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())

{'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	0.000301	False
1	0.750000	1.000000	dummy_constant	0.750000	1.000000	0.626888	0.312500	0.004003	False
2	1.710694	0.031409	dummy_constant	1.710694	0.031409	1.831745	1.685359	0.000314	False
3	1.363046	2.881184	dummy_constant	1.363046	2.881184	9.088519	6.414888	0.000281	False
4	0.028271	0.066820	dummy_constant	0.028271	0.066820	-1.094006	0.410174	0.000277	False
5	0.938202	0.485711	dummy_constant	0.938202	0.485711	0.095231	0.192225	0.000269	False
6	0.421738	1.039801	dummy_constant	0.421738	1.039801	0.159745	0.297510	0.000283	False
7	0.702589	0.000000	dummy_constant	0.702589	0.000000	-0.606369	0.291042	0.000296	False
8	1.027057	0.169517	dummy_constant	1.027057	0.169517	0.170147	0.387008	0.002777	False
9	0.100180	1.099378	dummy_constant	0.100180	1.099378	0.207013	0.519110	0.000311	False
10	0.090453	1.059998	dummy_constant	0.090453	1.059998	0.111053	0.481327	0.000267	False
11	0.976991	0.013542	dummy_constant	0.976991	0.013542	-0.142856	0.464162	0.000286	False
12	0.754457	0.754296	dummy_constant	0.754457	0.754296	0.038168	0.129415	0.000277	False
13	1.026842	0.061200	dummy_constant	1.026842	0.061200	0.000184	0.470108	0.000295	False
14	0.174488	1.100582	dummy_constant	0.174488	1.100582	0.322772	0.466657	0.000279	False
15	0.011836	0.991323	dummy_constant	0.011836	0.991323	-0.115319	0.479702	0.000322	False
16	0.050333	1.031271	dummy_constant	0.050333	1.031271	-0.005018	0.484449	0.000257	False
17	0.044903	1.036874	dummy_constant	0.044903	1.036874	0.000155	0.495347	0.000293	False
18	1.007436	0.000000	dummy_constant	1.007436	0.000000	-0.085072	0.507492	0.000272	False
19	0.064045	1.014245	dummy_constant	0.064045	1.014245	-0.020478	0.454505	0.001297	False
20	0.868258	0.566461	dummy_constant	0.868258	0.566461	0.173211	0.140031	0.000305	False
21	0.081431	1.044240	dummy_constant	0.081431	1.044240	0.065078	0.471396	0.002757	False
22	0.497309	0.980768	dummy_constant	0.497309	0.980768	0.175356	0.231145	0.000287	False
23	0.035558	1.013911	dummy_constant	0.035558	1.013911	-0.055399	0.479811	0.003868	False
24	0.631485	0.876174	dummy_constant	0.631485	0.876174	0.250790	0.158795	0.000293	False
25	0.375418	0.967959	dummy_constant	0.375418	0.967959	-0.015584	0.234507	0.001825	False
26	0.843748	0.537622	dummy_constant	0.843748	0.537622	0.094962	0.119578	0.000290	False
27	1.030837	0.000341	dummy_constant	1.030837	0.000341	-0.037373	0.531447	0.000317	False
28	0.715949	0.730737	dummy_constant	0.715949	0.730737	-0.052107	0.099873	0.000296	False
29	0.930923	0.389554	dummy_constant	0.930923	0.389554	-0.081462	0.197893	0.001965	False
30	1.021484	0.044778	dummy_constant	1.021484	0.044778	-0.030990	0.479173	0.000284	False
31	0.000000	0.033836	dummy_constant	0.000000	0.033836	-1.098855	0.467309	0.000274	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	0.663927	2
6	6	0.814993	3
7	7	0.814993	3
8	8	0.964534	4
9	9	0.964534	4
10	10	1.110300	5
11	11	1.110300	5
12	12	1.155447	6
13	13	1.206766	6
14	14	1.206766	6
15	15	1.206766	6
16	16	1.206766	6
17	17	1.236483	5
18	18	1.236483	5
19	19	1.236483	5
20	20	1.249621	6
21	21	1.249621	6
22	22	1.263884	6
23	23	1.263884	6
24	24	1.276746	7
25	25	1.276746	7
26	26	1.284074	7
27	27	1.284074	7
28	28	1.284074	7
29	29	1.284074	7
30	30	1.284074	7
31	31	1.284074	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	0.000301	False
1	0.750000	1.000000	dummy_constant	0.750000	1.000000	0.626888	0.312500	0.004003	False
2	1.710694	0.031409	dummy_constant	1.710694	0.031409	1.831745	1.685359	0.000314	False
3	1.363046	2.881184	dummy_constant	1.363046	2.881184	9.088519	6.414888	0.000281	False
4	0.028271	0.066820	dummy_constant	0.028271	0.066820	-1.094006	0.410174	0.000277	False
5	0.938202	0.485711	dummy_constant	0.938202	0.485711	0.095231	0.192225	0.000269	False
6	0.421738	1.039801	dummy_constant	0.421738	1.039801	0.159745	0.297510	0.000283	False
7	0.702589	0.000000	dummy_constant	0.702589	0.000000	-0.606369	0.291042	0.000296	False
8	1.027057	0.169517	dummy_constant	1.027057	0.169517	0.170147	0.387008	0.002777	False
9	0.100180	1.099378	dummy_constant	0.100180	1.099378	0.207013	0.519110	0.000311	False
10	0.090453	1.059998	dummy_constant	0.090453	1.059998	0.111053	0.481327	0.000267	False
11	0.976991	0.013542	dummy_constant	0.976991	0.013542	-0.142856	0.464162	0.000286	False
12	0.754457	0.754296	dummy_constant	0.754457	0.754296	0.038168	0.129415	0.000277	False
13	1.026842	0.061200	dummy_constant	1.026842	0.061200	0.000184	0.470108	0.000295	False
14	0.174488	1.100582	dummy_constant	0.174488	1.100582	0.322772	0.466657	0.000279	False
15	0.011836	0.991323	dummy_constant	0.011836	0.991323	-0.115319	0.479702	0.000322	False
16	0.050333	1.031271	dummy_constant	0.050333	1.031271	-0.005018	0.484449	0.000257	False
17	0.044903	1.036874	dummy_constant	0.044903	1.036874	0.000155	0.495347	0.000293	False
18	1.007436	0.000000	dummy_constant	1.007436	0.000000	-0.085072	0.507492	0.000272	False
19	0.064045	1.014245	dummy_constant	0.064045	1.014245	-0.020478	0.454505	0.001297	False
20	0.868258	0.566461	dummy_constant	0.868258	0.566461	0.173211	0.140031	0.000305	False
21	0.081431	1.044240	dummy_constant	0.081431	1.044240	0.065078	0.471396	0.002757	False
22	0.497309	0.980768	dummy_constant	0.497309	0.980768	0.175356	0.231145	0.000287	False
23	0.035558	1.013911	dummy_constant	0.035558	1.013911	-0.055399	0.479811	0.003868	False
24	0.631485	0.876174	dummy_constant	0.631485	0.876174	0.250790	0.158795	0.000293	False
25	0.375418	0.967959	dummy_constant	0.375418	0.967959	-0.015584	0.234507	0.001825	False
26	0.843748	0.537622	dummy_constant	0.843748	0.537622	0.094962	0.119578	0.000290	False
27	1.030837	0.000341	dummy_constant	1.030837	0.000341	-0.037373	0.531447	0.000317	False
28	0.715949	0.730737	dummy_constant	0.715949	0.730737	-0.052107	0.099873	0.000296	False
29	0.930923	0.389554	dummy_constant	0.930923	0.389554	-0.081462	0.197893	0.001965	False
30	1.021484	0.044778	dummy_constant	1.021484	0.044778	-0.030990	0.479173	0.000284	False
31	0.000000	0.033836	dummy_constant	0.000000	0.033836	-1.098855	0.467309	0.000274	False