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	0.000155	False
1	0.750000	1.000000	dummy_constant	0.750000	1.000000	0.626888	0.312500	0.000135	False
2	1.399633	0.266099	dummy_constant	1.399633	0.266099	1.128863	0.864049	0.000160	False
3	2.797334	0.852486	dummy_constant	2.797334	0.852486	7.549755	5.401989	0.000153	False
4	0.550963	0.000000	dummy_constant	0.550963	0.000000	-0.796440	0.252597	0.000146	False
5	1.107526	0.177343	dummy_constant	1.107526	0.177343	0.340533	0.473195	0.000150	False
6	0.000000	3.141590	dummy_constant	0.000000	3.141590	8.769588	7.227998	0.000150	False
7	0.389903	1.059314	dummy_constant	0.389903	1.059314	0.193978	0.324954	0.000153	False
8	1.016162	0.099108	dummy_constant	1.016162	0.099108	0.040887	0.427138	0.000151	False
9	0.697516	0.821777	dummy_constant	0.697516	0.821777	0.135646	0.142553	0.000148	False
10	1.002548	0.010400	dummy_constant	1.002548	0.010400	-0.093415	0.492263	0.000148	False
11	0.172140	1.090533	dummy_constant	0.172140	1.090533	0.299304	0.456221	0.000149	False
12	0.901752	0.540057	dummy_constant	0.901752	0.540057	0.175128	0.163009	0.000167	False
13	0.026244	1.007488	dummy_constant	0.026244	1.007488	-0.075722	0.481989	0.000149	False
14	0.093131	1.048678	dummy_constant	0.093131	1.048678	0.093100	0.466590	0.000148	False
15	1.048155	0.059815	dummy_constant	1.048155	0.059815	0.040996	0.494236	0.000150	False
16	0.437831	0.953846	dummy_constant	0.437831	0.953846	0.019100	0.209841	0.000149	False
17	0.039480	1.021959	dummy_constant	0.039480	1.021959	-0.035556	0.484520	0.000153	False
18	0.760770	0.779738	dummy_constant	0.760770	0.779738	0.088696	0.146254	0.000154	False
19	0.945413	0.318184	dummy_constant	0.945413	0.318184	-0.051305	0.231450	0.000152	False
20	0.056750	1.034561	dummy_constant	0.056750	1.034561	0.009575	0.482226	0.000152	False
21	1.033318	0.063656	dummy_constant	1.033318	0.063656	0.016463	0.474824	0.000154	False
22	0.956416	0.366340	dummy_constant	0.956416	0.366340	-0.041941	0.226181	0.000150	False
23	1.031065	0.035002	dummy_constant	1.031065	0.035002	-0.021298	0.498254	0.000147	False
24	0.811105	0.651917	dummy_constant	0.811105	0.651917	0.099145	0.119865	0.000149	False
25	0.756591	0.708810	dummy_constant	0.756591	0.708810	-0.011865	0.109441	0.000149	False
26	1.000465	0.218076	dummy_constant	1.000465	0.218076	0.144246	0.329947	0.000157	False
27	0.574784	0.852321	dummy_constant	0.574784	0.852321	0.156594	0.129723	0.000125	False
28	0.284301	0.994790	dummy_constant	0.284301	0.994790	0.095994	0.291343	0.000149	False
29	0.045751	1.039497	dummy_constant	0.045751	1.039497	0.006405	0.497398	0.000149	False
30	1.037538	0.052504	dummy_constant	1.037538	0.052504	0.010219	0.489200	0.000149	False
31	0.000000	1.022215	dummy_constant	0.000000	1.022215	-0.055077	0.522708	0.000142	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]'}, ylabel='x2'>,
         <Axes: title={'center': 'Posterior SD [y1]'}>],
        [<Axes: title={'center': 'Posterior Mean [y2]'}, ylabel='x2'>,
         <Axes: title={'center': 'Posterior SD [y2]'}>],
        [<Axes: title={'center': 'Posterior Mean [c1]'}, ylabel='x2'>,
         <Axes: title={'center': 'Posterior SD [c1]'}>],
        [<Axes: title={'center': 'Posterior Mean [c2]'}, ylabel='x2'>,
         <Axes: title={'center': 'Posterior SD [c2]'}>],
        [<Axes: title={'center': 'Acq. Function'}, ylabel='x2'>, <Axes: >],
        [<Axes: title={'center': 'Feasibility'}, xlabel='x1', ylabel='x2'>,
         <Axes: xlabel='x1'>]], 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	0.724753	3
6	6	0.724753	3
7	7	0.883443	4
8	8	0.973616	4
9	9	1.030639	4
10	10	1.030639	4
11	11	1.119805	5
12	12	1.150877	5
13	13	1.150877	5
14	14	1.198922	4
15	15	1.216676	5
16	16	1.241303	6
17	17	1.241303	6
18	18	1.247229	7
19	19	1.247229	7
20	20	1.269028	7
21	21	1.269554	8
22	22	1.269554	8
23	23	1.269554	8
24	24	1.281141	9
25	25	1.281141	9
26	26	1.286195	10
27	27	1.298656	11
28	28	1.304762	12
29	29	1.309826	13
30	30	1.313248	13
31	31	1.313248	13

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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	0.000155	False
1	0.750000	1.000000	dummy_constant	0.750000	1.000000	0.626888	0.312500	0.000135	False
2	1.399633	0.266099	dummy_constant	1.399633	0.266099	1.128863	0.864049	0.000160	False
3	2.797334	0.852486	dummy_constant	2.797334	0.852486	7.549755	5.401989	0.000153	False
4	0.550963	0.000000	dummy_constant	0.550963	0.000000	-0.796440	0.252597	0.000146	False
5	1.107526	0.177343	dummy_constant	1.107526	0.177343	0.340533	0.473195	0.000150	False
6	0.000000	3.141590	dummy_constant	0.000000	3.141590	8.769588	7.227998	0.000150	False
7	0.389903	1.059314	dummy_constant	0.389903	1.059314	0.193978	0.324954	0.000153	False
8	1.016162	0.099108	dummy_constant	1.016162	0.099108	0.040887	0.427138	0.000151	False
9	0.697516	0.821777	dummy_constant	0.697516	0.821777	0.135646	0.142553	0.000148	False
10	1.002548	0.010400	dummy_constant	1.002548	0.010400	-0.093415	0.492263	0.000148	False
11	0.172140	1.090533	dummy_constant	0.172140	1.090533	0.299304	0.456221	0.000149	False
12	0.901752	0.540057	dummy_constant	0.901752	0.540057	0.175128	0.163009	0.000167	False
13	0.026244	1.007488	dummy_constant	0.026244	1.007488	-0.075722	0.481989	0.000149	False
14	0.093131	1.048678	dummy_constant	0.093131	1.048678	0.093100	0.466590	0.000148	False
15	1.048155	0.059815	dummy_constant	1.048155	0.059815	0.040996	0.494236	0.000150	False
16	0.437831	0.953846	dummy_constant	0.437831	0.953846	0.019100	0.209841	0.000149	False
17	0.039480	1.021959	dummy_constant	0.039480	1.021959	-0.035556	0.484520	0.000153	False
18	0.760770	0.779738	dummy_constant	0.760770	0.779738	0.088696	0.146254	0.000154	False
19	0.945413	0.318184	dummy_constant	0.945413	0.318184	-0.051305	0.231450	0.000152	False
20	0.056750	1.034561	dummy_constant	0.056750	1.034561	0.009575	0.482226	0.000152	False
21	1.033318	0.063656	dummy_constant	1.033318	0.063656	0.016463	0.474824	0.000154	False
22	0.956416	0.366340	dummy_constant	0.956416	0.366340	-0.041941	0.226181	0.000150	False
23	1.031065	0.035002	dummy_constant	1.031065	0.035002	-0.021298	0.498254	0.000147	False
24	0.811105	0.651917	dummy_constant	0.811105	0.651917	0.099145	0.119865	0.000149	False
25	0.756591	0.708810	dummy_constant	0.756591	0.708810	-0.011865	0.109441	0.000149	False
26	1.000465	0.218076	dummy_constant	1.000465	0.218076	0.144246	0.329947	0.000157	False
27	0.574784	0.852321	dummy_constant	0.574784	0.852321	0.156594	0.129723	0.000125	False
28	0.284301	0.994790	dummy_constant	0.284301	0.994790	0.095994	0.291343	0.000149	False
29	0.045751	1.039497	dummy_constant	0.045751	1.039497	0.006405	0.497398	0.000149	False
30	1.037538	0.052504	dummy_constant	1.037538	0.052504	0.010219	0.489200	0.000149	False
31	0.000000	1.022215	dummy_constant	0.000000	1.022215	-0.055077	0.522708	0.000142	False

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