Multi-objective Bayesian Optimization¶

TNK function $n=2$ variables: $x_i \in [0, \pi], i=1,2$

Objectives:

• $f_i(x) = x_i$

Constraints:

• $g_1(x) = -x_1^2 -x_2^2 + 1 + 0.1 \cos\left(16 \arctan \frac{x_1}{x_2}\right) \le 0$
• $g_2(x) = (x_1 - 1/2)^2 + (x_2-1/2)^2 \le 0.5$

In [1]:

# 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': []}

In [2]:

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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In [3]:

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	0.000122	False
1	0.750000	1.000000	dummy_constant	0.750000	1.000000	0.626888	0.312500	0.000135	False
2	0.264314	1.517085	dummy_constant	0.264314	1.517085	1.464212	1.090010	0.000152	False
3	0.121347	0.634433	dummy_constant	0.121347	0.634433	-0.483463	0.161450	0.000151	False
4	0.000000	0.000000	dummy_constant	0.000000	0.000000	-1.100000	0.500000	0.000151	False
5	0.005694	0.927092	dummy_constant	0.005694	0.927092	-0.239986	0.426746	0.000151	False
6	0.275152	0.977640	dummy_constant	0.275152	0.977640	0.063213	0.278697	0.000152	False
7	3.141590	0.000000	dummy_constant	3.141590	0.000000	8.769588	7.227998	0.000118	False
8	0.959644	0.204618	dummy_constant	0.959644	0.204618	0.060382	0.298523	0.000151	False
9	1.015889	0.124754	dummy_constant	1.015889	0.124754	0.085083	0.406951	0.000164	False
10	0.082141	1.024671	dummy_constant	0.082141	1.024671	0.028015	0.449886	0.000152	False
11	0.580189	0.817477	dummy_constant	0.580189	0.817477	0.094894	0.107222	0.000149	False
12	1.023306	0.051914	dummy_constant	1.023306	0.051914	-0.019026	0.474630	0.000108	False
13	0.850031	0.564281	dummy_constant	0.850031	0.564281	0.140850	0.126654	0.000148	False
14	0.074521	1.048222	dummy_constant	0.074521	1.048222	0.062161	0.481580	0.000152	False
15	0.412779	0.888111	dummy_constant	0.412779	0.888111	-0.118750	0.158238	0.000148	False
16	0.940702	0.404917	dummy_constant	0.940702	0.404917	-0.048705	0.203259	0.000148	False
17	0.030030	1.018220	dummy_constant	0.030030	1.018220	-0.051403	0.489424	0.000147	False
18	1.000415	0.002854	dummy_constant	1.000415	0.002854	-0.099057	0.497570	0.000147	False
19	0.982167	0.000000	dummy_constant	0.982167	0.000000	-0.135348	0.482485	0.000148	False
20	1.038867	0.171349	dummy_constant	1.038867	0.171349	0.195081	0.398389	0.000147	False
21	0.407737	0.918308	dummy_constant	0.407737	0.918308	-0.082465	0.183494	0.000148	False
22	0.000000	0.040452	dummy_constant	0.000000	0.040452	-1.098364	0.461184	0.000148	False
23	0.703809	0.738255	dummy_constant	0.703809	0.738255	-0.052421	0.098303	0.000149	False
24	0.000240	0.997206	dummy_constant	0.000240	0.997206	-0.105579	0.496974	0.000147	False
25	0.069914	1.056705	dummy_constant	0.069914	1.056705	0.072370	0.494895	0.000129	False
26	0.735013	0.741395	dummy_constant	0.735013	0.741395	-0.009851	0.113503	0.000148	False
27	1.025572	0.039442	dummy_constant	1.025572	0.039442	-0.028322	0.488339	0.000148	False
28	0.584059	0.851467	dummy_constant	0.584059	0.851467	0.164227	0.130595	0.000147	False
29	1.022825	0.294977	dummy_constant	1.022825	0.294977	0.155004	0.315381	0.000147	False
30	0.041583	1.034302	dummy_constant	0.041583	1.034302	-0.008526	0.495625	0.000148	False
31	0.000000	0.986735	dummy_constant	0.000000	0.986735	-0.126354	0.486911	0.000148	False

plot results¶

In [4]:

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")

Plot path through input space¶

In [5]:

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")

In [6]:

## visualize model
X.generator.visualize_model()

## visualize model X.generator.visualize_model()

Out[6]:

(<Figure size 800x1650 with 19 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'}, xlabel='x1', ylabel='x2'>,
         <Axes: xlabel='x1'>]], dtype=object))