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	0.000237	False
1	0.750000	1.000000	dummy_constant	0.750000	1.000000	0.626888	0.312500	0.000137	False
2	0.443152	1.573545	dummy_constant	0.443152	1.573545	1.703896	1.155731	0.000158	False
3	0.288393	0.518940	dummy_constant	0.288393	0.518940	-0.621650	0.045136	0.000150	False
4	0.000000	0.000000	dummy_constant	0.000000	0.000000	-1.100000	0.500000	0.000152	False
5	0.023187	0.906145	dummy_constant	0.023187	0.906145	-0.270103	0.392304	0.000162	False
6	3.141590	0.334401	dummy_constant	3.141590	0.334401	8.993970	7.005421	0.000153	False
7	0.153376	1.025569	dummy_constant	0.153376	1.025569	0.147361	0.396371	0.000152	False
8	0.928655	0.182506	dummy_constant	0.928655	0.182506	-0.004358	0.284548	0.000156	False
9	0.996561	0.092324	dummy_constant	0.996561	0.092324	-0.007601	0.412773	0.000155	False
10	0.768162	0.629417	dummy_constant	0.768162	0.629417	-0.012520	0.088660	0.000150	False
11	1.034047	0.113721	dummy_constant	1.034047	0.113721	0.100264	0.434417	0.000150	False
12	1.013023	0.033168	dummy_constant	1.013023	0.033168	-0.059282	0.481125	0.000150	False
13	0.467963	0.879828	dummy_constant	0.467963	0.879828	-0.010191	0.145295	0.000151	False
14	0.000322	1.008081	dummy_constant	0.000322	1.008081	-0.083771	0.507825	0.000149	False
15	0.631064	0.768268	dummy_constant	0.631064	0.768268	-0.012224	0.089145	0.000154	False
16	0.900051	0.471779	dummy_constant	0.900051	0.471779	0.019778	0.160838	0.000153	False
17	1.010050	0.012295	dummy_constant	1.010050	0.012295	-0.077757	0.498007	0.000150	False
18	0.708258	0.729352	dummy_constant	0.708258	0.729352	-0.063674	0.095974	0.000149	False
19	0.839502	0.582848	dummy_constant	0.839502	0.582848	0.140435	0.122126	0.000153	False
20	0.065480	1.044557	dummy_constant	0.065480	1.044557	0.041499	0.485350	0.000156	False
21	0.034678	0.007846	dummy_constant	0.034678	0.007846	-0.907373	0.458739	0.000152	False
22	0.331845	0.962579	dummy_constant	0.331845	0.962579	-0.019733	0.242256	0.000159	False
23	1.013091	0.004432	dummy_constant	1.013091	0.004432	-0.073381	0.508850	0.000169	False
24	1.036434	0.052532	dummy_constant	1.036434	0.052532	0.008026	0.487989	0.000154	False
25	0.951654	0.372634	dummy_constant	0.951654	0.372634	-0.050678	0.220213	0.000153	False
26	1.000375	0.005699	dummy_constant	1.000375	0.005699	-0.098802	0.494709	0.000153	False
27	0.095930	1.058826	dummy_constant	0.095930	1.058826	0.117835	0.475560	0.000154	False
28	0.087313	1.017469	dummy_constant	0.087313	1.017469	0.022888	0.438084	0.000157	False
29	0.041737	1.033315	dummy_constant	0.041737	1.033315	-0.010373	0.494430	0.000152	False
30	1.022290	0.032071	dummy_constant	1.022290	0.032071	-0.041567	0.491744	0.000147	False
31	1.047658	0.072998	dummy_constant	1.047658	0.072998	0.058722	0.482260	0.000148	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	0.500000	2
7	7	0.783057	3
8	8	0.783057	3
9	9	0.783057	3
10	10	0.783057	3
11	11	1.079533	4
12	12	1.079533	4
13	13	1.079533	4
14	14	1.079533	4
15	15	1.079533	4
16	16	1.141800	4
17	17	1.141800	4
18	18	1.141800	4
19	19	1.167058	5
20	20	1.207090	6
21	21	1.207090	6
22	22	1.207090	6
23	23	1.207090	6
24	24	1.235455	7
25	25	1.235455	7
26	26	1.235455	7
27	27	1.235455	7
28	28	1.242078	7
29	29	1.242078	7
30	30	1.242078	7
31	31	1.242078	7

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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	0.000237	False
1	0.750000	1.000000	dummy_constant	0.750000	1.000000	0.626888	0.312500	0.000137	False
2	0.443152	1.573545	dummy_constant	0.443152	1.573545	1.703896	1.155731	0.000158	False
3	0.288393	0.518940	dummy_constant	0.288393	0.518940	-0.621650	0.045136	0.000150	False
4	0.000000	0.000000	dummy_constant	0.000000	0.000000	-1.100000	0.500000	0.000152	False
5	0.023187	0.906145	dummy_constant	0.023187	0.906145	-0.270103	0.392304	0.000162	False
6	3.141590	0.334401	dummy_constant	3.141590	0.334401	8.993970	7.005421	0.000153	False
7	0.153376	1.025569	dummy_constant	0.153376	1.025569	0.147361	0.396371	0.000152	False
8	0.928655	0.182506	dummy_constant	0.928655	0.182506	-0.004358	0.284548	0.000156	False
9	0.996561	0.092324	dummy_constant	0.996561	0.092324	-0.007601	0.412773	0.000155	False
10	0.768162	0.629417	dummy_constant	0.768162	0.629417	-0.012520	0.088660	0.000150	False
11	1.034047	0.113721	dummy_constant	1.034047	0.113721	0.100264	0.434417	0.000150	False
12	1.013023	0.033168	dummy_constant	1.013023	0.033168	-0.059282	0.481125	0.000150	False
13	0.467963	0.879828	dummy_constant	0.467963	0.879828	-0.010191	0.145295	0.000151	False
14	0.000322	1.008081	dummy_constant	0.000322	1.008081	-0.083771	0.507825	0.000149	False
15	0.631064	0.768268	dummy_constant	0.631064	0.768268	-0.012224	0.089145	0.000154	False
16	0.900051	0.471779	dummy_constant	0.900051	0.471779	0.019778	0.160838	0.000153	False
17	1.010050	0.012295	dummy_constant	1.010050	0.012295	-0.077757	0.498007	0.000150	False
18	0.708258	0.729352	dummy_constant	0.708258	0.729352	-0.063674	0.095974	0.000149	False
19	0.839502	0.582848	dummy_constant	0.839502	0.582848	0.140435	0.122126	0.000153	False
20	0.065480	1.044557	dummy_constant	0.065480	1.044557	0.041499	0.485350	0.000156	False
21	0.034678	0.007846	dummy_constant	0.034678	0.007846	-0.907373	0.458739	0.000152	False
22	0.331845	0.962579	dummy_constant	0.331845	0.962579	-0.019733	0.242256	0.000159	False
23	1.013091	0.004432	dummy_constant	1.013091	0.004432	-0.073381	0.508850	0.000169	False
24	1.036434	0.052532	dummy_constant	1.036434	0.052532	0.008026	0.487989	0.000154	False
25	0.951654	0.372634	dummy_constant	0.951654	0.372634	-0.050678	0.220213	0.000153	False
26	1.000375	0.005699	dummy_constant	1.000375	0.005699	-0.098802	0.494709	0.000153	False
27	0.095930	1.058826	dummy_constant	0.095930	1.058826	0.117835	0.475560	0.000154	False
28	0.087313	1.017469	dummy_constant	0.087313	1.017469	0.022888	0.438084	0.000157	False
29	0.041737	1.033315	dummy_constant	0.041737	1.033315	-0.010373	0.494430	0.000152	False
30	1.022290	0.032071	dummy_constant	1.022290	0.032071	-0.041567	0.491744	0.000147	False
31	1.047658	0.072998	dummy_constant	1.047658	0.072998	0.058722	0.482260	0.000148	False

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