synthetic.py

#!/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

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


class SyntheticTestFunction(BaseTestProblem, ABC):
    r"""Base class for synthetic test functions."""

    _optimal_value: float | None = None
    _optimizers: list[tuple[float, ...]] | None = None
    num_objectives: int = 1

    def __init__(
        self,
        noise_std: None | float | list[float] = None,
        negate: bool = False,
        bounds: list[tuple[float, float]] | None = None,
        dtype: torch.dtype = torch.double,
    ) -> 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.
            dtype: The dtype that is used for the bounds of the function.
        """
        if bounds is not None:
            self._bounds = bounds
        super().__init__(noise_std=noise_std, negate=negate, dtype=dtype)
        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."
            )


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: float | None = None,
        negate: bool = False,
        bounds: list[tuple[float, float]] | None = None,
        dtype: torch.dtype = torch.double,
    ) -> 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, dtype=dtype)
        self.a = 20
        self.b = 0.2
        self.c = 2 * math.pi

    def evaluate_true(self, X: Tensor) -> Tensor:
        a, b, c = self.a, self.b, self.c
        part1 = -a * torch.exp(-torch.linalg.norm(X, dim=-1) * b / math.sqrt(self.dim))
        part2 = -(torch.exp(torch.mean(torch.cos(c * X), dim=-1)))
        return part1 + part2 + a + math.e


class Beale(SyntheticTestFunction):
    dim = 2
    _optimal_value = 0.0
    _bounds = [(-4.5, 4.5), (-4.5, 4.5)]
    _optimizers = [(3.0, 0.5)]

    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


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)]

    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


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

    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


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))]

    def evaluate_true(self, X: Tensor) -> Tensor:
        return torch.sum(0.1 * torch.cos(5 * math.pi * X) - X.pow(2), dim=-1)


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

    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


class DixonPrice(SyntheticTestFunction):
    _optimal_value = 0.0

    def __init__(
        self,
        dim=2,
        noise_std: float | None = None,
        negate: bool = False,
        bounds: list[tuple[float, float]] | None = None,
        dtype: torch.dtype = torch.double,
    ) -> None:
        r"""
        Args:
            dim: The (input) dimension.
            noise_std: Standard deviation of the observation noise.
            negate: If True, negate the function.
            dtype: The dtype that is used for the bounds of 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, dtype=dtype)

    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


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

    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


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: float | None = None,
        negate: bool = False,
        bounds: list[tuple[float, float]] | None = None,
        dtype: torch.dtype = torch.double,
    ) -> 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.
            dtype: The dtype that is used for the bounds of the function.
        """
        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, dtype=dtype)

    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


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: float | None = None,
        negate: bool = False,
        bounds: list[tuple[float, float]] | None = None,
        dtype: torch.dtype = torch.double,
    ) -> 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.
            dtype: The dtype that is used for the bounds of the function.
        """
        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, dtype=dtype)
        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

    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


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),
    ]

    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))
        )


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: float | None = None,
        negate: bool = False,
        bounds: list[tuple[float, float]] | None = None,
        dtype: torch.dtype = torch.double,
    ) -> 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.
            dtype: The dtype that is used for the bounds of the function.
        """
        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, dtype=dtype)

    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


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: float | None = None,
        negate: bool = False,
        bounds: list[tuple[float, float]] | None = None,
        dtype: torch.dtype = torch.double,
    ) -> 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, dtype=dtype)
        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

    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
            )
        )


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: float | None = None,
        negate: bool = False,
        bounds: list[tuple[float, float]] | None = None,
        dtype: torch.dtype = torch.double,
    ) -> 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.
            dtype: The dtype that is used for the bounds of the function.
        """
        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, dtype=dtype)

    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


class Rastrigin(SyntheticTestFunction):
    _optimal_value = 0.0

    def __init__(
        self,
        dim=2,
        noise_std: float | None = None,
        negate: bool = False,
        bounds: list[tuple[float, float]] | None = None,
        dtype: torch.dtype = torch.double,
    ) -> 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.
            dtype: The dtype that is used for the bounds of the function.
        """
        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, dtype=dtype)

    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
        )


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: float | None = None,
        negate: bool = False,
        bounds: list[tuple[float, float]] | None = None,
        dtype: torch.dtype = torch.double,
    ) -> 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.
            dtype: The dtype that is used for the bounds of the function.
        """
        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, dtype=dtype)

    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,
        )


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: float | None = None,
        negate: bool = False,
        bounds: list[tuple[float, float]] | None = None,
        dtype: torch.dtype = torch.double,
    ) -> 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.
            dtype: The dtype that is used for the bounds of the function.
        """
        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, dtype=dtype)

        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))

    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


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)]

    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)
        )


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: float | None = None,
        negate: bool = False,
        bounds: list[tuple[float, float]] | None = None,
        dtype: torch.dtype = torch.double,
    ) -> 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.
            dtype: The dtype that is used for the bounds of the function.
        """
        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, dtype=dtype)

    def evaluate_true(self, X: Tensor) -> Tensor:
        return 0.5 * (X.pow(4) - 16 * X.pow(2) + 5 * X).sum(dim=-1)


class ThreeHumpCamel(SyntheticTestFunction):
    dim = 2
    _bounds = [(-5.0, 5.0), (-5.0, 5.0)]
    _optimal_value = 0.0
    _optimizers = [(0.0, 0.0)]

    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 ----------- #


class ConstrainedSyntheticTestFunction(
    ConstrainedBaseTestProblem, SyntheticTestFunction, ABC
):
    r"""Base class for constrained synthetic test functions."""

    def __init__(
        self,
        noise_std: None | float | list[float] = None,
        constraint_noise_std: None | float | list[float] = None,
        negate: bool = False,
        bounds: list[tuple[float, float]] | None = None,
        dtype: torch.dtype = torch.double,
    ) -> 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.
            dtype: The dtype that is used for the bounds of the function.
        """
        SyntheticTestFunction.__init__(
            self, noise_std=noise_std, negate=negate, bounds=bounds, dtype=dtype
        )
        self.constraint_noise_std = self._validate_constraint_noise(
            constraint_noise_std
        )

    def _validate_constraint_noise(
        self, constraint_noise_std
    ) -> 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


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]_

    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)

    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)


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: None | float = None,
        constraint_noise_std: None | float | list[float] = None,
        negate: bool = False,
        bounds: list[tuple[float, float]] | None = None,
        dtype: torch.dtype = torch.double,
    ) -> 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.
            dtype: The dtype that is used for the bounds of the function.
        """
        Hartmann.__init__(
            self,
            dim=dim,
            noise_std=noise_std,
            negate=negate,
            bounds=bounds,
            dtype=dtype,
        )
        self.constraint_noise_std = self._validate_constraint_noise(
            constraint_noise_std
        )

    def evaluate_slack_true(self, X: Tensor) -> Tensor:
        return -X.norm(dim=-1, keepdim=True) + 1


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: None | float = None,
        constraint_noise_std: None | float | list[float] = None,
        negate: bool = False,
        bounds: list[tuple[float, float]] | None = None,
        dtype: torch.dtype = torch.double,
    ) -> 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.
            dtype: The dtype that is used for the bounds of the function.
        """
        Hartmann.__init__(
            self,
            dim=dim,
            noise_std=noise_std,
            negate=negate,
            bounds=bounds,
            dtype=dtype,
        )
        self.constraint_noise_std = self._validate_constraint_noise(
            constraint_noise_std
        )

    def evaluate_slack_true(self, X: Tensor) -> Tensor:
        return -X.pow(2).sum(dim=-1, keepdim=True) + 1


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)]
    _optimal_value = 6059.946341  # from [CoelloCoello2002constraint]

    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
        )

    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,
        )


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)]
    _optimal_value = 1.728226  # from [CoelloCoello2002constraint]

    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)

    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)


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)]
    _optimal_value = 0.012681  # from [CoelloCoello2002constraint]

    def evaluate_true(self, X: Tensor) -> Tensor:
        x1, x2, x3 = X.unbind(-1)
        return x1.pow(2) * x2 * (x3 + 2)

    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)


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),
    ]

    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))
        )

    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,
        )