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	0.000200	False
1	0.750000	1.000000	dummy_constant	0.750000	1.000000	0.626888	0.312500	0.000135	False
2	0.180169	1.548583	dummy_constant	0.180169	1.548583	1.458435	1.201819	0.000150	False
3	1.924156	2.116347	dummy_constant	1.924156	2.116347	7.108850	4.640798	0.000145	False
4	0.000000	0.148425	dummy_constant	0.000000	0.148425	-1.077970	0.373605	0.000147	False
5	0.000000	1.184310	dummy_constant	0.000000	1.184310	0.302591	0.718281	0.000146	False
6	0.165457	1.086007	dummy_constant	0.165457	1.086007	0.281799	0.455323	0.000148	False
7	1.046943	0.464814	dummy_constant	1.046943	0.464814	0.220116	0.300384	0.000150	False
8	0.711759	0.775986	dummy_constant	0.711759	0.775986	0.031649	0.121010	0.000151	False
9	1.123748	0.325700	dummy_constant	1.123748	0.325700	0.388633	0.419442	0.000157	False
10	1.001257	0.147011	dummy_constant	1.001257	0.147011	0.093147	0.375860	0.000151	False
11	0.959095	0.000000	dummy_constant	0.959095	0.000000	-0.180137	0.460768	0.000129	False
12	1.039889	0.063793	dummy_constant	1.039889	0.063793	0.029761	0.481757	0.000150	False
13	0.076191	1.040563	dummy_constant	0.076191	1.040563	0.049510	0.471823	0.000146	False
14	0.497332	0.919030	dummy_constant	0.497332	0.919030	0.100183	0.175593	0.000151	False
15	0.890767	0.575425	dummy_constant	0.890767	0.575425	0.221521	0.158388	0.000152	False
16	0.025857	1.010881	dummy_constant	0.025857	1.010881	-0.069195	0.485811	0.000153	False
17	1.059342	0.082829	dummy_constant	1.059342	0.082829	0.097390	0.486896	0.000148	False
18	0.968735	0.372650	dummy_constant	0.968735	0.372650	-0.014493	0.235931	0.000148	False
19	1.010516	0.018821	dummy_constant	1.010516	0.018821	-0.074096	0.492160	0.000151	False
20	0.780664	0.672429	dummy_constant	0.780664	0.672429	0.024392	0.108504	0.000147	False
21	0.044968	1.031170	dummy_constant	0.044968	1.031170	-0.011324	0.489195	0.000150	False
22	0.294111	0.976060	dummy_constant	0.294111	0.976060	0.042159	0.269024	0.000149	False
23	0.062680	1.049911	dummy_constant	0.062680	1.049911	0.048405	0.493650	0.000154	False
24	0.042801	1.039112	dummy_constant	0.042801	1.039112	0.002505	0.499673	0.000151	False
25	0.574103	0.870758	dummy_constant	0.574103	0.870758	0.187329	0.142953	0.000153	False
26	1.001316	0.000000	dummy_constant	1.001316	0.000000	-0.097366	0.501318	0.000152	False
27	1.032450	0.038947	dummy_constant	1.032450	0.038947	-0.014878	0.496073	0.000151	False
28	0.905340	0.460582	dummy_constant	0.905340	0.460582	-0.000074	0.165854	0.000156	False
29	0.674412	0.763969	dummy_constant	0.674412	0.763969	-0.015977	0.100099	0.000148	False
30	0.982430	0.314489	dummy_constant	0.982430	0.314489	0.039865	0.267153	0.000148	False
31	0.125124	1.024890	dummy_constant	0.125124	1.024890	0.102491	0.416041	0.000153	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	0.741997	3
7	7	0.871203	4
8	8	0.939062	4
9	9	0.991403	5
10	10	1.110593	4
11	11	1.110593	4
12	12	1.148882	5
13	13	1.214721	5
14	14	1.240781	6
15	15	1.262908	6
16	16	1.262908	6
17	17	1.262908	6
18	18	1.262908	6
19	19	1.262908	6
20	20	1.274310	7
21	21	1.274310	7
22	22	1.287418	8
23	23	1.293499	9
24	24	1.303123	8
25	25	1.309768	9
26	26	1.309768	9
27	27	1.309768	9
28	28	1.309768	9
29	29	1.309768	9
30	30	1.314681	10
31	31	1.317084	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	0.000200	False
1	0.750000	1.000000	dummy_constant	0.750000	1.000000	0.626888	0.312500	0.000135	False
2	0.180169	1.548583	dummy_constant	0.180169	1.548583	1.458435	1.201819	0.000150	False
3	1.924156	2.116347	dummy_constant	1.924156	2.116347	7.108850	4.640798	0.000145	False
4	0.000000	0.148425	dummy_constant	0.000000	0.148425	-1.077970	0.373605	0.000147	False
5	0.000000	1.184310	dummy_constant	0.000000	1.184310	0.302591	0.718281	0.000146	False
6	0.165457	1.086007	dummy_constant	0.165457	1.086007	0.281799	0.455323	0.000148	False
7	1.046943	0.464814	dummy_constant	1.046943	0.464814	0.220116	0.300384	0.000150	False
8	0.711759	0.775986	dummy_constant	0.711759	0.775986	0.031649	0.121010	0.000151	False
9	1.123748	0.325700	dummy_constant	1.123748	0.325700	0.388633	0.419442	0.000157	False
10	1.001257	0.147011	dummy_constant	1.001257	0.147011	0.093147	0.375860	0.000151	False
11	0.959095	0.000000	dummy_constant	0.959095	0.000000	-0.180137	0.460768	0.000129	False
12	1.039889	0.063793	dummy_constant	1.039889	0.063793	0.029761	0.481757	0.000150	False
13	0.076191	1.040563	dummy_constant	0.076191	1.040563	0.049510	0.471823	0.000146	False
14	0.497332	0.919030	dummy_constant	0.497332	0.919030	0.100183	0.175593	0.000151	False
15	0.890767	0.575425	dummy_constant	0.890767	0.575425	0.221521	0.158388	0.000152	False
16	0.025857	1.010881	dummy_constant	0.025857	1.010881	-0.069195	0.485811	0.000153	False
17	1.059342	0.082829	dummy_constant	1.059342	0.082829	0.097390	0.486896	0.000148	False
18	0.968735	0.372650	dummy_constant	0.968735	0.372650	-0.014493	0.235931	0.000148	False
19	1.010516	0.018821	dummy_constant	1.010516	0.018821	-0.074096	0.492160	0.000151	False
20	0.780664	0.672429	dummy_constant	0.780664	0.672429	0.024392	0.108504	0.000147	False
21	0.044968	1.031170	dummy_constant	0.044968	1.031170	-0.011324	0.489195	0.000150	False
22	0.294111	0.976060	dummy_constant	0.294111	0.976060	0.042159	0.269024	0.000149	False
23	0.062680	1.049911	dummy_constant	0.062680	1.049911	0.048405	0.493650	0.000154	False
24	0.042801	1.039112	dummy_constant	0.042801	1.039112	0.002505	0.499673	0.000151	False
25	0.574103	0.870758	dummy_constant	0.574103	0.870758	0.187329	0.142953	0.000153	False
26	1.001316	0.000000	dummy_constant	1.001316	0.000000	-0.097366	0.501318	0.000152	False
27	1.032450	0.038947	dummy_constant	1.032450	0.038947	-0.014878	0.496073	0.000151	False
28	0.905340	0.460582	dummy_constant	0.905340	0.460582	-0.000074	0.165854	0.000156	False
29	0.674412	0.763969	dummy_constant	0.674412	0.763969	-0.015977	0.100099	0.000148	False
30	0.982430	0.314489	dummy_constant	0.982430	0.314489	0.039865	0.267153	0.000148	False
31	0.125124	1.024890	dummy_constant	0.125124	1.024890	0.102491	0.416041	0.000153	False

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