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())

/home/runner/work/Xopt/Xopt/.venv/lib/python3.12/site-packages/pyro/ops/stats.py:527: SyntaxWarning: invalid escape sequence '\g'
  we have :math:`ES^{*}(P,Q) \ge ES^{*}(Q,Q)` with equality holding if and only if :math:`P=Q`, i.e.

{'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	0.000200	False
1	0.750000	1.000000	dummy_constant	0.750000	1.000000	0.626888	0.312500	0.000147	False
2	1.200572	0.209261	dummy_constant	1.200572	0.209261	0.578010	0.575330	0.005630	False
3	0.622550	2.614008	dummy_constant	0.622550	2.614008	6.303182	4.484049	0.006105	False
4	0.152128	0.421849	dummy_constant	0.152128	0.421849	-0.872375	0.127123	0.005201	False
5	0.000000	1.014355	dummy_constant	0.000000	1.014355	-0.071084	0.514561	0.008362	False
6	3.141590	0.000000	dummy_constant	3.141590	0.000000	8.769588	7.227998	0.002934	False
7	1.091615	0.066447	dummy_constant	1.091615	0.066447	0.139733	0.537977	0.005341	False
8	0.190775	1.070569	dummy_constant	0.190775	1.070569	0.277436	0.421169	0.000150	False
9	1.059136	0.046217	dummy_constant	1.059136	0.046217	0.047277	0.518552	0.000250	False
10	1.056371	0.411265	dummy_constant	1.056371	0.411265	0.190882	0.317423	0.004777	False
11	1.081684	0.122833	dummy_constant	1.081684	0.122833	0.208740	0.480611	0.005176	False
12	0.718721	0.796147	dummy_constant	0.718721	0.796147	0.081973	0.135542	0.002375	False
13	0.066638	1.032700	dummy_constant	0.066638	1.032700	0.019515	0.471572	0.002819	False
14	0.949060	0.516100	dummy_constant	0.949060	0.516100	0.178566	0.201914	0.000207	False
15	0.687087	0.844516	dummy_constant	0.687087	0.844516	0.192094	0.153693	0.000147	False
16	0.749951	0.718992	dummy_constant	0.749951	0.718992	-0.014993	0.110433	0.000197	False
17	1.041533	0.054539	dummy_constant	1.041533	0.054539	0.020802	0.491694	0.002497	False
18	0.042266	1.024486	dummy_constant	0.042266	1.024486	-0.027659	0.484605	0.002675	False
19	0.000000	0.000000	dummy_constant	0.000000	0.000000	-1.100000	0.500000	0.002263	False
20	1.027577	0.408301	dummy_constant	1.027577	0.408301	0.125298	0.286746	0.002465	False
21	0.187828	1.001253	dummy_constant	0.187828	1.001253	0.136267	0.348706	0.002709	False
22	0.902561	0.534154	dummy_constant	0.902561	0.534154	0.164064	0.163222	0.003056	False
23	0.397843	0.925733	dummy_constant	0.397843	0.925733	-0.082519	0.191684	0.000167	False
24	0.027704	1.024683	dummy_constant	0.027704	1.024683	-0.040050	0.498355	0.002115	False
25	0.504045	0.891383	dummy_constant	0.504045	0.891383	0.085739	0.153197	0.000158	False
26	0.655820	0.830485	dummy_constant	0.655820	0.830485	0.149442	0.133500	0.000170	False
27	0.022662	1.017569	dummy_constant	0.022662	1.017569	-0.057760	0.495729	0.000166	False
28	1.023018	0.008732	dummy_constant	1.023018	0.008732	-0.052427	0.514892	0.000167	False
29	1.028859	0.068942	dummy_constant	1.028859	0.068942	0.015339	0.465503	0.000158	False
30	1.005291	0.261901	dummy_constant	1.005291	0.261901	0.138494	0.312010	0.000154	False
31	0.980664	0.000000	dummy_constant	0.980664	0.000000	-0.138298	0.481038	0.000183	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	0.500000	2
8	8	0.740149	3
9	9	0.740149	3
10	10	0.890421	4
11	11	1.011077	5
12	12	1.070624	5
13	13	1.148626	5
14	14	1.176076	5
15	15	1.182030	6
16	16	1.182030	6
17	17	1.226476	5
18	18	1.226476	5
19	19	1.226476	5
20	20	1.227981	6
21	21	1.243681	7
22	22	1.255863	8
23	23	1.255863	8
24	24	1.255863	8
25	25	1.275974	9
26	26	1.278322	9
27	27	1.278322	9
28	28	1.278322	9
29	29	1.282623	10
30	30	1.288476	10
31	31	1.288476	10

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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	0.000200	False
1	0.750000	1.000000	dummy_constant	0.750000	1.000000	0.626888	0.312500	0.000147	False
2	1.200572	0.209261	dummy_constant	1.200572	0.209261	0.578010	0.575330	0.005630	False
3	0.622550	2.614008	dummy_constant	0.622550	2.614008	6.303182	4.484049	0.006105	False
4	0.152128	0.421849	dummy_constant	0.152128	0.421849	-0.872375	0.127123	0.005201	False
5	0.000000	1.014355	dummy_constant	0.000000	1.014355	-0.071084	0.514561	0.008362	False
6	3.141590	0.000000	dummy_constant	3.141590	0.000000	8.769588	7.227998	0.002934	False
7	1.091615	0.066447	dummy_constant	1.091615	0.066447	0.139733	0.537977	0.005341	False
8	0.190775	1.070569	dummy_constant	0.190775	1.070569	0.277436	0.421169	0.000150	False
9	1.059136	0.046217	dummy_constant	1.059136	0.046217	0.047277	0.518552	0.000250	False
10	1.056371	0.411265	dummy_constant	1.056371	0.411265	0.190882	0.317423	0.004777	False
11	1.081684	0.122833	dummy_constant	1.081684	0.122833	0.208740	0.480611	0.005176	False
12	0.718721	0.796147	dummy_constant	0.718721	0.796147	0.081973	0.135542	0.002375	False
13	0.066638	1.032700	dummy_constant	0.066638	1.032700	0.019515	0.471572	0.002819	False
14	0.949060	0.516100	dummy_constant	0.949060	0.516100	0.178566	0.201914	0.000207	False
15	0.687087	0.844516	dummy_constant	0.687087	0.844516	0.192094	0.153693	0.000147	False
16	0.749951	0.718992	dummy_constant	0.749951	0.718992	-0.014993	0.110433	0.000197	False
17	1.041533	0.054539	dummy_constant	1.041533	0.054539	0.020802	0.491694	0.002497	False
18	0.042266	1.024486	dummy_constant	0.042266	1.024486	-0.027659	0.484605	0.002675	False
19	0.000000	0.000000	dummy_constant	0.000000	0.000000	-1.100000	0.500000	0.002263	False
20	1.027577	0.408301	dummy_constant	1.027577	0.408301	0.125298	0.286746	0.002465	False
21	0.187828	1.001253	dummy_constant	0.187828	1.001253	0.136267	0.348706	0.002709	False
22	0.902561	0.534154	dummy_constant	0.902561	0.534154	0.164064	0.163222	0.003056	False
23	0.397843	0.925733	dummy_constant	0.397843	0.925733	-0.082519	0.191684	0.000167	False
24	0.027704	1.024683	dummy_constant	0.027704	1.024683	-0.040050	0.498355	0.002115	False
25	0.504045	0.891383	dummy_constant	0.504045	0.891383	0.085739	0.153197	0.000158	False
26	0.655820	0.830485	dummy_constant	0.655820	0.830485	0.149442	0.133500	0.000170	False
27	0.022662	1.017569	dummy_constant	0.022662	1.017569	-0.057760	0.495729	0.000166	False
28	1.023018	0.008732	dummy_constant	1.023018	0.008732	-0.052427	0.514892	0.000167	False
29	1.028859	0.068942	dummy_constant	1.028859	0.068942	0.015339	0.465503	0.000158	False
30	1.005291	0.261901	dummy_constant	1.005291	0.261901	0.138494	0.312010	0.000154	False
31	0.980664	0.000000	dummy_constant	0.980664	0.000000	-0.138298	0.481038	0.000183	False