Note

Click here to download the full example code

# Quadratic approximations¶

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

plot
on the Sobieski’s SSBJ problem.

```
from __future__ import division, unicode_literals
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.

```
from gemseo.api import configure_logger, create_discipline, create_scenario
from gemseo.problems.sobieski.core import SobieskiProblem
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().read_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 - 14:42:25:
INFO - 14:42:25: *** Start MDO Scenario execution ***
INFO - 14:42:25: MDOScenario
INFO - 14:42:25: Disciplines: SobieskiPropulsion SobieskiAerodynamics SobieskiStructure SobieskiMission
INFO - 14:42:25: MDOFormulation: MDF
INFO - 14:42:25: Algorithm: SLSQP
INFO - 14:42:25: Optimization problem:
INFO - 14:42:25: Minimize: -y_4(x_shared, x_1, x_2, x_3)
INFO - 14:42:25: With respect to: x_shared, x_1, x_2, x_3
INFO - 14:42:25: Subject to constraints:
INFO - 14:42:25: g_1(x_shared, x_1, x_2, x_3) <= 0.0
INFO - 14:42:25: g_2(x_shared, x_1, x_2, x_3) <= 0.0
INFO - 14:42:25: g_3(x_shared, x_1, x_2, x_3) <= 0.0
INFO - 14:42:25: Design space:
INFO - 14:42:25: +----------+-------------+-------+-------------+-------+
INFO - 14:42:25: | name | lower_bound | value | upper_bound | type |
INFO - 14:42:25: +----------+-------------+-------+-------------+-------+
INFO - 14:42:25: | x_shared | 0.01 | 0.05 | 0.09 | float |
INFO - 14:42:25: | x_shared | 30000 | 45000 | 60000 | float |
INFO - 14:42:25: | x_shared | 1.4 | 1.6 | 1.8 | float |
INFO - 14:42:25: | x_shared | 2.5 | 5.5 | 8.5 | float |
INFO - 14:42:25: | x_shared | 40 | 55 | 70 | float |
INFO - 14:42:25: | x_shared | 500 | 1000 | 1500 | float |
INFO - 14:42:25: | x_1 | 0.1 | 0.25 | 0.4 | float |
INFO - 14:42:25: | x_1 | 0.75 | 1 | 1.25 | float |
INFO - 14:42:25: | x_2 | 0.75 | 1 | 1.25 | float |
INFO - 14:42:25: | x_3 | 0.1 | 0.5 | 1 | float |
INFO - 14:42:25: +----------+-------------+-------+-------------+-------+
INFO - 14:42:25: Optimization: 0%| | 0/10 [00:00<?, ?it]
/home/docs/checkouts/readthedocs.org/user_builds/gemseo/conda/3.2.2/lib/python3.8/site-packages/scipy/sparse/linalg/dsolve/linsolve.py:407: SparseEfficiencyWarning: splu requires CSC matrix format
warn('splu requires CSC matrix format', SparseEfficiencyWarning)
INFO - 14:42:25: Optimization: 20%|██ | 2/10 [00:00<00:00, 52.47 it/sec, obj=2.12e+3]
INFO - 14:42:25: Optimization: 40%|████ | 4/10 [00:00<00:00, 21.10 it/sec, obj=3.97e+3]
INFO - 14:42:25: Optimization: 50%|█████ | 5/10 [00:00<00:00, 16.44 it/sec, obj=3.96e+3]
INFO - 14:42:26: Optimization: 60%|██████ | 6/10 [00:00<00:00, 13.49 it/sec, obj=3.96e+3]
INFO - 14:42:26: Optimization: 70%|███████ | 7/10 [00:00<00:00, 11.43 it/sec, obj=3.96e+3]
INFO - 14:42:26: Optimization: 90%|█████████ | 9/10 [00:01<00:00, 9.76 it/sec, obj=3.96e+3]
INFO - 14:42:26: Optimization: 100%|██████████| 10/10 [00:01<00:00, 9.09 it/sec, obj=3.96e+3]
INFO - 14:42:26: Optimization result:
INFO - 14:42:26: Objective value = 3963.595455433326
INFO - 14:42:26: The result is feasible.
INFO - 14:42:26: Status: None
INFO - 14:42:26: Optimizer message: Maximum number of iterations reached. GEMSEO Stopped the driver
INFO - 14:42:26: Number of calls to the objective function by the optimizer: 12
INFO - 14:42:26: Constraints values:
INFO - 14:42:26: g_1 = [-0.01814919 -0.03340982 -0.04429875 -0.05187486 -0.05736009 -0.13720854
INFO - 14:42:26: -0.10279146]
INFO - 14:42:26: g_2 = 3.236261671801799e-05
INFO - 14:42:26: g_3 = [-7.67067574e-01 -2.32932426e-01 -9.19662628e-05 -1.83255000e-01]
INFO - 14:42:26: Design space:
INFO - 14:42:26: +----------+-------------+--------------------+-------------+-------+
INFO - 14:42:26: | name | lower_bound | value | upper_bound | type |
INFO - 14:42:26: +----------+-------------+--------------------+-------------+-------+
INFO - 14:42:26: | x_shared | 0.01 | 0.0600080906541795 | 0.09 | float |
INFO - 14:42:26: | x_shared | 30000 | 60000 | 60000 | float |
INFO - 14:42:26: | x_shared | 1.4 | 1.4 | 1.8 | float |
INFO - 14:42:26: | x_shared | 2.5 | 2.5 | 8.5 | float |
INFO - 14:42:26: | x_shared | 40 | 70 | 70 | float |
INFO - 14:42:26: | x_shared | 500 | 1500 | 1500 | float |
INFO - 14:42:26: | x_1 | 0.1 | 0.3999993439500847 | 0.4 | float |
INFO - 14:42:26: | x_1 | 0.75 | 0.75 | 1.25 | float |
INFO - 14:42:26: | x_2 | 0.75 | 0.75 | 1.25 | float |
INFO - 14:42:26: | x_3 | 0.1 | 0.156230376400943 | 1 | float |
INFO - 14:42:26: +----------+-------------+--------------------+-------------+-------+
INFO - 14:42:26: *** MDO Scenario run terminated in 0:00:01.110369 ***
{'algo': 'SLSQP', 'max_iter': 10}
```

## 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:
Options for 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 1.840 seconds)