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.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.add_constraint(constraint, "ineq")
scenario.execute({"algo": "SLSQP", "max_iter": 10})
```

```
INFO - 16:24:42:
INFO - 16:24:42: *** Start MDOScenario execution ***
INFO - 16:24:42: MDOScenario
INFO - 16:24:42: Disciplines: SobieskiAerodynamics SobieskiMission SobieskiPropulsion SobieskiStructure
INFO - 16:24:42: MDO formulation: MDF
INFO - 16:24:42: Optimization problem:
INFO - 16:24:42: minimize -y_4(x_shared, x_1, x_2, x_3)
INFO - 16:24:42: with respect to x_1, x_2, x_3, x_shared
INFO - 16:24:42: subject to constraints:
INFO - 16:24:42: g_1(x_shared, x_1, x_2, x_3) <= 0.0
INFO - 16:24:42: g_2(x_shared, x_1, x_2, x_3) <= 0.0
INFO - 16:24:42: g_3(x_shared, x_1, x_2, x_3) <= 0.0
INFO - 16:24:42: over the design space:
INFO - 16:24:42: +-------------+-------------+-------+-------------+-------+
INFO - 16:24:42: | name | lower_bound | value | upper_bound | type |
INFO - 16:24:42: +-------------+-------------+-------+-------------+-------+
INFO - 16:24:42: | x_shared[0] | 0.01 | 0.05 | 0.09 | float |
INFO - 16:24:42: | x_shared[1] | 30000 | 45000 | 60000 | float |
INFO - 16:24:42: | x_shared[2] | 1.4 | 1.6 | 1.8 | float |
INFO - 16:24:42: | x_shared[3] | 2.5 | 5.5 | 8.5 | float |
INFO - 16:24:42: | x_shared[4] | 40 | 55 | 70 | float |
INFO - 16:24:42: | x_shared[5] | 500 | 1000 | 1500 | float |
INFO - 16:24:42: | x_1[0] | 0.1 | 0.25 | 0.4 | float |
INFO - 16:24:42: | x_1[1] | 0.75 | 1 | 1.25 | float |
INFO - 16:24:42: | x_2 | 0.75 | 1 | 1.25 | float |
INFO - 16:24:42: | x_3 | 0.1 | 0.5 | 1 | float |
INFO - 16:24:42: +-------------+-------------+-------+-------------+-------+
INFO - 16:24:42: Solving optimization problem with algorithm SLSQP:
INFO - 16:24:42: ... 0%| | 0/10 [00:00<?, ?it]
INFO - 16:24:42: ... 10%|█ | 1/10 [00:00<00:00, 10.69 it/sec, obj=-536]
INFO - 16:24:43: ... 20%|██ | 2/10 [00:00<00:01, 7.63 it/sec, obj=-2.12e+3]
WARNING - 16:24:43: MDAJacobi has reached its maximum number of iterations but the normed residual 1.4486313079508508e-06 is still above the tolerance 1e-06.
INFO - 16:24:43: ... 30%|███ | 3/10 [00:00<00:01, 6.55 it/sec, obj=-3.75e+3]
INFO - 16:24:43: ... 40%|████ | 4/10 [00:00<00:00, 6.31 it/sec, obj=-4.01e+3]
WARNING - 16:24:43: MDAJacobi has reached its maximum number of iterations but the normed residual 2.928004141058104e-06 is still above the tolerance 1e-06.
INFO - 16:24:43: ... 50%|█████ | 5/10 [00:00<00:00, 6.00 it/sec, obj=-4.49e+3]
INFO - 16:24:43: ... 60%|██████ | 6/10 [00:00<00:00, 6.06 it/sec, obj=-3.4e+3]
INFO - 16:24:43: ... 70%|███████ | 7/10 [00:01<00:00, 6.64 it/sec, obj=-4.93e+3]
INFO - 16:24:43: ... 80%|████████ | 8/10 [00:01<00:00, 6.99 it/sec, obj=-4.76e+3]
INFO - 16:24:43: ... 90%|█████████ | 9/10 [00:01<00:00, 7.35 it/sec, obj=-4.62e+3]
INFO - 16:24:44: ... 100%|██████████| 10/10 [00:01<00:00, 7.71 it/sec, obj=-4.56e+3]
INFO - 16:24:44: Optimization result:
INFO - 16:24:44: Optimizer info:
INFO - 16:24:44: Status: None
INFO - 16:24:44: Message: Maximum number of iterations reached. GEMSEO Stopped the driver
INFO - 16:24:44: Number of calls to the objective function by the optimizer: 12
INFO - 16:24:44: Solution:
INFO - 16:24:44: The solution is feasible.
INFO - 16:24:44: Objective: -3749.8868975554387
INFO - 16:24:44: Standardized constraints:
INFO - 16:24:44: g_1 = [-0.01671296 -0.03238836 -0.04350867 -0.05123129 -0.05681738 -0.13780658
INFO - 16:24:44: -0.10219342]
INFO - 16:24:44: g_2 = -0.0004062839430756249
INFO - 16:24:44: g_3 = [-0.66482546 -0.33517454 -0.11023156 -0.183255 ]
INFO - 16:24:44: Design space:
INFO - 16:24:44: +-------------+-------------+---------------------+-------------+-------+
INFO - 16:24:44: | name | lower_bound | value | upper_bound | type |
INFO - 16:24:44: +-------------+-------------+---------------------+-------------+-------+
INFO - 16:24:44: | x_shared[0] | 0.01 | 0.05989842901423112 | 0.09 | float |
INFO - 16:24:44: | x_shared[1] | 30000 | 59853.73840058666 | 60000 | float |
INFO - 16:24:44: | x_shared[2] | 1.4 | 1.4 | 1.8 | float |
INFO - 16:24:44: | x_shared[3] | 2.5 | 2.527371250092273 | 8.5 | float |
INFO - 16:24:44: | x_shared[4] | 40 | 69.86825198198687 | 70 | float |
INFO - 16:24:44: | x_shared[5] | 500 | 1495.734648986894 | 1500 | float |
INFO - 16:24:44: | x_1[0] | 0.1 | 0.4 | 0.4 | float |
INFO - 16:24:44: | x_1[1] | 0.75 | 0.7521124139939552 | 1.25 | float |
INFO - 16:24:44: | x_2 | 0.75 | 0.7520888531444992 | 1.25 | float |
INFO - 16:24:44: | x_3 | 0.1 | 0.1398000762238233 | 1 | float |
INFO - 16:24:44: +-------------+-------------+---------------------+-------------+-------+
INFO - 16:24:44: *** End MDOScenario execution (time: 0:00:01.315489) ***
{'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)
```

```
/home/docs/checkouts/readthedocs.org/user_builds/gemseo/envs/stable/lib/python3.9/site-packages/genson/schema/strategies/base.py:42: UserWarning: Schema incompatible. Keyword 'description' has conflicting values ('The width and height of the figure in inches, e.g. ``(w, h)``.\nIf ``None``, use the :attr:`.OptPostProcessor.DEFAULT_FIG_SIZE`\nof the post-processor.' vs. 'The width and height of the figure in inches, e.g. `(w, h)`.\nIf ``None``, use the :attr:`.OptPostProcessor.DEFAULT_FIG_SIZE`\nof the post-processor.'). Using 'The width and height of the figure in inches, e.g. ``(w, h)``.\nIf ``None``, use the :attr:`.OptPostProcessor.DEFAULT_FIG_SIZE`\nof the post-processor.'
warn(('Schema incompatible. Keyword {0!r} has conflicting '
<gemseo.post.quad_approx.QuadApprox object at 0x7ffaf5f4f760>
```

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 2.237 seconds)