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 __future__ import annotations
from gemseo import configure_logger
from gemseo import create_discipline
from gemseo import create_scenario
from gemseo.problems.mdo.sobieski.core.design_space import SobieskiDesignSpace
```

## 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 create the `SobieskiDesignSpace`

.

```
design_space = SobieskiDesignSpace()
```

## 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,
"MDF",
"y_4",
design_space,
maximize_objective=True,
)
scenario.set_differentiation_method()
for constraint in ["g_1", "g_2", "g_3"]:
scenario.add_constraint(constraint, constraint_type="ineq")
scenario.execute({"algo": "SLSQP", "max_iter": 10})
```

```
INFO - 10:44:09:
INFO - 10:44:09: *** Start MDOScenario execution ***
INFO - 10:44:09: MDOScenario
INFO - 10:44:09: Disciplines: SobieskiAerodynamics SobieskiMission SobieskiPropulsion SobieskiStructure
INFO - 10:44:09: MDO formulation: MDF
INFO - 10:44:09: Optimization problem:
INFO - 10:44:09: minimize -y_4(x_shared, x_1, x_2, x_3)
INFO - 10:44:09: with respect to x_1, x_2, x_3, x_shared
INFO - 10:44:09: subject to constraints:
INFO - 10:44:09: g_1(x_shared, x_1, x_2, x_3) <= 0
INFO - 10:44:09: g_2(x_shared, x_1, x_2, x_3) <= 0
INFO - 10:44:09: g_3(x_shared, x_1, x_2, x_3) <= 0
INFO - 10:44:09: over the design space:
INFO - 10:44:09: +-------------+-------------+-------+-------------+-------+
INFO - 10:44:09: | Name | Lower bound | Value | Upper bound | Type |
INFO - 10:44:09: +-------------+-------------+-------+-------------+-------+
INFO - 10:44:09: | x_shared[0] | 0.01 | 0.05 | 0.09 | float |
INFO - 10:44:09: | x_shared[1] | 30000 | 45000 | 60000 | float |
INFO - 10:44:09: | x_shared[2] | 1.4 | 1.6 | 1.8 | float |
INFO - 10:44:09: | x_shared[3] | 2.5 | 5.5 | 8.5 | float |
INFO - 10:44:09: | x_shared[4] | 40 | 55 | 70 | float |
INFO - 10:44:09: | x_shared[5] | 500 | 1000 | 1500 | float |
INFO - 10:44:09: | x_1[0] | 0.1 | 0.25 | 0.4 | float |
INFO - 10:44:09: | x_1[1] | 0.75 | 1 | 1.25 | float |
INFO - 10:44:09: | x_2 | 0.75 | 1 | 1.25 | float |
INFO - 10:44:09: | x_3 | 0.1 | 0.5 | 1 | float |
INFO - 10:44:09: +-------------+-------------+-------+-------------+-------+
INFO - 10:44:09: Solving optimization problem with algorithm SLSQP:
INFO - 10:44:09: 10%|█ | 1/10 [00:00<00:00, 11.38 it/sec, obj=-536]
INFO - 10:44:09: 20%|██ | 2/10 [00:00<00:00, 8.18 it/sec, obj=-2.12e+3]
WARNING - 10:44:09: MDAJacobi has reached its maximum number of iterations but the normed residual 2.338273970736908e-06 is still above the tolerance 1e-06.
INFO - 10:44:09: 30%|███ | 3/10 [00:00<00:01, 6.92 it/sec, obj=-3.56e+3]
INFO - 10:44:10: 40%|████ | 4/10 [00:00<00:00, 6.60 it/sec, obj=-3.96e+3]
INFO - 10:44:10: 50%|█████ | 5/10 [00:00<00:00, 6.45 it/sec, obj=-3.96e+3]
INFO - 10:44:10: Optimization result:
INFO - 10:44:10: Optimizer info:
INFO - 10:44:10: Status: 8
INFO - 10:44:10: Message: Positive directional derivative for linesearch
INFO - 10:44:10: Number of calls to the objective function by the optimizer: 6
INFO - 10:44:10: Solution:
INFO - 10:44:10: The solution is feasible.
INFO - 10:44:10: Objective: -3963.403105287515
INFO - 10:44:10: Standardized constraints:
INFO - 10:44:10: g_1 = [-0.01806054 -0.03334606 -0.04424918 -0.05183437 -0.05732588 -0.13720864
INFO - 10:44:10: -0.10279136]
INFO - 10:44:10: g_2 = 3.1658077606078194e-06
INFO - 10:44:10: g_3 = [-7.67177346e-01 -2.32822654e-01 -5.57051011e-06 -1.83255000e-01]
INFO - 10:44:10: Design space:
INFO - 10:44:10: +-------------+-------------+---------------------+-------------+-------+
INFO - 10:44:10: | Name | Lower bound | Value | Upper bound | Type |
INFO - 10:44:10: +-------------+-------------+---------------------+-------------+-------+
INFO - 10:44:10: | x_shared[0] | 0.01 | 0.06000079145194018 | 0.09 | float |
INFO - 10:44:10: | x_shared[1] | 30000 | 60000 | 60000 | float |
INFO - 10:44:10: | x_shared[2] | 1.4 | 1.4 | 1.8 | float |
INFO - 10:44:10: | x_shared[3] | 2.5 | 2.5 | 8.5 | float |
INFO - 10:44:10: | x_shared[4] | 40 | 70 | 70 | float |
INFO - 10:44:10: | x_shared[5] | 500 | 1500 | 1500 | float |
INFO - 10:44:10: | x_1[0] | 0.1 | 0.3999999322608766 | 0.4 | float |
INFO - 10:44:10: | x_1[1] | 0.75 | 0.75 | 1.25 | float |
INFO - 10:44:10: | x_2 | 0.75 | 0.75 | 1.25 | float |
INFO - 10:44:10: | x_3 | 0.1 | 0.1562438752833519 | 1 | float |
INFO - 10:44:10: +-------------+-------------+---------------------+-------------+-------+
INFO - 10:44:10: *** End MDOScenario execution (time: 0:00:00.890466) ***
{'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 0x7f1303ba3910>
```

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 1.824 seconds)