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	0.004124	False
1	0.750000	1.000000	dummy_constant	0.750000	1.000000	0.626888	0.312500	0.000274	False
2	0.533506	1.467016	dummy_constant	0.533506	1.467016	1.360435	0.936242	0.002221	False
3	0.424530	0.575105	dummy_constant	0.424530	0.575105	-0.415826	0.011336	0.005146	False
4	0.006439	0.027089	dummy_constant	0.006439	0.027089	-0.916261	0.467247	0.002707	False
5	3.141590	0.000000	dummy_constant	3.141590	0.000000	8.769588	7.227998	0.005047	False
6	1.075811	0.054487	dummy_constant	1.075811	0.054487	0.091365	0.530040	0.002462	False
7	1.050112	0.077596	dummy_constant	1.050112	0.077596	0.070678	0.481048	0.000295	False
8	0.000000	0.709647	dummy_constant	0.000000	0.709647	-0.596401	0.293952	0.000503	False
9	0.675077	0.719971	dummy_constant	0.675077	0.719971	-0.112956	0.079039	0.000289	False
10	0.061557	1.029337	dummy_constant	0.061557	1.029337	0.005620	0.472430	0.000258	False
11	0.918135	0.547919	dummy_constant	0.918135	0.547919	0.211696	0.177133	0.000257	False
12	0.203425	1.067807	dummy_constant	0.203425	1.067807	0.280756	0.410362	0.000274	False
13	1.003244	0.000000	dummy_constant	1.003244	0.000000	-0.093502	0.503254	0.000274	False
14	0.929191	0.433088	dummy_constant	0.929191	0.433088	-0.025825	0.188682	0.000211	False
15	0.429184	0.918798	dummy_constant	0.429184	0.918798	-0.047531	0.180406	0.000277	False
16	0.749176	0.711948	dummy_constant	0.749176	0.711948	-0.023674	0.107011	0.000282	False
17	0.000000	0.040147	dummy_constant	0.000000	0.040147	-1.098388	0.461464	0.000282	False
18	0.841903	0.601629	dummy_constant	0.841903	0.601629	0.158391	0.127226	0.000267	False
19	0.000000	0.998973	dummy_constant	0.000000	0.998973	-0.102053	0.498974	0.000267	False
20	1.016075	0.277714	dummy_constant	1.016075	0.277714	0.152447	0.315745	0.000263	False
21	0.566981	0.846027	dummy_constant	0.566981	0.846027	0.137205	0.124221	0.000268	False
22	0.093192	1.059117	dummy_constant	0.093192	1.059117	0.113834	0.478105	0.000201	False
23	1.028702	0.043706	dummy_constant	1.028702	0.043706	-0.017659	0.487730	0.000264	False
24	0.030233	1.022616	dummy_constant	0.030233	1.022616	-0.042368	0.493809	0.000265	False
25	1.020429	0.028379	dummy_constant	1.020429	0.028379	-0.048186	0.493273	0.000270	False
26	0.308193	0.960113	dummy_constant	0.308193	0.960113	-0.008650	0.248494	0.000208	False
27	1.039267	0.052099	dummy_constant	1.039267	0.052099	0.013221	0.491424	0.000270	False
28	0.602232	0.812353	dummy_constant	0.602232	0.812353	0.093537	0.108016	0.000262	False
29	0.037359	1.024318	dummy_constant	0.037359	1.024318	-0.032841	0.488946	0.000261	False
30	1.046122	0.059531	dummy_constant	1.046122	0.059531	0.036503	0.492262	0.000270	False
31	0.777920	0.655526	dummy_constant	0.777920	0.655526	0.014229	0.101428	0.000262	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	0.802506	3
8	8	0.802506	3
9	9	0.802506	3
10	10	1.126531	4
11	11	1.173667	4
12	12	1.173667	4
13	13	1.173667	4
14	14	1.173667	4
15	15	1.173667	4
16	16	1.173667	4
17	17	1.173667	4
18	18	1.204036	5
19	19	1.204036	5
20	20	1.213233	6
21	21	1.260933	6
22	22	1.260933	6
23	23	1.260933	6
24	24	1.260933	6
25	25	1.260933	6
26	26	1.260933	6
27	27	1.274850	6
28	28	1.282921	7
29	29	1.282921	7
30	30	1.282921	7
31	31	1.292955	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	0.004124	False
1	0.750000	1.000000	dummy_constant	0.750000	1.000000	0.626888	0.312500	0.000274	False
2	0.533506	1.467016	dummy_constant	0.533506	1.467016	1.360435	0.936242	0.002221	False
3	0.424530	0.575105	dummy_constant	0.424530	0.575105	-0.415826	0.011336	0.005146	False
4	0.006439	0.027089	dummy_constant	0.006439	0.027089	-0.916261	0.467247	0.002707	False
5	3.141590	0.000000	dummy_constant	3.141590	0.000000	8.769588	7.227998	0.005047	False
6	1.075811	0.054487	dummy_constant	1.075811	0.054487	0.091365	0.530040	0.002462	False
7	1.050112	0.077596	dummy_constant	1.050112	0.077596	0.070678	0.481048	0.000295	False
8	0.000000	0.709647	dummy_constant	0.000000	0.709647	-0.596401	0.293952	0.000503	False
9	0.675077	0.719971	dummy_constant	0.675077	0.719971	-0.112956	0.079039	0.000289	False
10	0.061557	1.029337	dummy_constant	0.061557	1.029337	0.005620	0.472430	0.000258	False
11	0.918135	0.547919	dummy_constant	0.918135	0.547919	0.211696	0.177133	0.000257	False
12	0.203425	1.067807	dummy_constant	0.203425	1.067807	0.280756	0.410362	0.000274	False
13	1.003244	0.000000	dummy_constant	1.003244	0.000000	-0.093502	0.503254	0.000274	False
14	0.929191	0.433088	dummy_constant	0.929191	0.433088	-0.025825	0.188682	0.000211	False
15	0.429184	0.918798	dummy_constant	0.429184	0.918798	-0.047531	0.180406	0.000277	False
16	0.749176	0.711948	dummy_constant	0.749176	0.711948	-0.023674	0.107011	0.000282	False
17	0.000000	0.040147	dummy_constant	0.000000	0.040147	-1.098388	0.461464	0.000282	False
18	0.841903	0.601629	dummy_constant	0.841903	0.601629	0.158391	0.127226	0.000267	False
19	0.000000	0.998973	dummy_constant	0.000000	0.998973	-0.102053	0.498974	0.000267	False
20	1.016075	0.277714	dummy_constant	1.016075	0.277714	0.152447	0.315745	0.000263	False
21	0.566981	0.846027	dummy_constant	0.566981	0.846027	0.137205	0.124221	0.000268	False
22	0.093192	1.059117	dummy_constant	0.093192	1.059117	0.113834	0.478105	0.000201	False
23	1.028702	0.043706	dummy_constant	1.028702	0.043706	-0.017659	0.487730	0.000264	False
24	0.030233	1.022616	dummy_constant	0.030233	1.022616	-0.042368	0.493809	0.000265	False
25	1.020429	0.028379	dummy_constant	1.020429	0.028379	-0.048186	0.493273	0.000270	False
26	0.308193	0.960113	dummy_constant	0.308193	0.960113	-0.008650	0.248494	0.000208	False
27	1.039267	0.052099	dummy_constant	1.039267	0.052099	0.013221	0.491424	0.000270	False
28	0.602232	0.812353	dummy_constant	0.602232	0.812353	0.093537	0.108016	0.000262	False
29	0.037359	1.024318	dummy_constant	0.037359	1.024318	-0.032841	0.488946	0.000261	False
30	1.046122	0.059531	dummy_constant	1.046122	0.059531	0.036503	0.492262	0.000270	False
31	0.777920	0.655526	dummy_constant	0.777920	0.655526	0.014229	0.101428	0.000262	False