import torch
from nubo.test_functions import TestFunction
from torch import Tensor
from typing import Optional
[docs]
class Levy(TestFunction):
r"""
:math:`d`-dimensional Levy function.
The Levy function has many local minima and one global minimum
:math:`f(\boldsymbol x^*) = 0` at :math:`\boldsymbol x^* = (1, ..., 1)`. It
is usually evaluated on the hypercube :math:`\boldsymbol x \in [-10, 10]^d`.
.. math::
f(\boldsymbol 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 :math:`w_i = 1 + \frac{x_i - 1}{4}`, for all :math:`i = 1, ..., d`.
Attributes
----------
dims : ``int``
Number of input dimensions.
noise_std : ``float``
Standard deviation of Gaussian noise.
minimise : ``bool``
Minimisation problem if true, maximisation problem if false.
bounds : ``torch.Tensor``
(size 2 x `dims`) Bounds of input space.
optimum : ``dict``
Contains inputs and output of global maximum.
"""
def __init__(self,
dims: int,
noise_std: Optional[float]=0.0,
minimise: Optional[bool]=True) -> None:
"""
Parameters
----------
dims : ``int``
Number of input dimensions.
noise_std : ``float``, optional
Standard deviation of Gaussian noise, default is 0.0.
minimise : ``bool``, optional
Minimisation problem if true (default), maximisation problem if
false.
"""
self.dims = dims
self.bounds = Tensor([[-10.0, ] * dims, [10.0, ] * dims])
self.optimum = {"inputs": Tensor([[1.0, ] * dims]), "ouput": Tensor([[0.0]])}
self.noise_std = noise_std
self.minimise = minimise
[docs]
def eval(self, x: Tensor) -> Tensor:
"""
Compute output of Levy function for some test points `x`.
Parameters
----------
x : ``torch.Tensor``
(size n x `dims`) Test points.
"""
# compute output
w = 1.0 + (x - 1.0)/4.0
term_1 = torch.sin(torch.pi * w[:, 0])**2
term_2 = torch.sum((w[:, :-1] - 1.0)**2 * (1.0 + 10.0 * torch.sin(torch.pi * w[:, :-1] + 1.0)**2), dim=-1)
term_3 = (w[:, -1] - 1.0)**2 * (1.0 + torch.sin(2.0 * torch.pi * w[:, -1])**2)
y = term_1 + term_2 + term_3
# turn into maximisation problem
if not self.minimise:
y = -y
# add noise
noise = torch.normal(mean=0, std=self.noise_std, size=y.size())
f = y + noise
return f