#!/usr/bin/env python3
# Copyright (c) Meta Platforms, Inc. and affiliates.
#
# This source code is licensed under the MIT license found in the
# LICENSE file in the root directory of this source tree.
r"""
Synthetic functions for optimization benchmarks.
Most test functions (if not indicated otherwise) are taken from
[Bingham2013virtual]_.
References:
.. [Bingham2013virtual]
D. Bingham, S. Surjanovic. Virtual Library of Simulation Experiments.
https://www.sfu.ca/~ssurjano/optimization.html
.. [CoelloCoello2002constraint]
C. A. Coello Coello and E. Mezura Montes. Constraint-handling in genetic
algorithms through the use of dominance-based tournament selection.
Advanced Engineering Informatics, 16(3):193–203, 2002.
.. [Hedar2006derivfree]
A.-R. Hedar and M. Fukushima. Derivative-free filter simulated annealing
method for constrained continuous global optimization. Journal of Global
Optimization, 35(4):521–549, 2006.
.. [Lemonge2010constrained]
A. C. C. Lemonge, H. J. C. Barbosa, C. C. H. Borges, and F. B. dos Santos
Silva. Constrained optimization problems in mechanical engineering design
using a real-coded steady-state genetic algorithm. Mecánica Computacional,
XXIX:9287–9303, 2010.
.. [Letham2019]
B. Letham, B. Karrer, G. Ottoni, and E. Bakshy. Constrained Bayesian
Optimization with Noisy Experiments. Bayesian Analysis, Bayesian Anal.
14(2), 495-519, 2019.
.. [Gramacy2016]
R. Gramacy, G. Gray, S. Le Digabel, H. Lee, P. Ranjan, G. Wells & S. Wild.
Modeling an Augmented Lagrangian for Blackbox Constrained Optimization,
Technometrics, 2016.
"""
from __future__ import annotations
import math
from abc import ABC
from typing import List, Optional, Tuple, Union
import torch
from botorch.exceptions.errors import InputDataError
from botorch.test_functions.base import BaseTestProblem, ConstrainedBaseTestProblem
from botorch.test_functions.utils import round_nearest
from torch import Tensor
[docs]
class SyntheticTestFunction(BaseTestProblem, ABC):
r"""Base class for synthetic test functions."""
_optimal_value: Optional[float] = None
_optimizers: Optional[List[Tuple[float, ...]]] = None
num_objectives: int = 1
def __init__(
self,
noise_std: Union[None, float, List[float]] = None,
negate: bool = False,
bounds: Optional[List[Tuple[float, float]]] = None,
) -> None:
r"""
Args:
noise_std: Standard deviation of the observation noise. If a list is
provided, specifies separate noise standard deviations for each
objective in a multiobjective problem.
negate: If True, negate the function.
bounds: Custom bounds for the function specified as (lower, upper) pairs.
"""
if bounds is not None:
self._bounds = bounds
super().__init__(noise_std=noise_std, negate=negate)
if self._optimizers is not None:
if bounds is not None:
# Ensure at least one optimizer lies within the custom bounds
def in_bounds(
optimizer: Tuple[float, ...], bounds: List[Tuple[float, float]]
) -> bool:
for i, xopt in enumerate(optimizer):
lower, upper = bounds[i]
if xopt < lower or xopt > upper:
return False
return True
if not any(
in_bounds(optimizer=optimizer, bounds=bounds)
for optimizer in self._optimizers
):
raise ValueError(
"No global optimum found within custom bounds. Please specify "
"bounds which include at least one point in "
f"`{self.__class__.__name__}._optimizers`."
)
self.register_buffer(
"optimizers", torch.tensor(self._optimizers, dtype=self.bounds.dtype)
)
@property
def optimal_value(self) -> float:
r"""The global minimum (maximum if negate=True) of the function."""
if self._optimal_value is not None:
return -self._optimal_value if self.negate else self._optimal_value
else:
raise NotImplementedError(
f"Problem {self.__class__.__name__} does not specify an optimal value."
)
[docs]
class Ackley(SyntheticTestFunction):
r"""Ackley test function.
d-dimensional function (usually evaluated on `[-32.768, 32.768]^d`):
f(x) = -A exp(-B sqrt(1/d sum_{i=1}^d x_i^2)) -
exp(1/d sum_{i=1}^d cos(c x_i)) + A + exp(1)
f has one minimizer for its global minimum at `z_1 = (0, 0, ..., 0)` with
`f(z_1) = 0`.
"""
_optimal_value = 0.0
_check_grad_at_opt: bool = False
def __init__(
self,
dim: int = 2,
noise_std: Optional[float] = None,
negate: bool = False,
bounds: Optional[List[Tuple[float, float]]] = None,
) -> None:
r"""
Args:
dim: The (input) dimension.
noise_std: Standard deviation of the observation noise.
negate: If True, negate the function.
bounds: Custom bounds for the function specified as (lower, upper) pairs.
"""
self.dim = dim
if bounds is None:
bounds = [(-32.768, 32.768) for _ in range(self.dim)]
self._optimizers = [tuple(0.0 for _ in range(self.dim))]
super().__init__(noise_std=noise_std, negate=negate, bounds=bounds)
self.a = 20
self.b = 0.2
self.c = 2 * math.pi
[docs]
def evaluate_true(self, X: Tensor) -> Tensor:
a, b, c = self.a, self.b, self.c
part1 = -a * torch.exp(-b / math.sqrt(self.dim) * torch.linalg.norm(X, dim=-1))
part2 = -(torch.exp(torch.mean(torch.cos(c * X), dim=-1)))
return part1 + part2 + a + math.e
[docs]
class Beale(SyntheticTestFunction):
dim = 2
_optimal_value = 0.0
_bounds = [(-4.5, 4.5), (-4.5, 4.5)]
_optimizers = [(3.0, 0.5)]
[docs]
def evaluate_true(self, X: Tensor) -> Tensor:
x1, x2 = X[..., 0], X[..., 1]
part1 = (1.5 - x1 + x1 * x2).pow(2)
part2 = (2.25 - x1 + x1 * x2.pow(2)).pow(2)
part3 = (2.625 - x1 + x1 * x2.pow(3)).pow(2)
return part1 + part2 + part3
[docs]
class Branin(SyntheticTestFunction):
r"""Branin test function.
Two-dimensional function (usually evaluated on `[-5, 10] x [0, 15]`):
B(x) = (x_2 - b x_1^2 + c x_1 - r)^2 + 10 (1-t) cos(x_1) + 10
Here `b`, `c`, `r` and `t` are constants where `b = 5.1 / (4 * math.pi ** 2)`
`c = 5 / math.pi`, `r = 6`, `t = 1 / (8 * math.pi)`
B has 3 minimizers for its global minimum at `z_1 = (-pi, 12.275)`,
`z_2 = (pi, 2.275)`, `z_3 = (9.42478, 2.475)` with `B(z_i) = 0.397887`.
"""
dim = 2
_bounds = [(-5.0, 10.0), (0.0, 15.0)]
_optimal_value = 0.397887
_optimizers = [(-math.pi, 12.275), (math.pi, 2.275), (9.42478, 2.475)]
[docs]
def evaluate_true(self, X: Tensor) -> Tensor:
t1 = (
X[..., 1]
- 5.1 / (4 * math.pi**2) * X[..., 0].pow(2)
+ 5 / math.pi * X[..., 0]
- 6
)
t2 = 10 * (1 - 1 / (8 * math.pi)) * torch.cos(X[..., 0])
return t1.pow(2) + t2 + 10
[docs]
class Bukin(SyntheticTestFunction):
dim = 2
_bounds = [(-15.0, -5.0), (-3.0, 3.0)]
_optimal_value = 0.0
_optimizers = [(-10.0, 1.0)]
_check_grad_at_opt: bool = False
[docs]
def evaluate_true(self, X: Tensor) -> Tensor:
part1 = 100.0 * torch.sqrt(torch.abs(X[..., 1] - 0.01 * X[..., 0].pow(2)))
part2 = 0.01 * torch.abs(X[..., 0] + 10.0)
return part1 + part2
[docs]
class Cosine8(SyntheticTestFunction):
r"""Cosine Mixture test function.
8-dimensional function (usually evaluated on `[-1, 1]^8`):
f(x) = 0.1 sum_{i=1}^8 cos(5 pi x_i) - sum_{i=1}^8 x_i^2
f has one maximizer for its global maximum at `z_1 = (0, 0, ..., 0)` with
`f(z_1) = 0.8`
"""
dim = 8
_bounds = [(-1.0, 1.0) for _ in range(8)]
_optimal_value = 0.8
_optimizers = [tuple(0.0 for _ in range(8))]
[docs]
def evaluate_true(self, X: Tensor) -> Tensor:
return torch.sum(0.1 * torch.cos(5 * math.pi * X) - X.pow(2), dim=-1)
[docs]
class DropWave(SyntheticTestFunction):
dim = 2
_bounds = [(-5.12, 5.12), (-5.12, 5.12)]
_optimal_value = -1.0
_optimizers = [(0.0, 0.0)]
_check_grad_at_opt = False
[docs]
def evaluate_true(self, X: Tensor) -> Tensor:
norm = torch.linalg.norm(X, dim=-1)
part1 = 1.0 + torch.cos(12.0 * norm)
part2 = 0.5 * norm.pow(2) + 2.0
return -part1 / part2
[docs]
class DixonPrice(SyntheticTestFunction):
_optimal_value = 0.0
def __init__(
self,
dim=2,
noise_std: Optional[float] = None,
negate: bool = False,
bounds: Optional[List[Tuple[float, float]]] = None,
) -> None:
r"""
Args:
dim: The (input) dimension.
noise_std: Standard deviation of the observation noise.
negate: If True, negate the function.
"""
self.dim = dim
if bounds is None:
bounds = [(-10.0, 10.0) for _ in range(self.dim)]
self._optimizers = [
tuple(
math.pow(2.0, -(1.0 - 2.0 ** (-(i - 1))))
for i in range(1, self.dim + 1)
)
]
super().__init__(noise_std=noise_std, negate=negate, bounds=bounds)
[docs]
def evaluate_true(self, X: Tensor) -> Tensor:
d = self.dim
part1 = (X[..., 0] - 1).pow(2)
i = X.new(range(2, d + 1))
part2 = torch.sum(i * (2.0 * X[..., 1:].pow(2) - X[..., :-1]).pow(2), dim=-1)
return part1 + part2
[docs]
class EggHolder(SyntheticTestFunction):
r"""Eggholder test function.
Two-dimensional function (usually evaluated on `[-512, 512]^2`):
E(x) = (x_2 + 47) sin(R1(x)) - x_1 * sin(R2(x))
where `R1(x) = sqrt(|x_2 + x_1 / 2 + 47|)`, `R2(x) = sqrt|x_1 - (x_2 + 47)|)`.
"""
dim = 2
_bounds = [(-512.0, 512.0), (-512.0, 512.0)]
_optimal_value = -959.6407
_optimizers = [(512.0, 404.2319)]
_check_grad_at_opt: bool = False
[docs]
def evaluate_true(self, X: Tensor) -> Tensor:
x1, x2 = X[..., 0], X[..., 1]
part1 = -(x2 + 47.0) * torch.sin(torch.sqrt(torch.abs(x2 + x1 / 2.0 + 47.0)))
part2 = -x1 * torch.sin(torch.sqrt(torch.abs(x1 - (x2 + 47.0))))
return part1 + part2
[docs]
class Griewank(SyntheticTestFunction):
r"""Griewank synthetic test function.
The Griewank function is defined for any `d`, is typically evaluated on
`[-600, 600]^d`, and given by:
G(x) = sum_{i=1}^d x_i**2 / 4000 - prod_{i=1}^d cos(x_i / sqrt(i)) + 1
G has many widespread local minima, which are regularly distributed.
The global minimum is at `z = (0, ..., 0)` with `G(z) = 0`.
"""
_optimal_value = 0.0
def __init__(
self,
dim=2,
noise_std: Optional[float] = None,
negate: bool = False,
bounds: Optional[List[Tuple[float, float]]] = None,
) -> None:
r"""
Args:
dim: The (input) dimension.
noise_std: Standard deviation of the observation noise.
negate: If True, negate the function.
bounds: Custom bounds for the function specified as (lower, upper) pairs.
"""
self.dim = dim
if bounds is None:
bounds = [(-600.0, 600.0) for _ in range(self.dim)]
self._optimizers = [tuple(0.0 for _ in range(self.dim))]
super().__init__(noise_std=noise_std, negate=negate, bounds=bounds)
[docs]
def evaluate_true(self, X: Tensor) -> Tensor:
part1 = torch.sum(X.pow(2) / 4000.0, dim=-1)
d = X.shape[-1]
part2 = -(torch.prod(torch.cos(X / torch.sqrt(X.new(range(1, d + 1)))), dim=-1))
return part1 + part2 + 1.0
[docs]
class Hartmann(SyntheticTestFunction):
r"""Hartmann synthetic test function.
Most commonly used is the six-dimensional version (typically evaluated on
`[0, 1]^6`):
H(x) = - sum_{i=1}^4 ALPHA_i exp( - sum_{j=1}^6 A_ij (x_j - P_ij)**2 )
H has a 6 local minima and a global minimum at
z = (0.20169, 0.150011, 0.476874, 0.275332, 0.311652, 0.6573)
with `H(z) = -3.32237`.
"""
def __init__(
self,
dim=6,
noise_std: Optional[float] = None,
negate: bool = False,
bounds: Optional[List[Tuple[float, float]]] = None,
) -> None:
r"""
Args:
dim: The (input) dimension.
noise_std: Standard deviation of the observation noise.
negate: If True, negate the function.
bounds: Custom bounds for the function specified as (lower, upper) pairs.
"""
if dim not in (3, 4, 6):
raise ValueError(f"Hartmann with dim {dim} not defined")
self.dim = dim
if bounds is None:
bounds = [(0.0, 1.0) for _ in range(self.dim)]
# optimizers and optimal values for dim=4 not implemented
optvals = {3: -3.86278, 6: -3.32237}
optimizers = {
3: [(0.114614, 0.555649, 0.852547)],
6: [(0.20169, 0.150011, 0.476874, 0.275332, 0.311652, 0.6573)],
}
self._optimal_value = optvals.get(self.dim)
self._optimizers = optimizers.get(self.dim)
super().__init__(noise_std=noise_std, negate=negate, bounds=bounds)
self.register_buffer("ALPHA", torch.tensor([1.0, 1.2, 3.0, 3.2]))
if dim == 3:
A = [[3.0, 10, 30], [0.1, 10, 35], [3.0, 10, 30], [0.1, 10, 35]]
P = [
[3689, 1170, 2673],
[4699, 4387, 7470],
[1091, 8732, 5547],
[381, 5743, 8828.0],
]
elif dim == 4:
A = [
[10, 3, 17, 3.5],
[0.05, 10, 17, 0.1],
[3, 3.5, 1.7, 10],
[17, 8, 0.05, 10],
]
P = [
[1312, 1696, 5569, 124.0],
[2329, 4135, 8307, 3736],
[2348, 1451, 3522, 2883],
[4047, 8828, 8732, 5743],
]
elif dim == 6:
A = [
[10, 3, 17, 3.5, 1.7, 8],
[0.05, 10, 17, 0.1, 8, 14],
[3, 3.5, 1.7, 10, 17, 8],
[17, 8, 0.05, 10, 0.1, 14],
]
P = [
[1312, 1696, 5569, 124, 8283, 5886],
[2329, 4135, 8307, 3736, 1004, 9991],
[2348, 1451, 3522, 2883, 3047, 6650],
[4047, 8828, 8732, 5743, 1091, 381.0],
]
else: # pragma: no cover -- unreacheable code for pyre.
raise NotImplementedError
self.register_buffer("A", torch.tensor(A))
self.register_buffer("P", torch.tensor(P))
@property
def optimizers(self) -> Tensor:
if self.dim == 4:
raise NotImplementedError()
return super().optimizers
[docs]
def evaluate_true(self, X: Tensor) -> Tensor:
self.to(device=X.device, dtype=X.dtype)
inner_sum = torch.sum(
self.A * (X.unsqueeze(-2) - 0.0001 * self.P).pow(2), dim=-1
)
H = -(torch.sum(self.ALPHA * torch.exp(-inner_sum), dim=-1))
if self.dim == 4:
H = (1.1 + H) / 0.839
return H
[docs]
class HolderTable(SyntheticTestFunction):
r"""Holder Table synthetic test function.
Two-dimensional function (typically evaluated on `[0, 10] x [0, 10]`):
`H(x) = - | sin(x_1) * cos(x_2) * exp(| 1 - ||x|| / pi | ) |`
H has 4 global minima with `H(z_i) = -19.2085` at
z_1 = ( 8.05502, 9.66459)
z_2 = (-8.05502, -9.66459)
z_3 = (-8.05502, 9.66459)
z_4 = ( 8.05502, -9.66459)
"""
dim = 2
_bounds = [(-10.0, 10.0), (-10.0, 10.0)]
_optimal_value = -19.2085
_optimizers = [
(8.05502, 9.66459),
(-8.05502, -9.66459),
(-8.05502, 9.66459),
(8.05502, -9.66459),
]
[docs]
def evaluate_true(self, X: Tensor) -> Tensor:
term = torch.abs(1 - torch.linalg.norm(X, dim=-1) / math.pi)
return -(
torch.abs(torch.sin(X[..., 0]) * torch.cos(X[..., 1]) * torch.exp(term))
)
[docs]
class Levy(SyntheticTestFunction):
r"""Levy synthetic test function.
d-dimensional function (usually evaluated on `[-10, 10]^d`):
f(x) = sin^2(pi w_1) +
sum_{i=1}^{d-1} (w_i-1)^2 (1 + 10 sin^2(pi w_i + 1)) +
(w_d - 1)^2 (1 + sin^2(2 pi w_d))
where `w_i = 1 + (x_i - 1) / 4` for all `i`.
f has one minimizer for its global minimum at `z_1 = (1, 1, ..., 1)` with
`f(z_1) = 0`.
"""
_optimal_value = 0.0
def __init__(
self,
dim=2,
noise_std: Optional[float] = None,
negate: bool = False,
bounds: Optional[List[Tuple[float, float]]] = None,
) -> None:
r"""
Args:
dim: The (input) dimension.
noise_std: Standard deviation of the observation noise.
negate: If True, negate the function.
bounds: Custom bounds for the function specified as (lower, upper) pairs.
"""
self.dim = dim
if bounds is None:
bounds = [(-10.0, 10.0) for _ in range(self.dim)]
self._optimizers = [tuple(1.0 for _ in range(self.dim))]
super().__init__(noise_std=noise_std, negate=negate, bounds=bounds)
[docs]
def evaluate_true(self, X: Tensor) -> Tensor:
w = 1.0 + (X - 1.0) / 4.0
part1 = torch.sin(math.pi * w[..., 0]).pow(2)
part2 = torch.sum(
(w[..., :-1] - 1.0).pow(2)
* (1.0 + 10.0 * torch.sin(math.pi * w[..., :-1] + 1.0).pow(2)),
dim=-1,
)
part3 = (w[..., -1] - 1.0).pow(2) * (
1.0 + torch.sin(2.0 * math.pi * w[..., -1]).pow(2)
)
return part1 + part2 + part3
[docs]
class Michalewicz(SyntheticTestFunction):
r"""Michalewicz synthetic test function.
d-dim function (usually evaluated on hypercube [0, pi]^d):
M(x) = sum_{i=1}^d sin(x_i) (sin(i x_i^2 / pi)^20)
"""
def __init__(
self,
dim=2,
noise_std: Optional[float] = None,
negate: bool = False,
bounds: Optional[List[Tuple[float, float]]] = None,
) -> None:
r"""
Args:
dim: The (input) dimension.
noise_std: Standard deviation of the observation noise.
negate: If True, negate the function.
bounds: Custom bounds for the function specified as (lower, upper) pairs.
"""
self.dim = dim
if bounds is None:
bounds = [(0.0, math.pi) for _ in range(self.dim)]
optvals = {2: -1.80130341, 5: -4.687658, 10: -9.66015}
optimizers = {2: [(2.20290552, 1.57079633)]}
self._optimal_value = optvals.get(self.dim)
self._optimizers = optimizers.get(self.dim)
super().__init__(noise_std=noise_std, negate=negate, bounds=bounds)
self.register_buffer(
"i", torch.tensor(tuple(range(1, self.dim + 1)), dtype=self.bounds.dtype)
)
@property
def optimizers(self) -> Tensor:
if self.dim in (5, 10):
raise NotImplementedError()
return super().optimizers
[docs]
def evaluate_true(self, X: Tensor) -> Tensor:
self.to(device=X.device, dtype=X.dtype)
m = 10
return -(
torch.sum(
torch.sin(X) * torch.sin(self.i * X.pow(2) / math.pi).pow(2 * m), dim=-1
)
)
[docs]
class Powell(SyntheticTestFunction):
r"""Powell synthetic test function.
`d`-dim function (usually evaluated on the hypercube `[-4, 5]^d`):
P(x) = sum_{i=1}^d/4 (
(x_{4i-3} + 10 x_{4i-2})**2
+ 5 (x_{4i-1} - x_{4i})**2
+ (x_{4i-2} - 2 x_{4i-1})**4
+ 10 (x_{4i-3} - x_{4i})**4
)
P has a global minimizer at `z = (0, ..., 0)` with `P(z) = 0`.
"""
_optimal_value = 0.0
def __init__(
self,
dim=4,
noise_std: Optional[float] = None,
negate: bool = False,
bounds: Optional[List[Tuple[float, float]]] = None,
) -> None:
r"""
Args:
dim: The (input) dimension.
noise_std: Standard deviation of the observation noise.
negate: If True, negate the function.
bounds: Custom bounds for the function specified as (lower, upper) pairs.
"""
self.dim = dim
if bounds is None:
bounds = [(-4.0, 5.0) for _ in range(self.dim)]
self._optimizers = [tuple(0.0 for _ in range(self.dim))]
super().__init__(noise_std=noise_std, negate=negate, bounds=bounds)
[docs]
def evaluate_true(self, X: Tensor) -> Tensor:
result = torch.zeros_like(X[..., 0])
for i in range(self.dim // 4):
i_ = i + 1
part1 = (X[..., 4 * i_ - 4] + 10.0 * X[..., 4 * i_ - 3]).pow(2)
part2 = 5.0 * (X[..., 4 * i_ - 2] - X[..., 4 * i_ - 1]).pow(2)
part3 = (X[..., 4 * i_ - 3] - 2.0 * X[..., 4 * i_ - 2]).pow(4)
part4 = 10.0 * (X[..., 4 * i_ - 4] - X[..., 4 * i_ - 1]).pow(4)
result += part1 + part2 + part3 + part4
return result
[docs]
class Rastrigin(SyntheticTestFunction):
_optimal_value = 0.0
def __init__(
self,
dim=2,
noise_std: Optional[float] = None,
negate: bool = False,
bounds: Optional[List[Tuple[float, float]]] = None,
) -> None:
r"""
Args:
dim: The (input) dimension.
noise_std: Standard deviation of the observation noise.
negate: If True, negate the function.
bounds: Custom bounds for the function specified as (lower, upper) pairs.
"""
self.dim = dim
if bounds is None:
bounds = [(-5.12, 5.12) for _ in range(self.dim)]
self._optimizers = [tuple(0.0 for _ in range(self.dim))]
super().__init__(noise_std=noise_std, negate=negate, bounds=bounds)
[docs]
def evaluate_true(self, X: Tensor) -> Tensor:
return 10.0 * self.dim + torch.sum(
X.pow(2) - 10.0 * torch.cos(2.0 * math.pi * X), dim=-1
)
[docs]
class Rosenbrock(SyntheticTestFunction):
r"""Rosenbrock synthetic test function.
d-dimensional function (usually evaluated on `[-5, 10]^d`):
f(x) = sum_{i=1}^{d-1} (100 (x_{i+1} - x_i^2)^2 + (x_i - 1)^2)
f has one minimizer for its global minimum at `z_1 = (1, 1, ..., 1)` with
`f(z_i) = 0.0`.
"""
_optimal_value = 0.0
def __init__(
self,
dim=2,
noise_std: Optional[float] = None,
negate: bool = False,
bounds: Optional[List[Tuple[float, float]]] = None,
) -> None:
r"""
Args:
dim: The (input) dimension.
noise_std: Standard deviation of the observation noise.
negate: If True, negate the function.
bounds: Custom bounds for the function specified as (lower, upper) pairs.
"""
self.dim = dim
if bounds is None:
bounds = [(-5.0, 10.0) for _ in range(self.dim)]
self._optimizers = [tuple(1.0 for _ in range(self.dim))]
super().__init__(noise_std=noise_std, negate=negate, bounds=bounds)
[docs]
def evaluate_true(self, X: Tensor) -> Tensor:
return torch.sum(
100.0 * (X[..., 1:] - X[..., :-1].pow(2)).pow(2) + (X[..., :-1] - 1).pow(2),
dim=-1,
)
[docs]
class Shekel(SyntheticTestFunction):
r"""Shekel synthtetic test function.
4-dimensional function (usually evaluated on `[0, 10]^4`):
f(x) = -sum_{i=1}^10 (sum_{j=1}^4 (x_j - A_{ji})^2 + C_i)^{-1}
f has one minimizer for its global minimum at `z_1 = (4, 4, 4, 4)` with
`f(z_1) = -10.5363`.
"""
dim = 4
_bounds = [(0.0, 10.0), (0.0, 10.0), (0.0, 10.0), (0.0, 10.0)]
_optimizers = [(4.000747, 3.99951, 4.00075, 3.99951)]
def __init__(
self,
m: int = 10,
noise_std: Optional[float] = None,
negate: bool = False,
bounds: Optional[List[Tuple[float, float]]] = None,
) -> None:
r"""
Args:
m: Defaults to 10.
noise_std: Standard deviation of the observation noise.
negate: If True, negate the function.
bounds: Custom bounds for the function specified as (lower, upper) pairs.
"""
self.m = m
optvals = {5: -10.1532, 7: -10.4029, 10: -10.536443}
self._optimal_value = optvals[self.m]
super().__init__(noise_std=noise_std, negate=negate, bounds=bounds)
self.register_buffer("beta", torch.tensor([1, 2, 2, 4, 4, 6, 3, 7, 5, 5.0]))
C_t = torch.tensor(
[
[4, 1, 8, 6, 3, 2, 5, 8, 6, 7],
[4, 1, 8, 6, 7, 9, 3, 1, 2, 3.6],
[4, 1, 8, 6, 3, 2, 5, 8, 6, 7],
[4, 1, 8, 6, 7, 9, 3, 1, 2, 3.6],
],
)
self.register_buffer("C", C_t.transpose(-1, -2))
[docs]
def evaluate_true(self, X: Tensor) -> Tensor:
self.to(device=X.device, dtype=X.dtype)
beta = self.beta / 10.0
result = -sum(
1 / (torch.sum((X - self.C[i]).pow(2), dim=-1) + beta[i])
for i in range(self.m)
)
return result
[docs]
class SixHumpCamel(SyntheticTestFunction):
dim = 2
_bounds = [(-3.0, 3.0), (-2.0, 2.0)]
_optimal_value = -1.0316
_optimizers = [(0.0898, -0.7126), (-0.0898, 0.7126)]
[docs]
def evaluate_true(self, X: Tensor) -> Tensor:
x1, x2 = X[..., 0], X[..., 1]
return (
(4 - 2.1 * x1.pow(2) + x1.pow(4) / 3) * x1.pow(2)
+ x1 * x2
+ (4 * x2.pow(2) - 4) * x2.pow(2)
)
[docs]
class StyblinskiTang(SyntheticTestFunction):
r"""Styblinski-Tang synthtetic test function.
d-dimensional function (usually evaluated on the hypercube `[-5, 5]^d`):
H(x) = 0.5 * sum_{i=1}^d (x_i^4 - 16 * x_i^2 + 5 * x_i)
H has a single global mininimum `H(z) = -39.166166 * d` at `z = [-2.903534]^d`
"""
def __init__(
self,
dim=2,
noise_std: Optional[float] = None,
negate: bool = False,
bounds: Optional[List[Tuple[float, float]]] = None,
) -> None:
r"""
Args:
dim: The (input) dimension.
noise_std: Standard deviation of the observation noise.
negate: If True, negate the function.
bounds: Custom bounds for the function specified as (lower, upper) pairs.
"""
self.dim = dim
if bounds is None:
bounds = [(-5.0, 5.0) for _ in range(self.dim)]
self._optimal_value = -39.166166 * self.dim
self._optimizers = [tuple(-2.903534 for _ in range(self.dim))]
super().__init__(noise_std=noise_std, negate=negate, bounds=bounds)
[docs]
def evaluate_true(self, X: Tensor) -> Tensor:
return 0.5 * (X.pow(4) - 16 * X.pow(2) + 5 * X).sum(dim=-1)
[docs]
class ThreeHumpCamel(SyntheticTestFunction):
dim = 2
_bounds = [(-5.0, 5.0), (-5.0, 5.0)]
_optimal_value = 0.0
_optimizers = [(0.0, 0.0)]
[docs]
def evaluate_true(self, X: Tensor) -> Tensor:
x1, x2 = X[..., 0], X[..., 1]
return (
2.0 * x1.pow(2) - 1.05 * x1.pow(4) + x1.pow(6) / 6.0 + x1 * x2 + x2.pow(2)
)
# ------------ Constrained synthetic test functions ----------- #
[docs]
class ConstrainedSyntheticTestFunction(
ConstrainedBaseTestProblem, SyntheticTestFunction, ABC
):
r"""Base class for constrained synthetic test functions."""
def __init__(
self,
noise_std: Union[None, float, List[float]] = None,
constraint_noise_std: Union[None, float, List[float]] = None,
negate: bool = False,
bounds: Optional[List[Tuple[float, float]]] = None,
) -> None:
r"""
Args:
noise_std: Standard deviation of the observation noise. If a list is
provided, specifies separate noise standard deviations for each
objective in a multiobjective problem.
constraint_noise_std: Standard deviation of the constraint noise.
If a list is provided, specifies separate noise standard
deviations for each constraint.
negate: If True, negate the function.
bounds: Custom bounds for the function specified as (lower, upper) pairs.
"""
SyntheticTestFunction.__init__(
self, noise_std=noise_std, negate=negate, bounds=bounds
)
self.constraint_noise_std = self._validate_constraint_noise(
constraint_noise_std
)
def _validate_constraint_noise(
self, constraint_noise_std
) -> Union[None, float, List[float]]:
"""
Validates that constraint_noise_std has length equal to
the number of constraints, if given as a list
Args:
constraint_noise_std: Standard deviation of the constraint noise.
If a list is provided, specifies separate noise standard
deviations for each constraint.
"""
if (
isinstance(constraint_noise_std, list)
and len(constraint_noise_std) != self.num_constraints
):
raise InputDataError(
"If specified as a list, length of constraint_noise_std "
f"({len(constraint_noise_std)}) must match the "
f"number of constraints ({self.num_constraints})"
)
return constraint_noise_std
[docs]
class ConstrainedGramacy(ConstrainedSyntheticTestFunction):
r"""Constrained Gramacy test function.
This problem comes from [Gramacy2016]_. The problem is defined
over the unit cube and the goal is to minimize x1+x2 subject to
1.5 - x1 - 2 * x2 - 0.5 * sin(2*pi*(x1^2 - 2 * x2)) <= 0
and x1^2 + x2^2 - 1.5 <= 0.
"""
num_objectives = 1
num_constraints = 2
dim = 2
_bounds = [(0.0, 1.0), (0.0, 1.0)]
_optimizers = [(0.1954, 0.4044)]
_optimal_value = 0.5998 # approximate from [Gramacy2016]_
[docs]
def evaluate_true(self, X: Tensor) -> Tensor:
"""
Evaluate the function (w/o observation noise) on a set of points.
Args:
X: A `batch_shape x d`-dim tensor of point(s) at which to evaluate the
function.
"""
return X.sum(dim=-1)
[docs]
def evaluate_slack_true(self, X: Tensor) -> Tensor:
x1, x2 = X.split(1, dim=-1)
c1 = 1.5 - x1 - 2 * x2 - 0.5 * torch.sin(2 * math.pi * (x1.pow(2) - 2 * x2))
c2 = x1.pow(2) + x2.pow(2) - 1.5
return torch.cat([-c1, -c2], dim=-1)
[docs]
class ConstrainedHartmann(Hartmann, ConstrainedSyntheticTestFunction):
r"""Constrained Hartmann test function.
This is a constrained version of the standard Hartmann test function that
uses `||x||_2 <= 1` as the constraint. This problem comes from [Letham2019]_.
"""
num_constraints = 1
def __init__(
self,
dim: int = 6,
noise_std: Union[None, float] = None,
constraint_noise_std: Union[None, float, List[float]] = None,
negate: bool = False,
bounds: Optional[List[Tuple[float, float]]] = None,
) -> None:
r"""
Args:
dim: The (input) dimension.
noise_std: Standard deviation of the observation noise.
constraint_noise_std: Standard deviation of the constraint noise.
If a list is provided, specifies separate noise standard
deviations for each constraint.
negate: If True, negate the function.
bounds: Custom bounds for the function specified as (lower, upper) pairs.
"""
Hartmann.__init__(
self, dim=dim, noise_std=noise_std, negate=negate, bounds=bounds
)
self.constraint_noise_std = self._validate_constraint_noise(
constraint_noise_std
)
[docs]
def evaluate_slack_true(self, X: Tensor) -> Tensor:
return -X.norm(dim=-1, keepdim=True) + 1
[docs]
class ConstrainedHartmannSmooth(Hartmann, ConstrainedSyntheticTestFunction):
r"""Smooth constrained Hartmann test function.
This is a constrained version of the standard Hartmann test function that
uses `||x||_2^2 <= 1` as the constraint to obtain smoother constraint slack.
"""
num_constraints = 1
def __init__(
self,
dim: int = 6,
noise_std: Union[None, float] = None,
constraint_noise_std: Union[None, float, List[float]] = None,
negate: bool = False,
bounds: Optional[List[Tuple[float, float]]] = None,
) -> None:
r"""
Args:
dim: The (input) dimension.
noise_std: Standard deviation of the observation noise.
constraint_noise_std: Standard deviation of the constraint noise.
If a list is provided, specifies separate noise standard
deviations for each constraint.
negate: If True, negate the function.
bounds: Custom bounds for the function specified as (lower, upper) pairs.
"""
Hartmann.__init__(
self, dim=dim, noise_std=noise_std, negate=negate, bounds=bounds
)
self.constraint_noise_std = self._validate_constraint_noise(
constraint_noise_std
)
[docs]
def evaluate_slack_true(self, X: Tensor) -> Tensor:
return -X.pow(2).sum(dim=-1, keepdim=True) + 1
[docs]
class PressureVessel(ConstrainedSyntheticTestFunction):
r"""Pressure vessel design problem with constraints.
The four-dimensional pressure vessel design problem with four black-box
constraints from [CoelloCoello2002constraint]_.
"""
dim = 4
num_constraints = 4
_bounds = [(0.0, 10.0), (0.0, 10.0), (10.0, 50.0), (150.0, 200.0)]
[docs]
def evaluate_true(self, X: Tensor) -> Tensor:
x1, x2, x3, x4 = X.unbind(-1)
x1 = round_nearest(x1, increment=0.0625, bounds=self._bounds[0])
x2 = round_nearest(x2, increment=0.0625, bounds=self._bounds[1])
return (
0.6224 * x1 * x3 * x4
+ 1.7781 * x2 * x3.pow(2)
+ 3.1661 * x1.pow(2) * x4
+ 19.84 * x1.pow(2) * x3
)
[docs]
def evaluate_slack_true(self, X: Tensor) -> Tensor:
x1, x2, x3, x4 = X.unbind(-1)
return -torch.stack(
[
-x1 + 0.0193 * x3,
-x2 + 0.00954 * x3,
-math.pi * x3.pow(2) * x4 - (4 / 3) * math.pi * x3.pow(3) + 1296000.0,
x4 - 240.0,
],
dim=-1,
)
[docs]
class WeldedBeamSO(ConstrainedSyntheticTestFunction):
r"""Welded beam design problem with constraints (single-outcome).
The four-dimensional welded beam design proble problem with six
black-box constraints from [CoelloCoello2002constraint]_.
For a (somewhat modified) multi-objective version, see
`botorch.test_functions.multi_objective.WeldedBeam`.
"""
dim = 4
num_constraints = 6
_bounds = [(0.125, 10.0), (0.1, 10.0), (0.1, 10.0), (0.1, 10.0)]
[docs]
def evaluate_true(self, X: Tensor) -> Tensor:
x1, x2, x3, x4 = X.unbind(-1)
return 1.10471 * x1.pow(2) * x2 + 0.04811 * x3 * x4 * (14.0 + x2)
[docs]
def evaluate_slack_true(self, X: Tensor) -> Tensor:
x1, x2, x3, x4 = X.unbind(-1)
P = 6000.0
L = 14.0
E = 30e6
G = 12e6
t_max = 13600.0
s_max = 30000.0
d_max = 0.25
M = P * (L + x2 / 2)
R = torch.sqrt(0.25 * (x2.pow(2) + (x1 + x3).pow(2)))
J = 2 * math.sqrt(2) * x1 * x2 * (x2.pow(2) / 12 + 0.25 * (x1 + x3).pow(2))
P_c = (
4.013
* E
* x3
* x4.pow(3)
* 6
/ (L**2)
* (1 - 0.25 * x3 * math.sqrt(E / G) / L)
)
t1 = P / (math.sqrt(2) * x1 * x2)
t2 = M * R / J
t = torch.sqrt(t1.pow(2) + t1 * t2 * x2 / R + t2.pow(2))
s = 6 * P * L / (x4 * x3.pow(2))
d = 4 * P * L**3 / (E * x3.pow(3) * x4)
g1 = t - t_max
g2 = s - s_max
g3 = x1 - x4
g4 = 0.10471 * x1.pow(2) + 0.04811 * x3 * x4 * (14.0 + x2) - 5.0
g5 = d - d_max
g6 = P - P_c
return -torch.stack([g1, g2, g3, g4, g5, g6], dim=-1)
[docs]
class TensionCompressionString(ConstrainedSyntheticTestFunction):
r"""Tension compression string optimization problem with constraints.
The three-dimensional tension compression string optimization problem with
four black-box constraints from [Hedar2006derivfree]_.
"""
dim = 3
num_constraints = 4
_bounds = [(0.01, 1.0), (0.01, 1.0), (0.01, 20.0)]
[docs]
def evaluate_true(self, X: Tensor) -> Tensor:
x1, x2, x3 = X.unbind(-1)
return x1.pow(2) * x2 * (x3 + 2)
[docs]
def evaluate_slack_true(self, X: Tensor) -> Tensor:
x1, x2, x3 = X.unbind(-1)
constraints = torch.stack(
[
1 - x2.pow(3) * x3 / (71785 * x1.pow(4)),
(4 * x2.pow(2) - x1 * x2) / (12566 * x1.pow(3) * (x2 - x1))
+ 1 / (5108 * x1.pow(2))
- 1,
1 - 140.45 * x1 / (x3 * x2.pow(2)),
(x1 + x2) / 1.5 - 1,
],
dim=-1,
)
return -constraints.clamp_max(100)
[docs]
class SpeedReducer(ConstrainedSyntheticTestFunction):
r"""Speed Reducer design problem with constraints.
The seven-dimensional speed reducer design problem with eleven black-box
constraints from [Lemonge2010constrained]_.
"""
dim = 7
num_constraints = 11
_bounds = [
(2.6, 3.6),
(0.7, 0.8),
(17.0, 28.0),
(7.3, 8.3),
(7.8, 8.3),
(2.9, 3.9),
(5.0, 5.5),
]
[docs]
def evaluate_true(self, X: Tensor) -> Tensor:
x1, x2, x3, x4, x5, x6, x7 = X.unbind(-1)
return (
0.7854 * x1 * x2.pow(2) * (3.3333 * x3.pow(2) + 14.9334 * x3 - 43.0934)
+ -1.508 * x1 * (x6.pow(2) + x7.pow(2))
+ 7.4777 * (x6.pow(3) + x7.pow(3))
+ 0.7854 * (x4 * x6.pow(2) + x5 * x7.pow(2))
)
[docs]
def evaluate_slack_true(self, X: Tensor) -> Tensor:
x1, x2, x3, x4, x5, x6, x7 = X.unbind(-1)
return -torch.stack(
[
27.0 * (1 / x1) * (1 / x2.pow(2)) * (1 / x3) - 1,
397.5 * (1 / x1) * (1 / x2.pow(2)) * (1 / x3.pow(2)) - 1,
1.93 * (1 / x2) * (1 / x3) * x4.pow(3) * (1 / x6.pow(4)) - 1,
1.93 * (1 / x2) * (1 / x3) * x5.pow(3) * (1 / x7.pow(4)) - 1,
1
/ (0.1 * x6.pow(3))
* torch.sqrt((745 * x4 / (x2 * x3)).pow(2) + 16.9 * 1e6)
- 1100,
1
/ (0.1 * x7.pow(3))
* torch.sqrt((745 * x5 / (x2 * x3)).pow(2) + 157.5 * 1e6)
- 850,
x2 * x3 - 40,
5 - x1 / x2,
x1 / x2 - 12,
(1.5 * x6 + 1.9) / x4 - 1,
(1.1 * x7 + 1.9) / x5 - 1,
],
dim=-1,
)
