# Tutorial: How to solve an optimization problem¶

Although the library **|g|** is dedicated to the MDO, it can also be used for mono-disciplinary optimization problems.
This tutorial presents some examples on analytical test cases.

## 1. Optimization based on a design of experiments¶

Let \((P)\) be a simple optimization problem:

In this subsection, we will see how to use **|g|** to solve this problem \((P)\) by means of a Design Of Experiments (DOE)

### 1.a. Define the objective function¶

Firstly, by means of the `create_discipline()`

API function,
we create a `MDODiscipline`

of `AutoPyDiscipline`

type
from a python function:

```
from gemseo.api import create_discipline
def f(x1=0., x2=0.):
y = x1 + x2
return y
discipline = create_discipline("AutoPyDiscipline", py_func=f)
```

Now, we want to minimize this `MDODiscipline`

over a design of experiments (DOE).

### 1.b. Define the design space¶

For that, by means of the `create_design_space()`

API function,
we define the `DesignSpace`

\([-5, 5]\times[-5, 5]\)
by using its `add_variable()`

method.

```
from gemseo.api import create_design_space
design_space = create_design_space()
design_space.add_variable("x1", 1, l_b=-5, u_b=5, var_type="integer")
design_space.add_variable("x2", 1, l_b=-5, u_b=5, var_type="integer")
```

### 1.c. Define the DOE scenario¶

Then, by means of the `create_scenario()`

API function,
we define a `DOEScenario`

from the `MDODiscipline`

and the `DesignSpace`

defined above:

```
from gemseo.api import create_scenario
scenario = create_scenario(
discipline, "DisciplinaryOpt", "y", design_space, scenario_type="DOE"
)
```

### 1.d. Execute the DOE scenario¶

Lastly, we solve the `OptimizationProblem`

included in the `DOEScenario`

defined above by minimizing the objective function over a design of experiments included in the `DesignSpace`

.
Precisely, we choose a full factorial design of size \(11^2\):

```
scenario.execute({"algo": "fullfact", "n_samples": 11**2})
```

The optimum results can be found in the execution log. It is also possible to
extract them by invoking the `get_optimum()`

method. It
returns a dictionary containing the optimum results for the
scenario under consideration:

```
opt_results = scenario.get_optimum()
print("The solution of P is (x*,f(x*)) = ({}, {})".format(
opt_results.x_opt, opt_results.f_opt
))
```

which yields:

```
The solution of P is (x*,f(x*)) = ([-5, -5], -10.0).
```

## 2. Optimization based on a quasi-Newton method by means of the library scipy¶

Let \((P)\) be a simple optimization problem:

In this subsection, we will see how to use **|g|** to solve this problem \((P)\) by means of an optimizer
directly used from the library scipy.

### 2.a. Define the objective function¶

Firstly, we create the objective function and its gradient as standard python functions:

```
import numpy as np
from gemseo.api import create_discipline
def g(x=0):
y = np.sin(x) - np.exp(x)
return y
def dgdx(x=0):
y = np.cos(x) - np.exp(x)
return y
```

### 2.b. Minimize the objective function¶

Now, we can to minimize this `MDODiscipline`

over its design space by means of
the L-BFGS-B algorithm implemented in the function `scipy.optimize.fmin_l_bfgs_b`

.

```
from scipy import optimize
x_0 = -0.5 * np.ones(1)
opt = optimize.fmin_l_bfgs_b(g, x_0, fprime=dgdx, bounds=[(-.2, 2.)])
x_opt, f_opt, _ = opt
```

Then, we can display the solution of our optimization problem with the following code:

```
print("The solution of P is (x*,f(x*)) = ({}, {})".format(x_opt[0], f_opt[0]))
```

which gives:

```
The solution of P is (x*,f(x*)) = (-0.2, -1.01740008).
```

See also

You can found the scipy implementation of the L-BFGS-B algorithm by clicking here.

## 3. Optimization based on a quasi-Newton method by means of the GEMSEO optimization interface¶

Let \((P)\) be a simple optimization problem:

In this subsection, we will see how to use **|g|** to solve this problem \((P)\) by means of an optimizer
from scipy called through the optimization interface of **|g|**.

### 3.a. Define the objective function¶

Firstly, by means of the `create_discipline()`

API function,
we create a `MDODiscipline`

of `AutoPyDiscipline`

type
from a python function:

```
import numpy as np
from gemseo.api import create_discipline
def g(x=0):
y = np.sin(x) - np.exp(x)
return y
def dgdx(x=0):
y = np.cos(x) - np.exp(x)
return y
discipline = create_discipline("AutoPyDiscipline", py_func=g, py_jac=dgdx)
```

Now, we can to minimize this `MDODiscipline`

over a design space,
by means of a quasi-Newton method from the initial point \(0.5\).

### 3.b. Define the design space¶

For that, by means of the `create_design_space()`

API function,
we define the `DesignSpace`

\([-2., 2.]\)
with initial value \(0.5\)
by using its `add_variable()`

method.

```
from gemseo.api import create_design_space
design_space = create_design_space()
design_space.add_variable("x", 1, l_b=-2., u_b=2., value=-0.5 * np.ones(1))
```

### 3.c. Define the optimization problem¶

Then, by means of the `create_scenario()`

API function,
we define a `MDOScenario`

from the `MDODiscipline`

and the `DesignSpace`

defined above:

```
from gemseo.api import create_scenario
scenario = create_scenario(
discipline, "DisciplinaryOpt", "y", design_space, scenario_type="MDO"
)
```

### 3.d. Execute the optimization problem¶

Lastly, we solve the `OptimizationProblem`

included in the `MDOScenario`

defined above by minimizing the objective function over the `DesignSpace`

.
Precisely, we choose the L-BFGS-B algorithm
implemented in the function `scipy.optimize.fmin_l_bfgs_b`

and
indirectly called by means of the class `OptimizersFactory`

and of its function `execute()`

:

```
scenario.execute({"algo": "L-BFGS-B", "max_iter": 100})
```

The optimization results are displayed in the log file. They can also be obtained using the following code:

```
opt_results = scenario.get_optimum()
print("The solution of P is (x*,f(x*)) = ({}, {})".format(
opt_results.x_opt, opt_results.f_opt
))
```

which yields:

```
The solution of P is (x*,f(x*)) = (-1.29, -1.24).
```

See also

You can found the scipy implementation of the L-BFGS-B algorithm algorithm by clicking here.

Tip

In order to get the list of available optimization algorithms, use:

```
from gemseo.api import get_available_opt_algorithms
algo_list = get_available_opt_algorithms()
print('Available algorithms: {}'.format(algo_list))
```

what gives:

```
Available algorithms: ['NLOPT_SLSQP', 'L-BFGS-B', 'SLSQP', 'NLOPT_COBYLA', 'NLOPT_BFGS', 'NLOPT_NEWUOA', 'TNC', 'P-L-BFGS-B', 'NLOPT_MMA', 'NLOPT_BOBYQA', 'ODD']
```

## 4. Saving and post-processing¶

After the resolution of the `OptimizationProblem`

, we can export the results into a HDF file:

```
problem = scenario.formulation.opt_problem
problem.export_hdf("my_optim.hdf5")
```

We can also post-process the optimization history by means of the function `execute_post()`

,
either from the `OptimizationProblem`

:

```
from gemseo.api import execute_post
execute_post(problem, "OptHistoryView", save=True, file_path="opt_view_with_doe")
```

or from the HDF file created above:

```
from gemseo.api import execute_post
execute_post("my_optim.hdf5", "OptHistoryView", save=True, file_path="opt_view_from_disk")
```

This command produces a series of PDF files: