#!/usr/bin/env python3
# Copyright (c) Facebook, Inc. and its affiliates. All Rights Reserved
r"""
Synthetic functions for multi-fidelity optimization benchmarks.
"""
import math
from typing import Optional
import torch
from torch import Tensor
from .synthetic import SyntheticTestFunction
[docs]class AugmentedBranin(SyntheticTestFunction):
r"""Augmented Branin test function for multi-fidelity optimization.
3-dimensional function with domain `[-5, 10] x [0, 15] * [0,1]`, where
the last dimension of is the fidelity parameter:
B(x) = (x_2 - (b - 0.1 * (1 - x_3))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 infinitely many minimizers with `x_1 = -pi, pi, 3pi`
and `B_min = 0.397887`
"""
dim = 3
_bounds = [(-5.0, 10.0), (0.0, 15.0), (0.0, 1.0)]
_optimal_value = 0.397887
_optimizers = [ # this is a subset, ther are infinitely many optimizers
(-math.pi, 12.275, 1),
(math.pi, 1.3867356039019576, 0.1),
(math.pi, 1.781519779945532, 0.5),
(math.pi, 2.1763039559891064, 0.9),
]
[docs] def evaluate_true(self, X: Tensor) -> Tensor:
t1 = (
X[..., 1]
- (5.1 / (4 * math.pi ** 2) - 0.1 * (1 - X[:, 2])) * X[:, 0] ** 2
+ 5 / math.pi * X[..., 0]
- 6
)
t2 = 10 * (1 - 1 / (8 * math.pi)) * torch.cos(X[..., 0])
return t1 ** 2 + t2 + 10
[docs]class AugmentedHartmann(SyntheticTestFunction):
r"""Augmented Hartmann synthetic test function.
7-dimensional function (typically evaluated on `[0, 1]^7`), where the last
dimension is the fidelity parameter.
H(x) = -(ALPHA_1 - 0.1 * (1-x_7)) * exp(- sum_{j=1}^6 A_1j (x_j - P_1j) ** 2) -
sum_{i=2}^4 ALPHA_i exp( - sum_{j=1}^6 A_ij (x_j - P_ij) ** 2)
H has a unique global minimizer
`x = [0.20169, 0.150011, 0.476874, 0.275332, 0.311652, 0.6573, 1.0]`
with `H_min = -3.32237`
"""
dim = 7
_bounds = [(0.0, 1.0) for _ in range(7)]
_optimal_value = -3.32237
_optimizers = [(0.20169, 0.150011, 0.476874, 0.275332, 0.311652, 0.6573, 1.0)]
_check_grad_at_opt = False
def __init__(self, noise_std: Optional[float] = None, negate: bool = False) -> None:
super().__init__(noise_std=noise_std, negate=negate)
self.register_buffer("ALPHA", torch.tensor([1.0, 1.2, 3.0, 3.2]))
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],
]
self.register_buffer("A", torch.tensor(A, dtype=torch.float))
self.register_buffer("P", torch.tensor(P, dtype=torch.float))
[docs] def evaluate_true(self, X: Tensor) -> Tensor:
self.to(device=X.device, dtype=X.dtype)
inner_sum = torch.sum(
self.A * (X[..., :6].unsqueeze(1) - 0.0001 * self.P) ** 2, dim=2
)
alpha1 = self.ALPHA[0] - 0.1 * (1 - X[..., 6])
H = (
-torch.sum(self.ALPHA[1:] * torch.exp(-inner_sum)[..., 1:], dim=1)
- alpha1 * torch.exp(-inner_sum)[..., 0]
)
return H
[docs]class AugmentedRosenbrock(SyntheticTestFunction):
r"""Augmented Rosenbrock synthetic test function for multi-fidelity optimization.
d-dimensional function (usually evaluated on `[-5, 10]^(d-2) * [0, 1]^2`),
where the last two dimensions are the fidelity parameters:
f(x) = sum_{i=1}^{d-1} (100 (x_{i+1} - x_i^2 + 0.1 * (1-x_{d-1}))^2 +
(x_i - 1 + 0.1 * (1 - x_d)^2)^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=3, noise_std: Optional[float] = None, negate: bool = False
) -> None:
if dim < 3:
raise ValueError(
"AugmentedRosenbrock must be defined it at least 3 dimensions"
)
self.dim = dim
self._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)
[docs] def evaluate_true(self, X: Tensor) -> Tensor:
X_curr = X[..., :-3]
X_next = X[..., 1:-2]
t1 = 100 * (X_next - X_curr ** 2 + 0.1 * (1 - X[..., [-2]])) ** 2
t2 = (X_curr - 1 + 0.1 * (1 - X[..., [-1]]) ** 2) ** 2
return -(t1 + t2).sum(dim=-1)
