Multi-objective Bayesian Optimization¶

TNK function $n=2$ variables: $x_i \in [0, \pi], i=1,2$

Objectives:

$f_i(x) = x_i$

Constraints:

$g_1(x) = -x_1^2 -x_2^2 + 1 + 0.1 \cos\left(16 \arctan \frac{x_1}{x_2}\right) \le 0$
$g_2(x) = (x_1 - 1/2)^2 + (x_2-1/2)^2 \le 0.5$

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# set values if testing
import os

import pandas as pd
import numpy as np

from xopt import Xopt, Evaluator
from xopt.generators.bayesian import MOBOGenerator
from xopt.resources.test_functions.tnk import evaluate_TNK, tnk_vocs
from xopt.vocs import get_feasibility_data

import matplotlib.pyplot as plt

# Ignore all warnings
import warnings

warnings.filterwarnings("ignore")

SMOKE_TEST = os.environ.get("SMOKE_TEST")
N_MC_SAMPLES = 1 if SMOKE_TEST else 128
NUM_RESTARTS = 1 if SMOKE_TEST else 20
N_STEPS = 1 if SMOKE_TEST else 30
MAX_ITER = 1 if SMOKE_TEST else 200

evaluator = Evaluator(function=evaluate_TNK)
print(tnk_vocs.dict())
# set values if testing
import os

import pandas as pd
import numpy as np

from xopt import Xopt, Evaluator
from xopt.generators.bayesian import MOBOGenerator
from xopt.resources.test_functions.tnk import evaluate_TNK, tnk_vocs
from xopt.vocs import get_feasibility_data

import matplotlib.pyplot as plt

# Ignore all warnings
import warnings

warnings.filterwarnings("ignore")

SMOKE_TEST = os.environ.get("SMOKE_TEST")
N_MC_SAMPLES = 1 if SMOKE_TEST else 128
NUM_RESTARTS = 1 if SMOKE_TEST else 20
N_STEPS = 1 if SMOKE_TEST else 30
MAX_ITER = 1 if SMOKE_TEST else 200

evaluator = Evaluator(function=evaluate_TNK)
print(tnk_vocs.dict())

{'variables': {'x1': {'dtype': None, 'default_value': None, 'domain': [0.0, 3.14159], 'type': 'ContinuousVariable'}, 'x2': {'dtype': None, 'default_value': None, 'domain': [0.0, 3.14159], 'type': 'ContinuousVariable'}}, 'objectives': {'y1': {'dtype': None, 'type': 'MinimizeObjective'}, 'y2': {'dtype': None, 'type': 'MinimizeObjective'}}, 'constraints': {'c1': {'dtype': None, 'value': 0.0, 'type': 'GreaterThanConstraint'}, 'c2': {'dtype': None, 'value': 0.5, 'type': 'LessThanConstraint'}}, 'constants': {'a': {'dtype': None, 'value': 'dummy_constant', 'type': 'Constant'}}, 'observables': {}}

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generator = MOBOGenerator(vocs=tnk_vocs, reference_point={"y1": 1.5, "y2": 1.5})
generator.n_monte_carlo_samples = N_MC_SAMPLES
generator.numerical_optimizer.n_restarts = NUM_RESTARTS
generator.numerical_optimizer.max_iter = MAX_ITER
generator.gp_constructor.use_low_noise_prior = True


X = Xopt(generator=generator, evaluator=evaluator)
X.evaluate_data(pd.DataFrame({"x1": [1.0, 0.75], "x2": [0.75, 1.0]}))

for i in range(N_STEPS):
    print(i)
    X.step()
generator = MOBOGenerator(vocs=tnk_vocs, reference_point={"y1": 1.5, "y2": 1.5})
generator.n_monte_carlo_samples = N_MC_SAMPLES
generator.numerical_optimizer.n_restarts = NUM_RESTARTS
generator.numerical_optimizer.max_iter = MAX_ITER
generator.gp_constructor.use_low_noise_prior = True


X = Xopt(generator=generator, evaluator=evaluator)
X.evaluate_data(pd.DataFrame({"x1": [1.0, 0.75], "x2": [0.75, 1.0]}))

for i in range(N_STEPS):
    print(i)
    X.step()

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X.generator.data
X.generator.data

Out[3]:

	x1	x2	a	y1	y2	c1	c2	xopt_runtime	xopt_error
0	1.000000	0.750000	dummy_constant	1.000000	0.750000	0.626888	0.312500	0.001377	False
1	0.750000	1.000000	dummy_constant	0.750000	1.000000	0.626888	0.312500	0.000263	False
2	0.268736	1.600722	dummy_constant	0.268736	1.600722	1.723218	1.265073	0.000280	False
3	0.335127	0.792341	dummy_constant	0.335127	0.792341	-0.359178	0.112646	0.000305	False
4	0.000000	0.247857	dummy_constant	0.000000	0.247857	-1.038567	0.313576	0.000296	False
5	3.141590	0.000000	dummy_constant	3.141590	0.000000	8.769588	7.227998	0.000303	False
6	1.062598	0.000071	dummy_constant	1.062598	0.000071	0.029115	0.566446	0.000301	False
7	1.058239	0.175585	dummy_constant	1.058239	0.175585	0.237935	0.416876	0.000284	False
8	0.258806	1.025443	dummy_constant	0.258806	1.025443	0.187177	0.334265	0.000287	False
9	0.015228	0.966661	dummy_constant	0.015228	0.966661	-0.162176	0.452776	0.000283	False
10	1.038942	0.065072	dummy_constant	1.038942	0.065072	0.029673	0.479620	0.000287	False
11	0.085818	1.044497	dummy_constant	0.085818	1.044497	0.072712	0.468023	0.000294	False
12	0.944616	0.438159	dummy_constant	0.944616	0.438159	0.005638	0.201507	0.000290	False
13	1.048864	0.071598	dummy_constant	1.048864	0.071598	0.059038	0.484781	0.000337	False
14	0.402116	0.944841	dummy_constant	0.402116	0.944841	-0.044384	0.207465	0.003209	False
15	0.629293	0.848435	dummy_constant	0.629293	0.848435	0.186519	0.138124	0.000295	False
16	0.000000	0.072962	dummy_constant	0.000000	0.072962	-1.094677	0.432361	0.000301	False
17	0.000000	0.047063	dummy_constant	0.000000	0.047063	-1.097785	0.455152	0.000282	False
18	0.000000	0.034229	dummy_constant	0.000000	0.034229	-1.098828	0.466943	0.000287	False
19	0.791831	0.677957	dummy_constant	0.791831	0.677957	0.053874	0.116834	0.000287	False
20	1.033694	0.039858	dummy_constant	1.033694	0.039858	-0.011472	0.496560	0.000287	False
21	0.000000	0.024710	dummy_constant	0.000000	0.024710	-1.099389	0.475901	0.000283	False
22	0.127138	1.011808	dummy_constant	0.127138	1.011808	0.081534	0.400974	0.000309	False
23	0.038166	1.027454	dummy_constant	0.038166	1.027454	-0.025748	0.491498	0.000288	False
24	1.034665	0.051010	dummy_constant	1.034665	0.051010	0.002619	0.487458	0.000280	False
25	1.021935	0.022659	dummy_constant	1.021935	0.022659	-0.048910	0.500271	0.000297	False
26	0.017465	1.017213	dummy_constant	0.017465	1.017213	-0.061223	0.500350	0.000283	False
27	0.000000	0.021837	dummy_constant	0.000000	0.021837	-1.099523	0.478640	0.000303	False
28	0.703447	0.747443	dummy_constant	0.703447	0.747443	-0.034958	0.102619	0.000305	False
29	0.884126	0.532550	dummy_constant	0.884126	0.532550	0.138423	0.148612	0.000294	False
30	0.996220	0.340907	dummy_constant	0.996220	0.340907	0.055301	0.271544	0.000269	False
31	0.000013	0.016196	dummy_constant	0.000013	0.016196	-1.099730	0.484053	0.000295	False

plot results¶

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fig, ax = plt.subplots()

theta = np.linspace(0, np.pi / 2)
r = np.sqrt(1 + 0.1 * np.cos(16 * theta))
x_1 = r * np.sin(theta)
x_2_lower = r * np.cos(theta)
x_2_upper = (0.5 - (x_1 - 0.5) ** 2) ** 0.5 + 0.5

z = np.zeros_like(x_1)

# ax2.plot(x_1, x_2_lower,'r')
ax.fill_between(x_1, z, x_2_lower, fc="white")
circle = plt.Circle(
    (0.5, 0.5), 0.5**0.5, color="r", alpha=0.25, zorder=0, label="Valid Region"
)
ax.add_patch(circle)
history = pd.concat(
    [X.data, get_feasibility_data(tnk_vocs, X.data)], axis=1, ignore_index=False
)


ax.plot(*history[["x1", "x2"]][history["feasible"]].to_numpy().T, ".C1")
ax.plot(*history[["x1", "x2"]][~history["feasible"]].to_numpy().T, ".C2")

ax.set_xlim(0, 3.14)
ax.set_ylim(0, 3.14)
ax.set_xlabel("x1")
ax.set_ylabel("x2")
ax.set_aspect("equal")
fig, ax = plt.subplots()

theta = np.linspace(0, np.pi / 2)
r = np.sqrt(1 + 0.1 * np.cos(16 * theta))
x_1 = r * np.sin(theta)
x_2_lower = r * np.cos(theta)
x_2_upper = (0.5 - (x_1 - 0.5) ** 2) ** 0.5 + 0.5

z = np.zeros_like(x_1)

# ax2.plot(x_1, x_2_lower,'r')
ax.fill_between(x_1, z, x_2_lower, fc="white")
circle = plt.Circle(
    (0.5, 0.5), 0.5**0.5, color="r", alpha=0.25, zorder=0, label="Valid Region"
)
ax.add_patch(circle)
history = pd.concat(
    [X.data, get_feasibility_data(tnk_vocs, X.data)], axis=1, ignore_index=False
)


ax.plot(*history[["x1", "x2"]][history["feasible"]].to_numpy().T, ".C1")
ax.plot(*history[["x1", "x2"]][~history["feasible"]].to_numpy().T, ".C2")

ax.set_xlim(0, 3.14)
ax.set_ylim(0, 3.14)
ax.set_xlabel("x1")
ax.set_ylabel("x2")
ax.set_aspect("equal")

No description has been provided for this image

Plot path through input space¶

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ax = history.plot("x1", "x2")
ax.set_ylim(0, 3.14)
ax.set_xlim(0, 3.14)
ax.set_aspect("equal")
ax = history.plot("x1", "x2")
ax.set_ylim(0, 3.14)
ax.set_xlim(0, 3.14)
ax.set_aspect("equal")

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## visualize model
X.generator.visualize_model(show_feasibility=True)
## visualize model
X.generator.visualize_model(show_feasibility=True)

Out[6]:

(<Figure size 800x1980 with 22 Axes>,
 array([[<Axes: title={'center': 'Posterior Mean [y1]'}, xlabel='x1', ylabel='x2'>,
         <Axes: title={'center': 'Posterior SD [y1]'}, xlabel='x1', ylabel='x2'>],
        [<Axes: title={'center': 'Posterior Mean [y2]'}, xlabel='x1', ylabel='x2'>,
         <Axes: title={'center': 'Posterior SD [y2]'}, xlabel='x1', ylabel='x2'>],
        [<Axes: title={'center': 'Posterior Mean [c1]'}, xlabel='x1', ylabel='x2'>,
         <Axes: title={'center': 'Posterior SD [c1]'}, xlabel='x1', ylabel='x2'>],
        [<Axes: title={'center': 'Posterior Mean [c2]'}, xlabel='x1', ylabel='x2'>,
         <Axes: title={'center': 'Posterior SD [c2]'}, xlabel='x1', ylabel='x2'>],
        [<Axes: title={'center': 'Acq. Function'}, xlabel='x1', ylabel='x2'>,
         <Axes: >],
        [<Axes: title={'center': 'Feasibility'}, xlabel='x1', ylabel='x2'>,
         <Axes: >]], dtype=object))

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X.generator.update_pareto_front_history()
X.generator.pareto_front_history.plot(y="hypervolume", label="Hypervolume")
X.generator.update_pareto_front_history()
X.generator.pareto_front_history.plot(y="hypervolume", label="Hypervolume")

Out[7]:

<Axes: >

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X.generator.pareto_front_history
X.generator.pareto_front_history

Out[8]:

	iteration	hypervolume	n_non_dominated
0	0	0.375000	1
1	1	0.500000	2
2	2	0.500000	2
3	3	0.500000	2
4	4	0.500000	2
5	5	0.500000	2
6	6	0.500000	2
7	7	0.753754	3
8	8	0.986854	4
9	9	0.986854	4
10	10	1.048891	4
11	11	1.127688	5
12	12	1.170949	5
13	13	1.170949	5
14	14	1.170949	5
15	15	1.221812	5
16	16	1.221812	5
17	17	1.221812	5
18	18	1.221812	5
19	19	1.247858	6
20	20	1.247858	6
21	21	1.247858	6
22	22	1.257213	6
23	23	1.257213	6
24	24	1.265353	6
25	25	1.265353	6
26	26	1.265353	6
27	27	1.265353	6
28	28	1.265353	6
29	29	1.274148	7
30	30	1.277887	8
31	31	1.277887	8

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X.data
X.data

Out[9]:

	x1	x2	a	y1	y2	c1	c2	xopt_runtime	xopt_error
0	1.000000	0.750000	dummy_constant	1.000000	0.750000	0.626888	0.312500	0.001377	False
1	0.750000	1.000000	dummy_constant	0.750000	1.000000	0.626888	0.312500	0.000263	False
2	0.268736	1.600722	dummy_constant	0.268736	1.600722	1.723218	1.265073	0.000280	False
3	0.335127	0.792341	dummy_constant	0.335127	0.792341	-0.359178	0.112646	0.000305	False
4	0.000000	0.247857	dummy_constant	0.000000	0.247857	-1.038567	0.313576	0.000296	False
5	3.141590	0.000000	dummy_constant	3.141590	0.000000	8.769588	7.227998	0.000303	False
6	1.062598	0.000071	dummy_constant	1.062598	0.000071	0.029115	0.566446	0.000301	False
7	1.058239	0.175585	dummy_constant	1.058239	0.175585	0.237935	0.416876	0.000284	False
8	0.258806	1.025443	dummy_constant	0.258806	1.025443	0.187177	0.334265	0.000287	False
9	0.015228	0.966661	dummy_constant	0.015228	0.966661	-0.162176	0.452776	0.000283	False
10	1.038942	0.065072	dummy_constant	1.038942	0.065072	0.029673	0.479620	0.000287	False
11	0.085818	1.044497	dummy_constant	0.085818	1.044497	0.072712	0.468023	0.000294	False
12	0.944616	0.438159	dummy_constant	0.944616	0.438159	0.005638	0.201507	0.000290	False
13	1.048864	0.071598	dummy_constant	1.048864	0.071598	0.059038	0.484781	0.000337	False
14	0.402116	0.944841	dummy_constant	0.402116	0.944841	-0.044384	0.207465	0.003209	False
15	0.629293	0.848435	dummy_constant	0.629293	0.848435	0.186519	0.138124	0.000295	False
16	0.000000	0.072962	dummy_constant	0.000000	0.072962	-1.094677	0.432361	0.000301	False
17	0.000000	0.047063	dummy_constant	0.000000	0.047063	-1.097785	0.455152	0.000282	False
18	0.000000	0.034229	dummy_constant	0.000000	0.034229	-1.098828	0.466943	0.000287	False
19	0.791831	0.677957	dummy_constant	0.791831	0.677957	0.053874	0.116834	0.000287	False
20	1.033694	0.039858	dummy_constant	1.033694	0.039858	-0.011472	0.496560	0.000287	False
21	0.000000	0.024710	dummy_constant	0.000000	0.024710	-1.099389	0.475901	0.000283	False
22	0.127138	1.011808	dummy_constant	0.127138	1.011808	0.081534	0.400974	0.000309	False
23	0.038166	1.027454	dummy_constant	0.038166	1.027454	-0.025748	0.491498	0.000288	False
24	1.034665	0.051010	dummy_constant	1.034665	0.051010	0.002619	0.487458	0.000280	False
25	1.021935	0.022659	dummy_constant	1.021935	0.022659	-0.048910	0.500271	0.000297	False
26	0.017465	1.017213	dummy_constant	0.017465	1.017213	-0.061223	0.500350	0.000283	False
27	0.000000	0.021837	dummy_constant	0.000000	0.021837	-1.099523	0.478640	0.000303	False
28	0.703447	0.747443	dummy_constant	0.703447	0.747443	-0.034958	0.102619	0.000305	False
29	0.884126	0.532550	dummy_constant	0.884126	0.532550	0.138423	0.148612	0.000294	False
30	0.996220	0.340907	dummy_constant	0.996220	0.340907	0.055301	0.271544	0.000269	False
31	0.000013	0.016196	dummy_constant	0.000013	0.016196	-1.099730	0.484053	0.000295	False