#! /usr/bin/env python3
# Copyright (c) Facebook, Inc. and its affiliates.
#
# This source code is licensed under the MIT license found in the
# LICENSE file in the root directory of this source tree.
r"""
Multi-objective optimization benchmark problems.
References
.. [Deb2005dtlz]
K. Deb, L. Thiele, M. Laumanns, E. Zitzler, A. Abraham, L. Jain, R. Goldberg.
"Scalable test problems for evolutionary multi-objective optimization"
in Evolutionary Multiobjective Optimization, London, U.K.: Springer-Verlag,
pp. 105-145, 2005.
.. [GarridoMerchan2020]
E. C. Garrido-Merch ́an and D. Hern ́andez-Lobato. Parallel Predictive Entropy
Search for Multi-objective Bayesian Optimization with Constraints.
arXiv e-prints, arXiv:2004.00601, Apr. 2020.
.. [Gelbart2014]
Michael A. Gelbart, Jasper Snoek, and Ryan P. Adams. 2014. Bayesian
optimization with unknown constraints. In Proceedings of the Thirtieth
Conference on Uncertainty in Artificial Intelligence (UAI’14).
AUAI Press, Arlington, Virginia, USA, 250–259.
.. [Tanabe2020]
Ryoji Tanabe, Hisao Ishibuchi, An easy-to-use real-world multi-objective
optimization problem suite, Applied Soft Computing,Volume 89, 2020.
.. [Yang2019a]
K. Yang, M. Emmerich, A. Deutz, and T. Bäck. 2019.
"Multi-Objective Bayesian Global Optimization using expected hypervolume
improvement gradient" in Swarm and evolutionary computation 44, pp. 945--956,
2019.
.. [Zitzler2000]
E. Zitzler, K. Deb, and L. Thiele, “Comparison of multiobjective
evolutionary algorithms: Empirical results,” Evol. Comput., vol. 8, no. 2,
pp. 173–195, 2000.
"""
from __future__ import annotations
import math
from typing import Optional
import torch
from botorch.test_functions.base import (
ConstrainedBaseTestProblem,
MultiObjectiveTestProblem,
)
from botorch.test_functions.synthetic import Branin
from botorch.utils.sampling import sample_hypersphere, sample_simplex
from botorch.utils.transforms import unnormalize
from scipy.special import gamma
from torch import Tensor
[docs]class BraninCurrin(MultiObjectiveTestProblem):
r"""Two objective problem composed of the Branin and Currin functions.
Branin (rescaled):
f(x) = (
15*x_1 - 5.1 * (15 * x_0 - 5) ** 2 / (4 * pi ** 2) + 5 * (15 * x_0 - 5)
/ pi - 5
) ** 2 + (10 - 10 / (8 * pi)) * cos(15 * x_0 - 5))
Currin:
f(x) = (1 - exp(-1 / (2 * x_1))) * (
2300 * x_0 ** 3 + 1900 * x_0 ** 2 + 2092 * x_0 + 60
) / 100 * x_0 ** 3 + 500 * x_0 ** 2 + 4 * x_0 + 20
"""
dim = 2
num_objectives = 2
_bounds = [(0.0, 1.0), (0.0, 1.0)]
_ref_point = [18.0, 6.0]
_max_hv = 59.36011874867746 # this is approximated using NSGA-II
def __init__(self, noise_std: Optional[float] = None, negate: bool = False) -> None:
r"""Constructor for Branin-Currin.
Arguments:
noise_std: Standard deviation of the observation noise.
negate: If True, negate the objectives.
"""
super().__init__(noise_std=noise_std, negate=negate)
self._branin = Branin()
def _rescaled_branin(self, X: Tensor) -> Tensor:
# return to Branin bounds
x_0 = 15 * X[..., 0] - 5
x_1 = 15 * X[..., 1]
return self._branin(torch.stack([x_0, x_1], dim=-1))
@staticmethod
def _currin(X: Tensor) -> Tensor:
x_0 = X[..., 0]
x_1 = X[..., 1]
factor1 = 1 - torch.exp(-1 / (2 * x_1))
numer = 2300 * x_0.pow(3) + 1900 * x_0.pow(2) + 2092 * x_0 + 60
denom = 100 * x_0.pow(3) + 500 * x_0.pow(2) + 4 * x_0 + 20
return factor1 * numer / denom
[docs] def evaluate_true(self, X: Tensor) -> Tensor:
# branin rescaled with inputsto [0,1]^2
branin = self._rescaled_branin(X=X)
currin = self._currin(X=X)
return torch.stack([branin, currin], dim=-1)
[docs]class DTLZ(MultiObjectiveTestProblem):
r"""Base class for DTLZ problems.
See [Deb2005dtlz]_ for more details on DTLZ.
"""
def __init__(
self,
dim: int,
num_objectives: int = 2,
noise_std: Optional[float] = None,
negate: bool = False,
) -> None:
if dim <= num_objectives:
raise ValueError(
f"dim must be > num_objectives, but got {dim} and {num_objectives}"
)
self.num_objectives = num_objectives
self.dim = dim
self.k = self.dim - self.num_objectives + 1
self._bounds = [(0.0, 1.0) for _ in range(self.dim)]
self._ref_point = [self._ref_val for _ in range(num_objectives)]
super().__init__(noise_std=noise_std, negate=negate)
[docs]class DTLZ1(DTLZ):
r"""DLTZ1 test problem.
d-dimensional problem evaluated on `[0, 1]^d`:
f_0(x) = 0.5 * x_0 * (1 + g(x))
f_1(x) = 0.5 * (1 - x_0) * (1 + g(x))
g(x) = 100 * \sum_{i=m}^{n-1} (
k + (x_i - 0.5)^2 - cos(20 * pi * (x_i - 0.5))
)
where k = n - m + 1.
The pareto front is given by the line (or hyperplane) \sum_i f_i(x) = 0.5.
The goal is to minimize both objectives. The reference point comes from [Yang2019]_.
"""
_ref_val = 400.0
@property
def _max_hv(self) -> float:
return self._ref_val ** self.num_objectives - 1 / 2 ** self.num_objectives
[docs] def evaluate_true(self, X: Tensor) -> Tensor:
X_m = X[..., -self.k :]
X_m_minus_half = X_m - 0.5
sum_term = (
X_m_minus_half.pow(2) - torch.cos(20 * math.pi * X_m_minus_half)
).sum(dim=-1)
g_X_m = 100 * (self.k + sum_term)
g_X_m_term = 0.5 * (1 + g_X_m)
fs = []
for i in range(self.num_objectives):
idx = self.num_objectives - 1 - i
f_i = g_X_m_term * X[..., :idx].prod(dim=-1)
if i > 0:
f_i *= 1 - X[..., idx]
fs.append(f_i)
return torch.stack(fs, dim=-1)
[docs] def gen_pareto_front(self, n: int) -> Tensor:
r"""Generate `n` pareto optimal points.
The pareto points randomly sampled from the hyperplane sum_i f(x_i) = 0.5.
"""
f_X = 0.5 * sample_simplex(
n=n,
d=self.num_objectives,
qmc=True,
dtype=self.ref_point.dtype,
device=self.ref_point.device,
)
if self.negate:
f_X *= -1
return f_X
[docs]class DTLZ2(DTLZ):
r"""DLTZ2 test problem.
d-dimensional problem evaluated on `[0, 1]^d`:
f_0(x) = (1 + g(x)) * cos(x_0 * pi / 2)
f_1(x) = (1 + g(x)) * sin(x_0 * pi / 2)
g(x) = \sum_{i=m}^{n-1} (x_i - 0.5)^2
The pareto front is given by the unit hypersphere \sum{i} f_i^2 = 1.
Note: the pareto front is completely concave. The goal is to minimize
both objectives.
"""
_ref_val = 1.1
@property
def _max_hv(self) -> float:
# hypercube - volume of hypersphere in R^n such that all coordinates are
# positive
hypercube_vol = self._ref_val ** self.num_objectives
pos_hypersphere_vol = (
math.pi ** (self.num_objectives / 2)
/ gamma(self.num_objectives / 2 + 1)
/ 2 ** self.num_objectives
)
return hypercube_vol - pos_hypersphere_vol
[docs] def evaluate_true(self, X: Tensor) -> Tensor:
X_m = X[..., -self.k :]
g_X = (X_m - 0.5).pow(2).sum(dim=-1)
g_X_plus1 = 1 + g_X
fs = []
pi_over_2 = math.pi / 2
for i in range(self.num_objectives):
idx = self.num_objectives - 1 - i
f_i = g_X_plus1.clone()
f_i *= torch.cos(X[..., :idx] * pi_over_2).prod(dim=-1)
if i > 0:
f_i *= torch.sin(X[..., idx] * pi_over_2)
fs.append(f_i)
return torch.stack(fs, dim=-1)
[docs] def gen_pareto_front(self, n: int) -> Tensor:
r"""Generate `n` pareto optimal points.
The pareto points are randomly sampled from the hypersphere's
positive section.
"""
f_X = sample_hypersphere(
n=n,
d=self.num_objectives,
dtype=self.ref_point.dtype,
device=self.ref_point.device,
qmc=True,
).abs()
if self.negate:
f_X *= -1
return f_X
[docs]class VehicleSafety(MultiObjectiveTestProblem):
r"""Optimize Vehicle crash-worthiness.
See [Tanabe2020]_ for details.
The reference point is 1.1 * the nadir point from
approximate front provided by [Tanabe2020]_.
The maximum hypervolume is computed using the approximate
pareto front from [Tanabe2020]_.
"""
_ref_point = [1864.72022, 11.81993945, 0.2903999384]
_max_hv = 246.81607081187002
_bounds = [(1.0, 3.0)] * 5
dim = 5
num_objectives = 3
[docs] def evaluate_true(self, X: Tensor) -> Tensor:
X1, X2, X3, X4, X5 = torch.split(X, 1, -1)
f1 = (
1640.2823
+ 2.3573285 * X1
+ 2.3220035 * X2
+ 4.5688768 * X3
+ 7.7213633 * X4
+ 4.4559504 * X5
)
f2 = (
6.5856
+ 1.15 * X1
- 1.0427 * X2
+ 0.9738 * X3
+ 0.8364 * X4
- 0.3695 * X1 * X4
+ 0.0861 * X1 * X5
+ 0.3628 * X2 * X4
- 0.1106 * X1.pow(2)
- 0.3437 * X3.pow(2)
+ 0.1764 * X4.pow(2)
)
f3 = (
-0.0551
+ 0.0181 * X1
+ 0.1024 * X2
+ 0.0421 * X3
- 0.0073 * X1 * X2
+ 0.024 * X2 * X3
- 0.0118 * X2 * X4
- 0.0204 * X3 * X4
- 0.008 * X3 * X5
- 0.0241 * X2.pow(2)
+ 0.0109 * X4.pow(2)
)
f_X = torch.cat([f1, f2, f3], dim=-1)
return f_X
[docs]class ZDT(MultiObjectiveTestProblem):
r"""Base class for ZDT problems.
See [Zitzler2000]_ for more details on ZDT.
"""
_ref_point = [11.0, 11.0]
def __init__(
self,
dim: int,
num_objectives: int = 2,
noise_std: Optional[float] = None,
negate: bool = False,
) -> None:
if num_objectives != 2:
raise NotImplementedError(
f"{type(self).__name__} currently only supports 2 objectives."
)
if dim < num_objectives:
raise ValueError(
f"dim must be >= num_objectives, but got {dim} and {num_objectives}"
)
self.num_objectives = num_objectives
self.dim = dim
self._bounds = [(0.0, 1.0) for _ in range(self.dim)]
super().__init__(noise_std=noise_std, negate=negate)
@staticmethod
def _g(X: Tensor) -> Tensor:
return 1 + 9 * X[..., 1:].mean(dim=-1)
[docs]class ZDT1(ZDT):
r"""ZDT1 test problem.
d-dimensional problem evaluated on `[0, 1]^d`:
f_0(x) = x_0
f_1(x) = g(x) * (1 - sqrt(x_0 / g(x))
g(x) = 1 + 9 / (d - 1) * \sum_{i=1}^{d-1} x_i
The reference point comes from [Yang2019a]_.
The pareto front is convex.
"""
_max_hv = 120 + 2 / 3
[docs] def evaluate_true(self, X: Tensor) -> Tensor:
f_0 = X[..., 0]
g = self._g(X=X)
f_1 = g * (1 - (f_0 / g).sqrt())
return torch.stack([f_0, f_1], dim=-1)
[docs] def gen_pareto_front(self, n: int) -> Tensor:
f_0 = torch.linspace(
0, 1, n, dtype=self.bounds.dtype, device=self.bounds.device
)
f_1 = 1 - f_0.sqrt()
f_X = torch.stack([f_0, f_1], dim=-1)
if self.negate:
f_X *= -1
return f_X
[docs]class ZDT2(ZDT):
r"""ZDT2 test problem.
d-dimensional problem evaluated on `[0, 1]^d`:
f_0(x) = x_0
f_1(x) = g(x) * (1 - (x_0 / g(x))^2)
g(x) = 1 + 9 / (d - 1) * \sum_{i=1}^{d-1} x_i
The reference point comes from [Yang2019a]_.
The pareto front is concave.
"""
_max_hv = 120 + 1 / 3
[docs] def evaluate_true(self, X: Tensor) -> Tensor:
f_0 = X[..., 0]
g = self._g(X=X)
f_1 = g * (1 - (f_0 / g).pow(2))
return torch.stack([f_0, f_1], dim=-1)
[docs] def gen_pareto_front(self, n: int) -> Tensor:
f_0 = torch.linspace(
0, 1, n, dtype=self.bounds.dtype, device=self.bounds.device
)
f_1 = 1 - f_0.pow(2)
f_X = torch.stack([f_0, f_1], dim=-1)
if self.negate:
f_X *= -1
return f_X
[docs]class ZDT3(ZDT):
r"""ZDT3 test problem.
d-dimensional problem evaluated on `[0, 1]^d`:
f_0(x) = x_0
f_1(x) = 1 - sqrt(x_0 / g(x)) - x_0 / g * sin(10 * pi * x_0)
g(x) = 1 + 9 / (d - 1) * \sum_{i=1}^{d-1} x_i
The reference point comes from [Yang2019a]_.
The pareto front consists of several discontinuous convex parts.
"""
_max_hv = 128.77811613069076060
_parts = [
# this interval includes both end points
[0, 0.0830015349],
# this interval includes only the right end points
[0.1822287280, 0.2577623634],
[0.4093136748, 0.4538821041],
[0.6183967944, 0.6525117038],
[0.8233317983, 0.8518328654],
]
# nugget to make sure linspace returns elements within the specified range
_eps = 1e-6
[docs] def evaluate_true(self, X: Tensor) -> Tensor:
f_0 = X[..., 0]
g = self._g(X=X)
f_1 = 1 - (f_0 / g).sqrt() - f_0 / g * torch.sin(10 * math.pi * f_0)
return torch.stack([f_0, f_1], dim=-1)
[docs] def gen_pareto_front(self, n: int) -> Tensor:
n_parts = len(self._parts)
n_per_part = torch.full(
torch.Size([n_parts]),
n // n_parts,
dtype=torch.long,
device=self.bounds.device,
)
left_over = n % n_parts
n_per_part[:left_over] += 1
f_0s = []
for i, p in enumerate(self._parts):
left, right = p
f_0s.append(
torch.linspace(
left + self._eps,
right - self._eps,
n_per_part[i],
dtype=self.bounds.dtype,
device=self.bounds.device,
)
)
f_0 = torch.cat(f_0s, dim=0)
f_1 = 1 - f_0.sqrt() - f_0 * torch.sin(10 * math.pi * f_0)
f_X = torch.stack([f_0, f_1], dim=-1)
if self.negate:
f_X *= -1
return f_X
# ------ Constrained Multi-Objective Test Problems ----- #
[docs]class BNH(MultiObjectiveTestProblem, ConstrainedBaseTestProblem):
r"""The constrained BNH problem.
See [GarridoMerchan2020]_ for more details on this problem. Note that this is a
minimization problem.
"""
dim = 2
num_objectives = 2
num_constraints = 2
_bounds = [(0.0, 5.0), (0.0, 3.0)]
_ref_point = [0.0, 0.0] # TODO: Determine proper reference point
[docs] def evaluate_true(self, X: Tensor) -> Tensor:
return torch.stack(
[4.0 * (X ** 2).sum(dim=-1), ((X - 5.0) ** 2).sum(dim=-1)], dim=-1
)
[docs] def evaluate_slack_true(self, X: Tensor) -> Tensor:
c1 = 25.0 - (X[..., 0] - 5.0) ** 2 - X[..., 1] ** 2
c2 = (X[..., 0] - 8.0) ** 2 + (X[..., 1] + 3.0) ** 2 - 7.7
return torch.stack([c1, c2], dim=-1)
[docs]class SRN(MultiObjectiveTestProblem, ConstrainedBaseTestProblem):
r"""The constrained SRN problem.
See [GarridoMerchan2020]_ for more details on this problem. Note that this is a
minimization problem.
"""
dim = 2
num_objectives = 2
num_constraints = 2
_bounds = [(-20.0, 20.0), (-20.0, 20.0)]
_ref_point = [0.0, 0.0] # TODO: Determine proper reference point
[docs] def evaluate_true(self, X: Tensor) -> Tensor:
obj1 = 2.0 + ((X - 2.0) ** 2).sum(dim=-1)
obj2 = 9.0 * X[..., 0] - (X[..., 1] - 1.0) ** 2
return torch.stack([obj1, obj2], dim=-1)
[docs] def evaluate_slack_true(self, X: Tensor) -> Tensor:
c1 = 225.0 - ((X ** 2) ** 2).sum(dim=-1)
c2 = -10.0 - X[..., 0] + 3 * X[..., 1]
return torch.stack([c1, c2], dim=-1)
[docs]class CONSTR(MultiObjectiveTestProblem, ConstrainedBaseTestProblem):
r"""The constrained CONSTR problem.
See [GarridoMerchan2020]_ for more details on this problem. Note that this is a
minimization problem.
"""
dim = 2
num_objectives = 2
num_constraints = 2
_bounds = [(0.1, 10.0), (0.0, 5.0)]
_ref_point = [10.0, 10.0]
[docs] def evaluate_true(self, X: Tensor) -> Tensor:
obj1 = X[..., 0]
obj2 = (1.0 + X[..., 1]) / X[..., 0]
return torch.stack([obj1, obj2], dim=-1)
[docs] def evaluate_slack_true(self, X: Tensor) -> Tensor:
c1 = 9.0 * X[..., 0] + X[..., 1] - 6.0
c2 = 9.0 * X[..., 0] - X[..., 1] - 1.0
return torch.stack([c1, c2], dim=-1)
[docs]class ConstrainedBraninCurrin(BraninCurrin, ConstrainedBaseTestProblem):
r"""Constrained Branin Currin Function.
This uses the disk constraint from [Gelbart2014]_.
"""
dim = 2
num_objectives = 2
num_constraints = 1
_bounds = [(0.0, 1.0), (0.0, 1.0)]
_con_bounds = [(-5.0, 10.0), (0.0, 15.0)]
_ref_point = [90.0, 10.0]
_max_hv = 372.74230433212836 # from NSGA-II with 90k evaluations
def __init__(self, noise_std: Optional[float] = None, negate: bool = False) -> None:
super().__init__(noise_std=noise_std, negate=negate)
con_bounds = torch.tensor(self._con_bounds, dtype=torch.float)
self.register_buffer("con_bounds", con_bounds)
[docs] def evaluate_slack_true(self, X: Tensor) -> Tensor:
X_tf = unnormalize(X, self.con_bounds)
return 50 - (X_tf[..., 0:1] - 2.5).pow(2) - (X_tf[..., 1:2] - 7.5).pow(2)
[docs]class C2DTLZ2(DTLZ2, ConstrainedBaseTestProblem):
num_constraints = 1
_r = 0.2
_max_hv = 5.439801376693978 # approximate from nsga-ii, TODO: replace with analytic
[docs] def evaluate_slack_true(self, X: Tensor) -> Tensor:
if X.ndim > 2:
raise NotImplementedError("Batch X is not supported.")
f_X = self.evaluate_true(X)
term1 = (f_X - 1).pow(2)
mask = ~(torch.eye(f_X.shape[-1], device=f_X.device).bool())
indices = torch.arange(f_X.shape[1], device=f_X.device).repeat(f_X.shape[1], 1)
indexer = indices[mask].view(f_X.shape[1], f_X.shape[-1] - 1)
term2_inner = (
f_X.unsqueeze(1)
.expand(f_X.shape[0], f_X.shape[-1], f_X.shape[-1])
.gather(dim=-1, index=indexer.repeat(f_X.shape[0], 1, 1))
)
term2 = (term2_inner.pow(2) - self._r ** 2).sum(dim=-1)
min1 = (term1 + term2).min(dim=-1).values
min2 = ((f_X - 1 / math.sqrt(f_X.shape[-1])).pow(2) - self._r ** 2).sum(dim=-1)
return -torch.min(min1, min2).unsqueeze(-1)
