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	0.001598	False
1	0.750000	1.000000	dummy_constant	0.750000	1.000000	0.626888	0.312500	0.000147	False
2	0.138911	1.644980	dummy_constant	0.138911	1.644980	1.703152	1.441364	0.000162	False
3	2.894116	1.285349	dummy_constant	2.894116	1.285349	8.936084	6.348564	0.000162	False
4	0.252452	0.320440	dummy_constant	0.252452	0.320440	-0.802210	0.093522	0.000170	False
5	1.158878	0.152556	dummy_constant	1.158878	0.152556	0.416256	0.554837	0.000171	False
6	1.045769	0.103903	dummy_constant	1.045769	0.103903	0.105798	0.454757	0.000174	False
7	0.159611	1.032823	dummy_constant	0.159611	1.032823	0.169426	0.399765	0.000186	False
8	0.000000	0.955127	dummy_constant	0.000000	0.955127	-0.187732	0.457141	0.000166	False
9	0.628248	0.759271	dummy_constant	0.628248	0.759271	-0.035246	0.083669	0.000161	False
10	0.058097	1.023790	dummy_constant	0.058097	1.023790	-0.010091	0.469634	0.000154	False
11	0.464431	0.890838	dummy_constant	0.464431	0.890838	-0.007114	0.154020	0.000165	False
12	1.006591	0.045782	dummy_constant	1.006591	0.045782	-0.059382	0.462948	0.000165	False
13	0.129015	1.048317	dummy_constant	0.129015	1.048317	0.153488	0.438281	0.000165	False
14	0.041406	1.034937	dummy_constant	0.041406	1.034937	-0.007413	0.496465	0.000183	False
15	0.823942	0.640215	dummy_constant	0.823942	0.640215	0.130123	0.124598	0.000139	False
16	0.986225	0.269435	dummy_constant	0.986225	0.269435	0.088310	0.289574	0.000174	False
17	1.041219	0.022115	dummy_constant	1.041219	0.022115	-0.009656	0.521292	0.000162	False
18	0.059635	1.039836	dummy_constant	0.059635	1.039836	0.023963	0.485344	0.000179	False
19	1.036883	0.054146	dummy_constant	1.036883	0.054146	0.010923	0.487029	0.000159	False
20	0.886274	0.486915	dummy_constant	0.886274	0.486915	0.040871	0.149379	0.000165	False
21	1.032090	0.036066	dummy_constant	1.032090	0.036066	-0.018274	0.498355	0.000165	False
22	0.935262	0.358481	dummy_constant	0.935262	0.358481	-0.087806	0.209480	0.000192	False
23	1.045143	0.063150	dummy_constant	1.045143	0.063150	0.039417	0.488019	0.000170	False
24	0.087315	0.000000	dummy_constant	0.087315	0.000000	-1.092376	0.420309	0.012580	False
25	0.546880	0.748404	dummy_constant	0.546880	0.748404	-0.062560	0.063902	0.000149	False
26	0.433402	0.879872	dummy_constant	0.433402	0.879872	-0.088624	0.148738	0.000166	False
27	1.033746	0.041360	dummy_constant	1.033746	0.041360	-0.009878	0.495235	0.001919	False
28	0.029688	1.025382	dummy_constant	0.029688	1.025382	-0.037176	0.497220	0.000941	False
29	0.044617	0.000000	dummy_constant	0.044617	0.000000	-1.098009	0.457373	0.007866	False
30	1.005599	0.012585	dummy_constant	1.005599	0.012585	-0.086614	0.493204	0.003706	False
31	0.022270	0.000000	dummy_constant	0.022270	0.000000	-1.099504	0.478226	0.000160	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	0.793477	3
7	7	1.069294	4
8	8	1.069294	4
9	9	1.069294	4
10	10	1.069294	4
11	11	1.069294	4
12	12	1.069294	4
13	13	1.083113	5
14	14	1.083113	5
15	15	1.151481	5
16	16	1.173559	6
17	17	1.173559	6
18	18	1.205745	6
19	19	1.230259	6
20	20	1.245581	7
21	21	1.245581	7
22	22	1.245581	7
23	23	1.245581	7
24	24	1.245581	7
25	25	1.245581	7
26	26	1.245581	7
27	27	1.245581	7
28	28	1.245581	7
29	29	1.245581	7
30	30	1.245581	7
31	31	1.245581	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	0.001598	False
1	0.750000	1.000000	dummy_constant	0.750000	1.000000	0.626888	0.312500	0.000147	False
2	0.138911	1.644980	dummy_constant	0.138911	1.644980	1.703152	1.441364	0.000162	False
3	2.894116	1.285349	dummy_constant	2.894116	1.285349	8.936084	6.348564	0.000162	False
4	0.252452	0.320440	dummy_constant	0.252452	0.320440	-0.802210	0.093522	0.000170	False
5	1.158878	0.152556	dummy_constant	1.158878	0.152556	0.416256	0.554837	0.000171	False
6	1.045769	0.103903	dummy_constant	1.045769	0.103903	0.105798	0.454757	0.000174	False
7	0.159611	1.032823	dummy_constant	0.159611	1.032823	0.169426	0.399765	0.000186	False
8	0.000000	0.955127	dummy_constant	0.000000	0.955127	-0.187732	0.457141	0.000166	False
9	0.628248	0.759271	dummy_constant	0.628248	0.759271	-0.035246	0.083669	0.000161	False
10	0.058097	1.023790	dummy_constant	0.058097	1.023790	-0.010091	0.469634	0.000154	False
11	0.464431	0.890838	dummy_constant	0.464431	0.890838	-0.007114	0.154020	0.000165	False
12	1.006591	0.045782	dummy_constant	1.006591	0.045782	-0.059382	0.462948	0.000165	False
13	0.129015	1.048317	dummy_constant	0.129015	1.048317	0.153488	0.438281	0.000165	False
14	0.041406	1.034937	dummy_constant	0.041406	1.034937	-0.007413	0.496465	0.000183	False
15	0.823942	0.640215	dummy_constant	0.823942	0.640215	0.130123	0.124598	0.000139	False
16	0.986225	0.269435	dummy_constant	0.986225	0.269435	0.088310	0.289574	0.000174	False
17	1.041219	0.022115	dummy_constant	1.041219	0.022115	-0.009656	0.521292	0.000162	False
18	0.059635	1.039836	dummy_constant	0.059635	1.039836	0.023963	0.485344	0.000179	False
19	1.036883	0.054146	dummy_constant	1.036883	0.054146	0.010923	0.487029	0.000159	False
20	0.886274	0.486915	dummy_constant	0.886274	0.486915	0.040871	0.149379	0.000165	False
21	1.032090	0.036066	dummy_constant	1.032090	0.036066	-0.018274	0.498355	0.000165	False
22	0.935262	0.358481	dummy_constant	0.935262	0.358481	-0.087806	0.209480	0.000192	False
23	1.045143	0.063150	dummy_constant	1.045143	0.063150	0.039417	0.488019	0.000170	False
24	0.087315	0.000000	dummy_constant	0.087315	0.000000	-1.092376	0.420309	0.012580	False
25	0.546880	0.748404	dummy_constant	0.546880	0.748404	-0.062560	0.063902	0.000149	False
26	0.433402	0.879872	dummy_constant	0.433402	0.879872	-0.088624	0.148738	0.000166	False
27	1.033746	0.041360	dummy_constant	1.033746	0.041360	-0.009878	0.495235	0.001919	False
28	0.029688	1.025382	dummy_constant	0.029688	1.025382	-0.037176	0.497220	0.000941	False
29	0.044617	0.000000	dummy_constant	0.044617	0.000000	-1.098009	0.457373	0.007866	False
30	1.005599	0.012585	dummy_constant	1.005599	0.012585	-0.086614	0.493204	0.003706	False
31	0.022270	0.000000	dummy_constant	0.022270	0.000000	-1.099504	0.478226	0.000160	False