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

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

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': (0.0, 3.14159), 'x2': (0.0, 3.14159)}, 'constraints': {'c1': ('GREATER_THAN', 0.0), 'c2': ('LESS_THAN', 0.5)}, 'objectives': {'y1': 'MINIMIZE', 'y2': 'MINIMIZE'}, 'constants': {'a': 'dummy_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, vocs=tnk_vocs)
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, vocs=tnk_vocs)
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	0.000156	False
1	0.750000	1.000000	dummy_constant	0.750000	1.000000	0.626888	0.312500	0.000134	False
2	1.548480	0.275010	dummy_constant	1.548480	0.275010	1.568048	1.149932	0.000154	False
3	0.241011	0.000000	dummy_constant	0.241011	0.000000	-1.041914	0.317075	0.000157	False
4	0.031652	0.809338	dummy_constant	0.031652	0.809338	-0.425042	0.315040	0.000152	False
5	0.000000	3.110180	dummy_constant	0.000000	3.110180	8.573218	7.063039	0.000153	False
6	0.233733	1.000078	dummy_constant	0.233733	1.000078	0.140971	0.320976	0.000152	False
7	0.925835	0.164186	dummy_constant	0.925835	0.164186	-0.021378	0.294107	0.000154	False
8	0.967857	0.053549	dummy_constant	0.967857	0.053549	-0.123765	0.418209	0.000154	False
9	1.027448	0.096386	dummy_constant	1.027448	0.096386	0.057525	0.441106	0.000149	False
10	0.783281	0.553075	dummy_constant	0.783281	0.553075	0.011047	0.083065	0.000150	False
11	0.062426	0.992776	dummy_constant	0.062426	0.992776	-0.064126	0.434299	0.000176	False
12	0.647138	0.763456	dummy_constant	0.647138	0.763456	-0.023514	0.091059	0.000148	False
13	0.068081	1.032383	dummy_constant	0.068081	1.032383	0.021005	0.469986	0.000154	False
14	0.885582	0.446461	dummy_constant	0.885582	0.446461	-0.053756	0.151540	0.000168	False
15	0.765232	0.672965	dummy_constant	0.765232	0.672965	-0.013443	0.100265	0.000160	False
16	0.583791	0.857496	dummy_constant	0.583791	0.857496	0.175150	0.134825	0.000157	False
17	1.006785	0.016167	dummy_constant	1.006785	0.016167	-0.082842	0.490925	0.000149	False
18	1.029390	0.050055	dummy_constant	1.029390	0.050055	-0.009124	0.482705	0.000155	False
19	0.021907	1.023518	dummy_constant	0.021907	1.023518	-0.046127	0.502644	0.000166	False
20	0.947668	0.344929	dummy_constant	0.947668	0.344929	-0.059560	0.224453	0.000151	False
21	1.061133	0.158745	dummy_constant	1.061133	0.158745	0.223297	0.431325	0.000126	False
22	0.060807	1.044455	dummy_constant	0.060807	1.044455	0.034836	0.489322	0.000151	False
23	0.780059	0.622825	dummy_constant	0.780059	0.622825	0.017738	0.093519	0.000150	False
24	0.976133	0.288116	dummy_constant	0.976133	0.288116	0.047839	0.271598	0.000151	False
25	0.037181	0.030208	dummy_constant	0.037181	0.030208	-0.989816	0.434906	0.000150	False
26	0.995252	0.181084	dummy_constant	0.995252	0.181084	0.119908	0.346982	0.000160	False
27	0.006461	0.025072	dummy_constant	0.006461	0.025072	-0.936689	0.469137	0.000151	False
28	1.042925	0.059027	dummy_constant	1.042925	0.059027	0.029376	0.489224	0.000148	False
29	0.037069	1.024146	dummy_constant	0.037069	1.024146	-0.033458	0.489034	0.000149	False
30	0.869251	0.501421	dummy_constant	0.869251	0.501421	0.056491	0.136348	0.000150	False
31	1.023102	0.029574	dummy_constant	1.023102	0.029574	-0.041887	0.494937	0.000150	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, tnk_vocs.feasibility_data(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, tnk_vocs.feasibility_data(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	0.758094	3
7	7	0.758094	3
8	8	0.758094	3
9	9	1.066961	4
10	10	1.169223	4
11	11	1.169223	4
12	12	1.169223	4
13	13	1.246684	5
14	14	1.246684	5
15	15	1.246684	5
16	16	1.275125	5
17	17	1.275125	5
18	18	1.275125	5
19	19	1.275125	5
20	20	1.275125	5
21	21	1.275125	5
22	22	1.278439	6
23	23	1.279195	7
24	24	1.292791	8
25	25	1.292791	8
26	26	1.296237	9
27	27	1.296237	9
28	28	1.313313	10
29	29	1.313313	10
30	30	1.318834	11
31	31	1.318834	11

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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	0.000156	False
1	0.750000	1.000000	dummy_constant	0.750000	1.000000	0.626888	0.312500	0.000134	False
2	1.548480	0.275010	dummy_constant	1.548480	0.275010	1.568048	1.149932	0.000154	False
3	0.241011	0.000000	dummy_constant	0.241011	0.000000	-1.041914	0.317075	0.000157	False
4	0.031652	0.809338	dummy_constant	0.031652	0.809338	-0.425042	0.315040	0.000152	False
5	0.000000	3.110180	dummy_constant	0.000000	3.110180	8.573218	7.063039	0.000153	False
6	0.233733	1.000078	dummy_constant	0.233733	1.000078	0.140971	0.320976	0.000152	False
7	0.925835	0.164186	dummy_constant	0.925835	0.164186	-0.021378	0.294107	0.000154	False
8	0.967857	0.053549	dummy_constant	0.967857	0.053549	-0.123765	0.418209	0.000154	False
9	1.027448	0.096386	dummy_constant	1.027448	0.096386	0.057525	0.441106	0.000149	False
10	0.783281	0.553075	dummy_constant	0.783281	0.553075	0.011047	0.083065	0.000150	False
11	0.062426	0.992776	dummy_constant	0.062426	0.992776	-0.064126	0.434299	0.000176	False
12	0.647138	0.763456	dummy_constant	0.647138	0.763456	-0.023514	0.091059	0.000148	False
13	0.068081	1.032383	dummy_constant	0.068081	1.032383	0.021005	0.469986	0.000154	False
14	0.885582	0.446461	dummy_constant	0.885582	0.446461	-0.053756	0.151540	0.000168	False
15	0.765232	0.672965	dummy_constant	0.765232	0.672965	-0.013443	0.100265	0.000160	False
16	0.583791	0.857496	dummy_constant	0.583791	0.857496	0.175150	0.134825	0.000157	False
17	1.006785	0.016167	dummy_constant	1.006785	0.016167	-0.082842	0.490925	0.000149	False
18	1.029390	0.050055	dummy_constant	1.029390	0.050055	-0.009124	0.482705	0.000155	False
19	0.021907	1.023518	dummy_constant	0.021907	1.023518	-0.046127	0.502644	0.000166	False
20	0.947668	0.344929	dummy_constant	0.947668	0.344929	-0.059560	0.224453	0.000151	False
21	1.061133	0.158745	dummy_constant	1.061133	0.158745	0.223297	0.431325	0.000126	False
22	0.060807	1.044455	dummy_constant	0.060807	1.044455	0.034836	0.489322	0.000151	False
23	0.780059	0.622825	dummy_constant	0.780059	0.622825	0.017738	0.093519	0.000150	False
24	0.976133	0.288116	dummy_constant	0.976133	0.288116	0.047839	0.271598	0.000151	False
25	0.037181	0.030208	dummy_constant	0.037181	0.030208	-0.989816	0.434906	0.000150	False
26	0.995252	0.181084	dummy_constant	0.995252	0.181084	0.119908	0.346982	0.000160	False
27	0.006461	0.025072	dummy_constant	0.006461	0.025072	-0.936689	0.469137	0.000151	False
28	1.042925	0.059027	dummy_constant	1.042925	0.059027	0.029376	0.489224	0.000148	False
29	0.037069	1.024146	dummy_constant	0.037069	1.024146	-0.033458	0.489034	0.000149	False
30	0.869251	0.501421	dummy_constant	0.869251	0.501421	0.056491	0.136348	0.000150	False
31	1.023102	0.029574	dummy_constant	1.023102	0.029574	-0.041887	0.494937	0.000150	False

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