720865
INFO - 10:03:01: -0.10279135]
INFO - 10:03:01: g_2 = 2.9360600315442298e-06
INFO - 10:03:01: g_3 = [-0.76310174 -0.23689826 -0.00553375 -0.183255 ]
INFO - 10:03:01: Design space:
INFO - 10:03:01: +----------+-------------+---------------------+-------------+-------+
INFO - 10:03:01: | name | lower_bound | value | upper_bound | type |
INFO - 10:03:01: +----------+-------------+---------------------+-------------+-------+
INFO - 10:03:01: | x_shared | 0.01 | 0.06000073401500788 | 0.09 | float |
INFO - 10:03:01: | x_shared | 30000 | 60000 | 60000 | float |
INFO - 10:03:01: | x_shared | 1.4 | 1.4 | 1.8 | float |
INFO - 10:03:01: | x_shared | 2.5 | 2.5 | 8.5 | float |
INFO - 10:03:01: | x_shared | 40 | 70 | 70 | float |
INFO - 10:03:01: | x_shared | 500 | 1500 | 1500 | float |
INFO - 10:03:01: | x_1 | 0.1 | 0.4 | 0.4 | float |
INFO - 10:03:01: | x_1 | 0.75 | 0.75 | 1.25 | float |
INFO - 10:03:01: | x_2 | 0.75 | 0.75 | 1.25 | float |
INFO - 10:03:01: | x_3 | 0.1 | 0.1553801266337427 | 1 | float |
INFO - 10:03:01: +----------+-------------+---------------------+-------------+-------+
INFO - 10:03:01: *** End MDOScenario execution (time: 0:00:00.820089) ***
{'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 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.

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

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 1.661 seconds)