# Example for constraints in a hybrid searchspace

Example for optimizing a synthetic test functions in a hybrid space with one
constraint in the discrete subspace and one constraint in the continuous subspace.
All test functions that are available in BoTorch are also available here and wrapped
via the `botorch_function_wrapper`.
This example assumes some basic familiarity with using BayBE.
We thus refer to [`campaign`](./../Basics/campaign.md) for a basic example.
Also, there is a large overlap with other examples with regards to using the test
function.
We thus refer to [`discrete_space`](./../Searchspaces/discrete_space.md) for
details on this aspect.

## Necessary imports for this example


```python
import numpy as np
from botorch.test_functions import Rastrigin
```


```python
from baybe import Campaign
from baybe.constraints import (
    ContinuousLinearEqualityConstraint,
    DiscreteSumConstraint,
    ThresholdCondition,
)
from baybe.objectives import SingleTargetObjective
from baybe.parameters import NumericalContinuousParameter, NumericalDiscreteParameter
from baybe.searchspace import SearchSpace
from baybe.targets import NumericalTarget
from baybe.utils.botorch_wrapper import botorch_function_wrapper
```

## Defining the test function

See [`discrete_space`](./../Searchspaces/discrete_space.md) for details.


```python
DIMENSION = 4
TestFunctionClass = Rastrigin
```

Specify a numerical stride for discrete parameters.
If you make it too small, it will make calculations expensive.
If you make it too large, constraints might not be satisfied anywhere.


```python
STRIDE = 1.0
```


```python
if not hasattr(TestFunctionClass, "dim"):
    TestFunction = TestFunctionClass(dim=DIMENSION)
else:
    TestFunction = TestFunctionClass()
    DIMENSION = TestFunctionClass().dim
```


```python
BOUNDS = TestFunction.bounds
WRAPPED_FUNCTION = botorch_function_wrapper(test_function=TestFunction)
```

## Creating the searchspace and the objective

Since the searchspace is continuous, we construct `NumericalContinuousParameter`.
We use the data of the test function to deduce bounds and number of parameters.


```python
parameters = [
    NumericalDiscreteParameter(
        name=f"x_{k + 1}",
        values=np.arange(
            np.round(BOUNDS[0, k], 0),
            np.round(BOUNDS[1, k], 0) + STRIDE,
            STRIDE,
        ).tolist(),
    )
    for k in range(0, DIMENSION // 2)
] + [
    NumericalContinuousParameter(
        name=f"x_{k+1}",
        bounds=(BOUNDS[0, k], BOUNDS[1, k]),
    )
    for k in range(DIMENSION // 2, DIMENSION)
]
```

We model the following constraints:
`1.0*x_1 + 1.0*x_2 = 1.0`
`1.0*x_3 - 1.0*x_4 = 2.0`


```python
constraints = [
    DiscreteSumConstraint(
        parameters=["x_1", "x_2"],
        condition=ThresholdCondition(
            threshold=1.0, operator="==", tolerance=STRIDE / 2.0
        ),
    ),
    ContinuousLinearEqualityConstraint(
        parameters=["x_3", "x_4"], coefficients=[1.0, -1.0], rhs=2.0
    ),
]
```


```python
searchspace = SearchSpace.from_product(parameters=parameters, constraints=constraints)
objective = SingleTargetObjective(target=NumericalTarget(name="Target", mode="MIN"))
```

## Construct the campaign and run some iterations


```python
campaign = Campaign(
    searchspace=searchspace,
    objective=objective,
)
```


```python
BATCH_SIZE = 5
N_ITERATIONS = 2
```


```python
for k in range(N_ITERATIONS):
    recommendation = campaign.recommend(batch_size=BATCH_SIZE)

    # target value are looked up via the botorch wrapper
    target_values = []
    for index, row in recommendation.iterrows():
        target_values.append(WRAPPED_FUNCTION(*row.to_list()))

    recommendation["Target"] = target_values

    campaign.add_measurements(recommendation)
```


```python
### Verify the constraints
measurements = campaign.measurements
TOLERANCE = 0.01
```

`1.0*x_1 + 1.0*x_2 = 1.0`


```python
print(
    "1.0*x_1 + 1.0*x_2 = 1.0 satisfied in all recommendations? ",
    np.allclose(
        1.0 * measurements["x_1"] + 1.0 * measurements["x_2"], 1.0, atol=TOLERANCE
    ),
)
```

    1.0*x_1 + 1.0*x_2 = 1.0 satisfied in all recommendations?  True


`1.0*x_3 - 1.0*x_4 = 2.0`


```python
print(
    "1.0*x_3 - 1.0*x_4 = 2.0 satisfied in all recommendations? ",
    np.allclose(
        1.0 * measurements["x_3"] - 1.0 * measurements["x_4"], 2.0, atol=TOLERANCE
    ),
)
```

    1.0*x_3 - 1.0*x_4 = 2.0 satisfied in all recommendations?  True