Constrained optimization

Constrained Bayesian Optimization¶

In this tutorial we demonstrate the use of Xopt to perform Bayesian Optimization on a simple test problem subject to a single constraint.

Define the test problem¶

Here we define a simple optimization problem, where we attempt to minimize the sin function in the domian [0,2*pi], subject to a cos constraining function.

In [1]:

# Ignore all warnings
import warnings
warnings.filterwarnings("ignore")
import time
import math

from xopt.vocs import VOCS

# define variables, function objective and constraining function
vocs = VOCS(
    variables={"x": [0, 2 * math.pi]},
    objectives={"f": "MINIMIZE"},
    constraints={"c": ["LESS_THAN", 0]}
)

# Ignore all warnings import warnings warnings.filterwarnings("ignore") import time import math from xopt.vocs import VOCS # define variables, function objective and constraining function vocs = VOCS( variables={"x": [0, 2 * math.pi]}, objectives={"f": "MINIMIZE"}, constraints={"c": ["LESS_THAN", 0]} )

In [2]:

# define a test function to optimize
import numpy as np

def test_function(input_dict):
    return {"f": np.sin(input_dict["x"]),"c": np.cos(input_dict["x"])}

# define a test function to optimize import numpy as np def test_function(input_dict): return {"f": np.sin(input_dict["x"]),"c": np.cos(input_dict["x"])}

Create Xopt objects¶

Create the evaluator to evaluate our test function and create a generator that uses the Expected Improvement acquisition function to perform Bayesian Optimization.

In [3]:

from xopt.evaluator import Evaluator
from xopt.generators.bayesian import ExpectedImprovementGenerator
from xopt import Xopt

evaluator = Evaluator(function=test_function)
generator = ExpectedImprovementGenerator(vocs=vocs)
X = Xopt(evaluator=evaluator, generator=generator, vocs=vocs)

from xopt.evaluator import Evaluator from xopt.generators.bayesian import ExpectedImprovementGenerator from xopt import Xopt evaluator = Evaluator(function=test_function) generator = ExpectedImprovementGenerator(vocs=vocs) X = Xopt(evaluator=evaluator, generator=generator, vocs=vocs)

Generate and evaluate initial points¶

To begin optimization, we must generate some random initial data points. The first call to X.step() will generate and evaluate a number of randomly points specified by the generator. Note that if we add data to xopt before calling X.step() by assigning the data to X.data, calls to X.step() will ignore the random generation and proceed to generating points via Bayesian optimization.

In [4]:

# call X.random_evaluate(n_samples) to generate + evaluate initial points
X.random_evaluate(n_samples=2)

# inspect the gathered data
X.data

# call X.random_evaluate(n_samples) to generate + evaluate initial points X.random_evaluate(n_samples=2) # inspect the gathered data X.data

Out[4]:

	x	f	c	xopt_runtime	xopt_error
0	0.066292	0.066243	0.997804	0.000006	False
1	3.205246	-0.063610	-0.997975	0.000002	False

Do bayesian optimization steps¶

To perform optimization we simply call X.step() in a loop. This allows us to do intermediate tasks in between optimization steps, such as examining the model and acquisition function at each step (as we demonstrate here).

In [5]:

import time

n_steps = 5

# test points for plotting
test_x = np.linspace(*X.vocs.bounds.flatten(), 50)

for i in range(n_steps):
    start = time.perf_counter()
    model = X.generator.train_model()
    fig, ax = X.generator.visualize_model(n_grid=100)
    print(time.perf_counter() - start)

    # add ground truth functions to plots
    out = test_function({"x": test_x})
    ax[0].plot(test_x, out["f"], "C0-.")
    ax[1].plot(test_x, out["c"], "C2-.")

    # do the optimization step
    X.step()

import time n_steps = 5 # test points for plotting test_x = np.linspace(*X.vocs.bounds.flatten(), 50) for i in range(n_steps): start = time.perf_counter() model = X.generator.train_model() fig, ax = X.generator.visualize_model(n_grid=100) print(time.perf_counter() - start) # add ground truth functions to plots out = test_function({"x": test_x}) ax[0].plot(test_x, out["f"], "C0-.") ax[1].plot(test_x, out["c"], "C2-.") # do the optimization step X.step()

0.5273497909947764
0.14978720899671316
0.15450850001070648
0.15901533400756307
0.16169404098764062

In [6]:

# access the collected data
X.data

# access the collected data X.data

Out[6]:

	x	f	c	xopt_runtime	xopt_error
0	0.066292	0.066243	0.997804	0.000006	False
1	3.205246	-0.063610	-0.997975	0.000002	False
2	4.544182	-0.985887	-0.167415	0.000006	False
3	5.992520	-0.286590	0.958053	0.000005	False
4	4.663187	-0.998790	-0.049182	0.000007	False
5	4.693153	-0.999815	-0.019235	0.000006	False
6	4.699189	-0.999913	-0.013200	0.000009	False

In [ ]: