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	0.000231	False
1	0.750000	1.000000	dummy_constant	0.750000	1.000000	0.626888	0.312500	0.000137	False
2	0.338271	1.660682	dummy_constant	0.338271	1.660682	1.972021	1.373338	0.000153	False
3	0.205007	0.571904	dummy_constant	0.205007	0.571904	-0.702266	0.092191	0.000150	False
4	0.000000	0.000000	dummy_constant	0.000000	0.000000	-1.100000	0.500000	0.000152	False
5	0.000000	0.903686	dummy_constant	0.000000	0.903686	-0.283351	0.412963	0.000154	False
6	0.278085	0.979258	dummy_constant	0.278085	0.979258	0.064424	0.278934	0.000150	False
7	3.085734	0.000000	dummy_constant	3.085734	0.000000	8.421757	6.936022	0.000153	False
8	0.944973	0.140977	dummy_constant	0.944973	0.140977	-0.015507	0.326898	0.000152	False
9	0.985903	0.061507	dummy_constant	0.985903	0.061507	-0.078503	0.428378	0.000149	False
10	1.022044	0.104556	dummy_constant	1.022044	0.104556	0.061535	0.428906	0.000151	False
11	0.085747	1.044210	dummy_constant	0.085747	1.044210	0.072031	0.467770	0.000153	False
12	0.785835	0.643724	dummy_constant	0.785835	0.643724	0.033371	0.102358	0.000152	False
13	1.010642	0.032004	dummy_constant	1.010642	0.032004	-0.065023	0.479775	0.000152	False
14	0.585311	0.825508	dummy_constant	0.585311	0.825508	0.114381	0.113233	0.000106	False
15	1.039659	0.068549	dummy_constant	1.039659	0.068549	0.036130	0.477382	0.000150	False
16	0.046488	1.031537	dummy_constant	0.046488	1.031537	-0.008913	0.488205	0.000156	False
17	0.898019	0.466369	dummy_constant	0.898019	0.466369	0.005041	0.159550	0.000154	False
18	0.423012	0.894659	dummy_constant	0.423012	0.894659	-0.091491	0.161683	0.000154	False
19	0.971871	0.329689	dummy_constant	0.971871	0.329689	0.003507	0.251669	0.000153	False
20	1.023677	0.034798	dummy_constant	1.023677	0.034798	-0.036456	0.490650	0.000154	False
21	0.030619	0.000000	dummy_constant	0.030619	0.000000	-1.099063	0.470319	0.000156	False
22	0.857606	0.555516	dummy_constant	0.857606	0.555516	0.141497	0.130964	0.000156	False
23	0.800518	0.592865	dummy_constant	0.800518	0.592865	0.063789	0.098935	0.000153	False
24	0.393268	0.927788	dummy_constant	0.393268	0.927788	-0.083685	0.194394	0.000116	False
25	1.034970	0.043078	dummy_constant	1.034970	0.043078	-0.005638	0.494970	0.000154	False
26	0.000000	0.101538	dummy_constant	0.000000	0.101538	-1.089690	0.408772	0.000156	False
27	0.500129	0.884016	dummy_constant	0.500129	0.884016	0.069043	0.147468	0.000154	False
28	0.015160	1.018749	dummy_constant	0.015160	1.018749	-0.059100	0.504171	0.000153	False
29	1.002915	0.004609	dummy_constant	1.002915	0.004609	-0.093871	0.498336	0.000148	False
30	0.033497	1.027672	dummy_constant	0.033497	1.027672	-0.029484	0.496062	0.000143	False
31	0.697331	0.717353	dummy_constant	0.697331	0.717353	-0.096582	0.086182	0.000154	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	0.500000	2
6	6	0.750932	2
7	7	0.750932	2
8	8	0.750932	2
9	9	0.750932	2
10	10	1.059426	3
11	11	1.147092	4
12	12	1.221294	4
13	13	1.221294	4
14	14	1.252125	5
15	15	1.268700	6
16	16	1.268700	6
17	17	1.290696	7
18	18	1.290696	7
19	19	1.297554	8
20	20	1.297554	8
21	21	1.297554	8
22	22	1.301119	9
23	23	1.304022	10
24	24	1.304022	10
25	25	1.304022	10
26	26	1.304022	10
27	27	1.312135	11
28	28	1.312135	11
29	29	1.312135	11
30	30	1.312135	11
31	31	1.312135	11

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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	0.000231	False
1	0.750000	1.000000	dummy_constant	0.750000	1.000000	0.626888	0.312500	0.000137	False
2	0.338271	1.660682	dummy_constant	0.338271	1.660682	1.972021	1.373338	0.000153	False
3	0.205007	0.571904	dummy_constant	0.205007	0.571904	-0.702266	0.092191	0.000150	False
4	0.000000	0.000000	dummy_constant	0.000000	0.000000	-1.100000	0.500000	0.000152	False
5	0.000000	0.903686	dummy_constant	0.000000	0.903686	-0.283351	0.412963	0.000154	False
6	0.278085	0.979258	dummy_constant	0.278085	0.979258	0.064424	0.278934	0.000150	False
7	3.085734	0.000000	dummy_constant	3.085734	0.000000	8.421757	6.936022	0.000153	False
8	0.944973	0.140977	dummy_constant	0.944973	0.140977	-0.015507	0.326898	0.000152	False
9	0.985903	0.061507	dummy_constant	0.985903	0.061507	-0.078503	0.428378	0.000149	False
10	1.022044	0.104556	dummy_constant	1.022044	0.104556	0.061535	0.428906	0.000151	False
11	0.085747	1.044210	dummy_constant	0.085747	1.044210	0.072031	0.467770	0.000153	False
12	0.785835	0.643724	dummy_constant	0.785835	0.643724	0.033371	0.102358	0.000152	False
13	1.010642	0.032004	dummy_constant	1.010642	0.032004	-0.065023	0.479775	0.000152	False
14	0.585311	0.825508	dummy_constant	0.585311	0.825508	0.114381	0.113233	0.000106	False
15	1.039659	0.068549	dummy_constant	1.039659	0.068549	0.036130	0.477382	0.000150	False
16	0.046488	1.031537	dummy_constant	0.046488	1.031537	-0.008913	0.488205	0.000156	False
17	0.898019	0.466369	dummy_constant	0.898019	0.466369	0.005041	0.159550	0.000154	False
18	0.423012	0.894659	dummy_constant	0.423012	0.894659	-0.091491	0.161683	0.000154	False
19	0.971871	0.329689	dummy_constant	0.971871	0.329689	0.003507	0.251669	0.000153	False
20	1.023677	0.034798	dummy_constant	1.023677	0.034798	-0.036456	0.490650	0.000154	False
21	0.030619	0.000000	dummy_constant	0.030619	0.000000	-1.099063	0.470319	0.000156	False
22	0.857606	0.555516	dummy_constant	0.857606	0.555516	0.141497	0.130964	0.000156	False
23	0.800518	0.592865	dummy_constant	0.800518	0.592865	0.063789	0.098935	0.000153	False
24	0.393268	0.927788	dummy_constant	0.393268	0.927788	-0.083685	0.194394	0.000116	False
25	1.034970	0.043078	dummy_constant	1.034970	0.043078	-0.005638	0.494970	0.000154	False
26	0.000000	0.101538	dummy_constant	0.000000	0.101538	-1.089690	0.408772	0.000156	False
27	0.500129	0.884016	dummy_constant	0.500129	0.884016	0.069043	0.147468	0.000154	False
28	0.015160	1.018749	dummy_constant	0.015160	1.018749	-0.059100	0.504171	0.000153	False
29	1.002915	0.004609	dummy_constant	1.002915	0.004609	-0.093871	0.498336	0.000148	False
30	0.033497	1.027672	dummy_constant	0.033497	1.027672	-0.029484	0.496062	0.000143	False
31	0.697331	0.717353	dummy_constant	0.697331	0.717353	-0.096582	0.086182	0.000154	False

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