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

from copy import deepcopy

import pandas as pd
import numpy as np

from xopt import Xopt, Evaluator
from xopt.generators.bayesian.mggpo import MGGPOGenerator
from xopt.resources.test_functions.tnk import evaluate_TNK, tnk_vocs

from matplotlib import 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

evaluator = Evaluator(function=evaluate_TNK)
evaluator.max_workers = 10

# test check options
vocs = deepcopy(tnk_vocs)
gen = MGGPOGenerator(vocs=vocs, reference_point={"y1": 1.5, "y2": 1.5})
gen.n_monte_carlo_samples = N_MC_SAMPLES
gen.numerical_optimizer.n_restarts = NUM_RESTARTS
gen.gp_constructor.use_low_noise_prior = True
X = Xopt(evaluator=evaluator, generator=gen, vocs=vocs)
X.evaluate_data(pd.DataFrame({"x1": [1.0, 0.75], "x2": [0.75, 1.0]}))

X

# set values if testing import os from copy import deepcopy import pandas as pd import numpy as np from xopt import Xopt, Evaluator from xopt.generators.bayesian.mggpo import MGGPOGenerator from xopt.resources.test_functions.tnk import evaluate_TNK, tnk_vocs from matplotlib import 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 evaluator = Evaluator(function=evaluate_TNK) evaluator.max_workers = 10 # test check options vocs = deepcopy(tnk_vocs) gen = MGGPOGenerator(vocs=vocs, reference_point={"y1": 1.5, "y2": 1.5}) gen.n_monte_carlo_samples = N_MC_SAMPLES gen.numerical_optimizer.n_restarts = NUM_RESTARTS gen.gp_constructor.use_low_noise_prior = True X = Xopt(evaluator=evaluator, generator=gen, vocs=vocs) X.evaluate_data(pd.DataFrame({"x1": [1.0, 0.75], "x2": [0.75, 1.0]})) X

Out[1]:

            Xopt
________________________________
Version: 2.6.7.dev55+g7aa2f3618.d20250930
Data size: 2
Config as YAML:
dump_file: null
evaluator:
  function: xopt.resources.test_functions.tnk.evaluate_TNK
  function_kwargs:
    raise_probability: 0
    random_sleep: 0
    sleep: 0
  max_workers: 10
  vectorized: false
generator:
  computation_time: null
  custom_objective: null
  fixed_features: null
  ga_generator:
    crossover_probability: 0.9
    mutation_probability: 1.0
    output_path: null
    population: null
    population_file: null
    population_size: 64
    supports_constraints: true
    supports_multi_objective: true
    supports_single_objective: true
  gp_constructor:
    covar_modules: {}
    custom_noise_prior: null
    mean_modules: {}
    name: standard
    trainable_mean_keys: []
    transform_inputs: true
    use_cached_hyperparameters: false
    use_low_noise_prior: true
  max_travel_distances: null
  model: null
  n_candidates: 1
  n_interpolate_points: null
  n_monte_carlo_samples: 128
  name: mggpo
  numerical_optimizer:
    max_iter: 2000
    max_time: 5.0
    n_restarts: 20
    name: LBFGS
  population_size: 64
  reference_point:
    y1: 1.5
    y2: 1.5
  supports_batch_generation: true
  supports_constraints: true
  supports_multi_objective: true
  turbo_controller: null
  use_cuda: false
max_evaluations: null
serialize_inline: false
serialize_torch: false
strict: true
vocs:
  constants:
    a: dummy_constant
  constraints:
    c1:
    - GREATER_THAN
    - 0.0
    c2:
    - LESS_THAN
    - 0.5
  objectives:
    y1: MINIMIZE
    y2: MINIMIZE
  observables: []
  variables:
    x1:
    - 0.0
    - 3.14159
    x2:
    - 0.0
    - 3.14159

In [2]:

for i in range(10):
    print(i)
    X.step()

for i in range(10): print(i) X.step()

0

1

2

3

4

5

6

7

8

9

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	0.000160	False
1	0.750000	1.000000	dummy_constant	0.750000	1.000000	0.626888	0.312500	0.000136	False
2	0.095593	1.521237	dummy_constant	0.095593	1.521237	1.269615	1.206469	0.000147	False
3	1.479003	0.054022	dummy_constant	1.479003	0.054022	1.106950	1.157344	0.000130	False
4	1.674079	0.124703	dummy_constant	1.674079	0.124703	1.780894	1.519310	0.000126	False
...	...	...	...	...	...	...	...	...	...
97	0.701029	0.747417	dummy_constant	0.701029	0.747417	-0.037091	0.101628	0.000121	False
98	0.563335	0.822224	dummy_constant	0.563335	0.822224	0.091672	0.107840	0.000128	False
99	0.822158	0.591116	dummy_constant	0.822158	0.591116	0.110674	0.112088	0.000122	False
100	0.999358	0.243040	dummy_constant	0.999358	0.243040	0.135829	0.315387	0.000121	False
101	0.618541	0.826438	dummy_constant	0.618541	0.826438	0.131194	0.120614	0.000121	False

102 rows × 9 columns

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")

In [5]:

X.generator.visualize_model(show_feasibility=True)

X.generator.visualize_model(show_feasibility=True)

Out[5]:

(<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 [6]:

X.generator.update_pareto_front_history()
X.generator.pareto_front_history.plot(y="hypervolume", label="Hypervolume")
X.generator.pareto_front_history.plot(y="n_non_dominated", label="n_non_dominated")

X.generator.update_pareto_front_history() X.generator.pareto_front_history.plot(y="hypervolume", label="Hypervolume") X.generator.pareto_front_history.plot(y="n_non_dominated", label="n_non_dominated")

Out[6]:

<Axes: >

In [ ]: