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/torch/jit/_script.py:1491: FutureWarning: `torch.jit.script` is deprecated. Please switch to `torch.compile` or `torch.export`.
  warnings.warn(

{'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	0.000171	False
1	0.750000	1.000000	dummy_constant	0.750000	1.000000	0.626888	0.312500	0.000141	False
2	1.673149	0.372832	dummy_constant	1.673149	0.372832	2.031794	1.392451	0.000168	False
3	0.570814	0.294095	dummy_constant	0.570814	0.294095	-0.611642	0.047411	0.000164	False
4	0.024747	0.008495	dummy_constant	0.024747	0.008495	-1.053990	0.467443	0.000163	False
5	0.989565	0.002450	dummy_constant	0.989565	0.002450	-0.120678	0.487230	0.000166	False
6	0.419617	0.874213	dummy_constant	0.419617	0.874213	-0.123616	0.146497	0.000159	False
7	1.022063	0.192238	dummy_constant	1.022063	0.192238	0.180177	0.367267	0.000162	False
8	0.012211	3.141590	dummy_constant	0.012211	3.141590	8.769930	7.215936	0.000166	False
9	0.302307	1.046188	dummy_constant	0.302307	1.046188	0.206903	0.337404	0.000167	False
10	0.796411	0.622960	dummy_constant	0.796411	0.622960	0.058959	0.102979	0.000163	False
11	0.155367	1.015335	dummy_constant	0.155367	1.015335	0.130743	0.384342	0.000175	False
12	1.035883	0.095831	dummy_constant	1.035883	0.095831	0.072770	0.450524	0.000165	False
13	0.026549	0.981313	dummy_constant	0.026549	0.981313	-0.127100	0.455818	0.000166	False
14	0.042011	1.027173	dummy_constant	0.042011	1.027173	-0.022514	0.487666	0.000161	False
15	0.066857	1.037074	dummy_constant	0.066857	1.037074	0.028514	0.476061	0.000194	False
16	0.037551	0.004058	dummy_constant	0.037551	0.004058	-0.983492	0.459818	0.000159	False
17	0.434973	0.934133	dummy_constant	0.434973	0.934133	-0.015356	0.192700	0.000164	False
18	0.902080	0.383827	dummy_constant	0.902080	0.383827	-0.137754	0.175165	0.000157	False
19	1.026986	0.040093	dummy_constant	1.026986	0.040093	-0.024829	0.489228	0.000165	False
20	0.936553	0.403149	dummy_constant	0.936553	0.403149	-0.057916	0.199958	0.000168	False
21	0.684478	0.774763	dummy_constant	0.684478	0.774763	0.013789	0.109527	0.000158	False
22	0.014779	1.010722	dummy_constant	0.014779	1.010722	-0.075499	0.496276	0.000158	False
23	0.824602	0.666005	dummy_constant	0.824602	0.666005	0.136017	0.132924	0.000182	False
24	0.085083	1.047275	dummy_constant	0.085083	1.047275	0.076987	0.471666	0.000180	False
25	0.022017	0.000000	dummy_constant	0.022017	0.000000	-1.099515	0.478467	0.000177	False
26	1.050884	0.068011	dummy_constant	1.050884	0.068011	0.057848	0.490088	0.000177	False
27	0.963228	0.462706	dummy_constant	0.963228	0.462706	0.078341	0.215971	0.000161	False
28	0.022453	0.000078	dummy_constant	0.022453	0.000078	-1.099343	0.477974	0.000170	False
29	0.036150	1.033686	dummy_constant	0.036150	1.033686	-0.014947	0.499978	0.000331	False
30	1.016185	0.025547	dummy_constant	1.016185	0.025547	-0.058737	0.491552	0.000161	False
31	0.785797	0.690130	dummy_constant	0.785797	0.690130	0.042760	0.117830	0.000164	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	0.766575	3
8	8	0.766575	3
9	9	0.969744	4
10	10	1.049307	4
11	11	1.134337	4
12	12	1.179081	5
13	13	1.179081	5
14	14	1.179081	5
15	15	1.220055	6
16	16	1.220055	6
17	17	1.220055	6
18	18	1.220055	6
19	19	1.220055	6
20	20	1.220055	6
21	21	1.246271	6
22	22	1.246271	6
23	23	1.246271	6
24	24	1.246271	6
25	25	1.246271	6
26	26	1.258766	7
27	27	1.268194	8
28	28	1.268194	8
29	29	1.268194	8
30	30	1.268194	8
31	31	1.269092	9

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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	0.000171	False
1	0.750000	1.000000	dummy_constant	0.750000	1.000000	0.626888	0.312500	0.000141	False
2	1.673149	0.372832	dummy_constant	1.673149	0.372832	2.031794	1.392451	0.000168	False
3	0.570814	0.294095	dummy_constant	0.570814	0.294095	-0.611642	0.047411	0.000164	False
4	0.024747	0.008495	dummy_constant	0.024747	0.008495	-1.053990	0.467443	0.000163	False
5	0.989565	0.002450	dummy_constant	0.989565	0.002450	-0.120678	0.487230	0.000166	False
6	0.419617	0.874213	dummy_constant	0.419617	0.874213	-0.123616	0.146497	0.000159	False
7	1.022063	0.192238	dummy_constant	1.022063	0.192238	0.180177	0.367267	0.000162	False
8	0.012211	3.141590	dummy_constant	0.012211	3.141590	8.769930	7.215936	0.000166	False
9	0.302307	1.046188	dummy_constant	0.302307	1.046188	0.206903	0.337404	0.000167	False
10	0.796411	0.622960	dummy_constant	0.796411	0.622960	0.058959	0.102979	0.000163	False
11	0.155367	1.015335	dummy_constant	0.155367	1.015335	0.130743	0.384342	0.000175	False
12	1.035883	0.095831	dummy_constant	1.035883	0.095831	0.072770	0.450524	0.000165	False
13	0.026549	0.981313	dummy_constant	0.026549	0.981313	-0.127100	0.455818	0.000166	False
14	0.042011	1.027173	dummy_constant	0.042011	1.027173	-0.022514	0.487666	0.000161	False
15	0.066857	1.037074	dummy_constant	0.066857	1.037074	0.028514	0.476061	0.000194	False
16	0.037551	0.004058	dummy_constant	0.037551	0.004058	-0.983492	0.459818	0.000159	False
17	0.434973	0.934133	dummy_constant	0.434973	0.934133	-0.015356	0.192700	0.000164	False
18	0.902080	0.383827	dummy_constant	0.902080	0.383827	-0.137754	0.175165	0.000157	False
19	1.026986	0.040093	dummy_constant	1.026986	0.040093	-0.024829	0.489228	0.000165	False
20	0.936553	0.403149	dummy_constant	0.936553	0.403149	-0.057916	0.199958	0.000168	False
21	0.684478	0.774763	dummy_constant	0.684478	0.774763	0.013789	0.109527	0.000158	False
22	0.014779	1.010722	dummy_constant	0.014779	1.010722	-0.075499	0.496276	0.000158	False
23	0.824602	0.666005	dummy_constant	0.824602	0.666005	0.136017	0.132924	0.000182	False
24	0.085083	1.047275	dummy_constant	0.085083	1.047275	0.076987	0.471666	0.000180	False
25	0.022017	0.000000	dummy_constant	0.022017	0.000000	-1.099515	0.478467	0.000177	False
26	1.050884	0.068011	dummy_constant	1.050884	0.068011	0.057848	0.490088	0.000177	False
27	0.963228	0.462706	dummy_constant	0.963228	0.462706	0.078341	0.215971	0.000161	False
28	0.022453	0.000078	dummy_constant	0.022453	0.000078	-1.099343	0.477974	0.000170	False
29	0.036150	1.033686	dummy_constant	0.036150	1.033686	-0.014947	0.499978	0.000331	False
30	1.016185	0.025547	dummy_constant	1.016185	0.025547	-0.058737	0.491552	0.000161	False
31	0.785797	0.690130	dummy_constant	0.785797	0.690130	0.042760	0.117830	0.000164	False