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())

/home/runner/work/Xopt/Xopt/.venv/lib/python3.12/site-packages/pyro/ops/stats.py:527: SyntaxWarning: invalid escape sequence '\g'
  we have :math:`ES^{*}(P,Q) \ge ES^{*}(Q,Q)` with equality holding if and only if :math:`P=Q`, i.e.

{'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()

0

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	0.000405	False
1	0.750000	1.000000	dummy_constant	0.750000	1.000000	0.626888	0.312500	0.000141	False
2	0.168922	1.610441	dummy_constant	0.168922	1.610441	1.632173	1.342692	0.011439	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(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))

In [7]:

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: >

In [8]:

X.generator.pareto_front_history

X.generator.pareto_front_history

Out[8]:

	iteration	hypervolume	n_non_dominated
0	0	0.375	1
1	1	0.500	2
2	2	0.500	2

In [9]:

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	0.000405	False
1	0.750000	1.000000	dummy_constant	0.750000	1.000000	0.626888	0.312500	0.000141	False
2	0.168922	1.610441	dummy_constant	0.168922	1.610441	1.632173	1.342692	0.011439	False

In [ ]: