botorch.test_functions

Abstract Test Function API

Base class for test functions for optimization benchmarks.

class botorch.test_functions.base.BaseTestProblem(noise_std=None, negate=False)[source]

Bases: torch.nn.modules.module.Module, abc.ABC

Base class for test functions.

Base constructor for test functions.

Parameters
  • noise_std (Optional[float]) – Standard deviation of the observation noise.

  • negate (bool) – If True, negate the function.

Return type

None

dim: int
forward(X, noise=True)[source]

Evaluate the function on a set of points.

Parameters
  • X (torch.Tensor) – A batch_shape x d-dim tensor of point(s) at which to evaluate the function.

  • noise (bool) – If True, add observation noise as specified by noise_std.

Returns

A batch_shape-dim tensor ouf function evaluations.

Return type

torch.Tensor

abstract evaluate_true(X)[source]

Evaluate the function (w/o observation noise) on a set of points.

Parameters

X (torch.Tensor) –

Return type

torch.Tensor

class botorch.test_functions.base.ConstrainedBaseTestProblem(noise_std=None, negate=False)[source]

Bases: botorch.test_functions.base.BaseTestProblem, abc.ABC

Base class for test functions with constraints.

In addition to one or more objectives, a problem may have a number of outcome constraints of the form c_i(x) >= 0 for i=1, …, n_c.

This base class provides common functionality for such problems.

Base constructor for test functions.

Parameters
  • noise_std (Optional[float]) – Standard deviation of the observation noise.

  • negate (bool) – If True, negate the function.

Return type

None

num_constraints: int
evaluate_slack(X, noise=True)[source]

Evaluate the constraint slack on a set of points.

Constraints i is assumed to be feasible at x if the associated slack c_i(x) is positive. Zero slack means that the constraint is active. Negative slack means that the constraint is violated.

Parameters
  • X (torch.Tensor) – A batch_shape x d-dim tensor of point(s) at which to evaluate the constraint slacks: c_1(X), …., c_{n_c}(X).

  • noise (bool) – If True, add observation noise to the slack as specified by noise_std.

Returns

A batch_shape x n_c-dim tensor of constraint slack (where positive slack

corresponds to the constraint being feasible).

Return type

torch.Tensor

is_feasible(X, noise=True)[source]

Evaluate whether the constraints are feasible on a set of points.

Parameters
  • X (torch.Tensor) – A batch_shape x d-dim tensor of point(s) at which to evaluate the constraints.

  • noise (bool) – If True, add observation noise as specified by noise_std.

Returns

A batch_shape-dim boolean tensor that is True iff all constraint

slacks (potentially including observation noise) are positive.

Return type

torch.Tensor

abstract evaluate_slack_true(X)[source]

Evaluate the constraint slack (w/o observation noise) on a set of points.

Parameters

X (torch.Tensor) – A batch_shape x d-dim tensor of point(s) at which to evaluate the constraint slacks: c_1(X), …., c_{n_c}(X).

Returns

A batch_shape x n_c-dim tensor of constraint slack (where positive slack

corresponds to the constraint being feasible).

Return type

torch.Tensor

class botorch.test_functions.base.MultiObjectiveTestProblem(noise_std=None, negate=False)[source]

Bases: botorch.test_functions.base.BaseTestProblem

Base class for test multi-objective test functions.

TODO: add a pareto distance function that returns the distance between a provided point and the closest point on the true pareto front.

Base constructor for multi-objective test functions.

Parameters
  • noise_std (Optional[float]) – Standard deviation of the observation noise.

  • negate (bool) – If True, negate the objectives.

Return type

None

num_objectives: int
property max_hv: float
gen_pareto_front(n)[source]

Generate n pareto optimal points.

Parameters

n (int) –

Return type

torch.Tensor

Synthetic Test Functions

Synthetic functions for optimization benchmarks. Reference: https://www.sfu.ca/~ssurjano/optimization.html

class botorch.test_functions.synthetic.SyntheticTestFunction(noise_std=None, negate=False)[source]

Bases: botorch.test_functions.base.BaseTestProblem

Base class for synthetic test functions.

Base constructor for synthetic test functions.

Parameters
  • noise_std (Optional[float]) – Standard deviation of the observation noise.

  • negate (bool) – If True, negate the function.

Return type

None

num_objectives: int = 1
property optimal_value: float

The global minimum (maximum if negate=True) of the function.

class botorch.test_functions.synthetic.Ackley(dim=2, noise_std=None, negate=False)[source]

Bases: botorch.test_functions.synthetic.SyntheticTestFunction

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.

Base constructor for synthetic test functions.

Parameters
  • noise_std (Optional[float]) – Standard deviation of the observation noise.

  • negate (bool) – If True, negate the function.

  • dim (int) –

Return type

None

evaluate_true(X)[source]

Evaluate the function (w/o observation noise) on a set of points.

Parameters

X (torch.Tensor) –

Return type

torch.Tensor

class botorch.test_functions.synthetic.Beale(noise_std=None, negate=False)[source]

Bases: botorch.test_functions.synthetic.SyntheticTestFunction

Base constructor for synthetic test functions.

Parameters
  • noise_std (Optional[float]) – Standard deviation of the observation noise.

  • negate (bool) – If True, negate the function.

Return type

None

dim: int = 2
evaluate_true(X)[source]

Evaluate the function (w/o observation noise) on a set of points.

Parameters

X (torch.Tensor) –

Return type

torch.Tensor

training: bool
class botorch.test_functions.synthetic.Branin(noise_std=None, negate=False)[source]

Bases: botorch.test_functions.synthetic.SyntheticTestFunction

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.

Base constructor for synthetic test functions.

Parameters
  • noise_std (Optional[float]) – Standard deviation of the observation noise.

  • negate (bool) – If True, negate the function.

Return type

None

dim: int = 2
evaluate_true(X)[source]

Evaluate the function (w/o observation noise) on a set of points.

Parameters

X (torch.Tensor) –

Return type

torch.Tensor

training: bool
class botorch.test_functions.synthetic.Bukin(noise_std=None, negate=False)[source]

Bases: botorch.test_functions.synthetic.SyntheticTestFunction

Base constructor for synthetic test functions.

Parameters
  • noise_std (Optional[float]) – Standard deviation of the observation noise.

  • negate (bool) – If True, negate the function.

Return type

None

dim: int = 2
evaluate_true(X)[source]

Evaluate the function (w/o observation noise) on a set of points.

Parameters

X (torch.Tensor) –

Return type

torch.Tensor

class botorch.test_functions.synthetic.Cosine8(noise_std=None, negate=False)[source]

Bases: botorch.test_functions.synthetic.SyntheticTestFunction

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

Base constructor for synthetic test functions.

Parameters
  • noise_std (Optional[float]) – Standard deviation of the observation noise.

  • negate (bool) – If True, negate the function.

Return type

None

dim: int = 8
evaluate_true(X)[source]

Evaluate the function (w/o observation noise) on a set of points.

Parameters

X (torch.Tensor) –

Return type

torch.Tensor

training: bool
class botorch.test_functions.synthetic.DropWave(noise_std=None, negate=False)[source]

Bases: botorch.test_functions.synthetic.SyntheticTestFunction

Base constructor for synthetic test functions.

Parameters
  • noise_std (Optional[float]) – Standard deviation of the observation noise.

  • negate (bool) – If True, negate the function.

Return type

None

dim: int = 2
evaluate_true(X)[source]

Evaluate the function (w/o observation noise) on a set of points.

Parameters

X (torch.Tensor) –

Return type

torch.Tensor

training: bool
class botorch.test_functions.synthetic.DixonPrice(dim=2, noise_std=None, negate=False)[source]

Bases: botorch.test_functions.synthetic.SyntheticTestFunction

Base constructor for synthetic test functions.

Parameters
  • noise_std (Optional[float]) – Standard deviation of the observation noise.

  • negate (bool) – If True, negate the function.

Return type

None

dim: int
evaluate_true(X)[source]

Evaluate the function (w/o observation noise) on a set of points.

Parameters

X (torch.Tensor) –

Return type

torch.Tensor

training: bool
class botorch.test_functions.synthetic.EggHolder(noise_std=None, negate=False)[source]

Bases: botorch.test_functions.synthetic.SyntheticTestFunction

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)|).

Base constructor for synthetic test functions.

Parameters
  • noise_std (Optional[float]) – Standard deviation of the observation noise.

  • negate (bool) – If True, negate the function.

Return type

None

dim: int = 2
evaluate_true(X)[source]

Evaluate the function (w/o observation noise) on a set of points.

Parameters

X (torch.Tensor) –

Return type

torch.Tensor

class botorch.test_functions.synthetic.Griewank(dim=2, noise_std=None, negate=False)[source]

Bases: botorch.test_functions.synthetic.SyntheticTestFunction

Base constructor for synthetic test functions.

Parameters
  • noise_std (Optional[float]) – Standard deviation of the observation noise.

  • negate (bool) – If True, negate the function.

Return type

None

dim: int
evaluate_true(X)[source]

Evaluate the function (w/o observation noise) on a set of points.

Parameters

X (torch.Tensor) –

Return type

torch.Tensor

training: bool
class botorch.test_functions.synthetic.Hartmann(dim=6, noise_std=None, negate=False)[source]

Bases: botorch.test_functions.synthetic.SyntheticTestFunction

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.

Base constructor for synthetic test functions.

Parameters
  • noise_std (Optional[float]) – Standard deviation of the observation noise.

  • negate (bool) – If True, negate the function.

Return type

None

dim: int
property optimal_value: float

The global minimum (maximum if negate=True) of the function.

property optimizers: torch.Tensor
evaluate_true(X)[source]

Evaluate the function (w/o observation noise) on a set of points.

Parameters

X (torch.Tensor) –

Return type

torch.Tensor

training: bool
class botorch.test_functions.synthetic.HolderTable(noise_std=None, negate=False)[source]

Bases: botorch.test_functions.synthetic.SyntheticTestFunction

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)

Base constructor for synthetic test functions.

Parameters
  • noise_std (Optional[float]) – Standard deviation of the observation noise.

  • negate (bool) – If True, negate the function.

Return type

None

dim: int = 2
evaluate_true(X)[source]

Evaluate the function (w/o observation noise) on a set of points.

Parameters

X (torch.Tensor) –

Return type

torch.Tensor

training: bool
class botorch.test_functions.synthetic.Levy(dim=2, noise_std=None, negate=False)[source]

Bases: botorch.test_functions.synthetic.SyntheticTestFunction

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.

Base constructor for synthetic test functions.

Parameters
  • noise_std (Optional[float]) – Standard deviation of the observation noise.

  • negate (bool) – If True, negate the function.

Return type

None

dim: int
evaluate_true(X)[source]

Evaluate the function (w/o observation noise) on a set of points.

Parameters

X (torch.Tensor) –

Return type

torch.Tensor

training: bool
class botorch.test_functions.synthetic.Michalewicz(dim=2, noise_std=None, negate=False)[source]

Bases: botorch.test_functions.synthetic.SyntheticTestFunction

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)

Base constructor for synthetic test functions.

Parameters
  • noise_std (Optional[float]) – Standard deviation of the observation noise.

  • negate (bool) – If True, negate the function.

Return type

None

dim: int
property optimizers: torch.Tensor
evaluate_true(X)[source]

Evaluate the function (w/o observation noise) on a set of points.

Parameters

X (torch.Tensor) –

Return type

torch.Tensor

training: bool
class botorch.test_functions.synthetic.Powell(dim=4, noise_std=None, negate=False)[source]

Bases: botorch.test_functions.synthetic.SyntheticTestFunction

Base constructor for synthetic test functions.

Parameters
  • noise_std (Optional[float]) – Standard deviation of the observation noise.

  • negate (bool) – If True, negate the function.

Return type

None

dim: int
evaluate_true(X)[source]

Evaluate the function (w/o observation noise) on a set of points.

Parameters

X (torch.Tensor) –

Return type

torch.Tensor

training: bool
class botorch.test_functions.synthetic.Rastrigin(dim=2, noise_std=None, negate=False)[source]

Bases: botorch.test_functions.synthetic.SyntheticTestFunction

Base constructor for synthetic test functions.

Parameters
  • noise_std (Optional[float]) – Standard deviation of the observation noise.

  • negate (bool) – If True, negate the function.

Return type

None

dim: int
evaluate_true(X)[source]

Evaluate the function (w/o observation noise) on a set of points.

Parameters

X (torch.Tensor) –

Return type

torch.Tensor

training: bool
class botorch.test_functions.synthetic.Rosenbrock(dim=2, noise_std=None, negate=False)[source]

Bases: botorch.test_functions.synthetic.SyntheticTestFunction

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.

Base constructor for synthetic test functions.

Parameters
  • noise_std (Optional[float]) – Standard deviation of the observation noise.

  • negate (bool) – If True, negate the function.

Return type

None

dim: int
evaluate_true(X)[source]

Evaluate the function (w/o observation noise) on a set of points.

Parameters

X (torch.Tensor) –

Return type

torch.Tensor

training: bool
class botorch.test_functions.synthetic.Shekel(m=10, noise_std=None, negate=False)[source]

Bases: botorch.test_functions.synthetic.SyntheticTestFunction

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.

Base constructor for synthetic test functions.

Parameters
  • noise_std (Optional[float]) – Standard deviation of the observation noise.

  • negate (bool) – If True, negate the function.

  • m (int) –

Return type

None

dim: int = 4
evaluate_true(X)[source]

Evaluate the function (w/o observation noise) on a set of points.

Parameters

X (torch.Tensor) –

Return type

torch.Tensor

training: bool
class botorch.test_functions.synthetic.SixHumpCamel(noise_std=None, negate=False)[source]

Bases: botorch.test_functions.synthetic.SyntheticTestFunction

Base constructor for synthetic test functions.

Parameters
  • noise_std (Optional[float]) – Standard deviation of the observation noise.

  • negate (bool) – If True, negate the function.

Return type

None

dim: int = 2
evaluate_true(X)[source]

Evaluate the function (w/o observation noise) on a set of points.

Parameters

X (torch.Tensor) –

Return type

torch.Tensor

training: bool
class botorch.test_functions.synthetic.StyblinskiTang(dim=2, noise_std=None, negate=False)[source]

Bases: botorch.test_functions.synthetic.SyntheticTestFunction

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

Base constructor for synthetic test functions.

Parameters
  • noise_std (Optional[float]) – Standard deviation of the observation noise.

  • negate (bool) – If True, negate the function.

Return type

None

dim: int
evaluate_true(X)[source]

Evaluate the function (w/o observation noise) on a set of points.

Parameters

X (torch.Tensor) –

Return type

torch.Tensor

training: bool
class botorch.test_functions.synthetic.ThreeHumpCamel(noise_std=None, negate=False)[source]

Bases: botorch.test_functions.synthetic.SyntheticTestFunction

Base constructor for synthetic test functions.

Parameters
  • noise_std (Optional[float]) – Standard deviation of the observation noise.

  • negate (bool) – If True, negate the function.

Return type

None

dim: int = 2
evaluate_true(X)[source]

Evaluate the function (w/o observation noise) on a set of points.

Parameters

X (torch.Tensor) –

Return type

torch.Tensor

training: bool

Multi-Fidelity Synthetic Test Functions

Synthetic functions for multi-fidelity optimization benchmarks.

class botorch.test_functions.multi_fidelity.AugmentedBranin(noise_std=None, negate=False)[source]

Bases: botorch.test_functions.synthetic.SyntheticTestFunction

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

Base constructor for synthetic test functions.

Parameters
  • noise_std (Optional[float]) – Standard deviation of the observation noise.

  • negate (bool) – If True, negate the function.

Return type

None

dim: int = 3
evaluate_true(X)[source]

Evaluate the function (w/o observation noise) on a set of points.

Parameters

X (torch.Tensor) –

Return type

torch.Tensor

training: bool
class botorch.test_functions.multi_fidelity.AugmentedHartmann(noise_std=None, negate=False)[source]

Bases: botorch.test_functions.synthetic.SyntheticTestFunction

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

Base constructor for synthetic test functions.

Parameters
  • noise_std (Optional[float]) – Standard deviation of the observation noise.

  • negate (bool) – If True, negate the function.

Return type

None

dim: int = 7
evaluate_true(X)[source]

Evaluate the function (w/o observation noise) on a set of points.

Parameters

X (torch.Tensor) –

Return type

torch.Tensor

training: bool
class botorch.test_functions.multi_fidelity.AugmentedRosenbrock(dim=3, noise_std=None, negate=False)[source]

Bases: botorch.test_functions.synthetic.SyntheticTestFunction

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.

Base constructor for synthetic test functions.

Parameters
  • noise_std (Optional[float]) – Standard deviation of the observation noise.

  • negate (bool) – If True, negate the function.

Return type

None

dim: int
evaluate_true(X)[source]

Evaluate the function (w/o observation noise) on a set of points.

Parameters

X (torch.Tensor) –

Return type

torch.Tensor

training: bool

Multi-Objective Synthetic Test Functions

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(1,2,3)

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.

Oszycka1995

A. Osyczka, S. Kundu. 1995. A new method to solve generalized multicriteria optimization problems using the simple genetic algorithm. In Structural Optimization 10. 94–99.

Tanabe2020(1,2,3)

Ryoji Tanabe, Hisao Ishibuchi, An easy-to-use real-world multi-objective optimization problem suite, Applied Soft Computing,Volume 89, 2020.

Yang2019a(1,2,3)

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.

class botorch.test_functions.multi_objective.BraninCurrin(noise_std=None, negate=False)[source]

Bases: botorch.test_functions.base.MultiObjectiveTestProblem

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

Constructor for Branin-Currin.

Parameters
  • noise_std (Optional[float]) – Standard deviation of the observation noise.

  • negate (bool) – If True, negate the objectives.

Return type

None

dim: int = 2
num_objectives: int = 2
evaluate_true(X)[source]

Evaluate the function (w/o observation noise) on a set of points.

Parameters

X (torch.Tensor) –

Return type

torch.Tensor

training: bool
class botorch.test_functions.multi_objective.DTLZ(dim, num_objectives=2, noise_std=None, negate=False)[source]

Bases: botorch.test_functions.base.MultiObjectiveTestProblem

Base class for DTLZ problems.

See [Deb2005dtlz] for more details on DTLZ.

Base constructor for multi-objective test functions.

Parameters
  • noise_std (Optional[float]) – Standard deviation of the observation noise.

  • negate (bool) – If True, negate the objectives.

  • dim (int) –

  • num_objectives (int) –

Return type

None

num_objectives: int
dim: int
training: bool
class botorch.test_functions.multi_objective.DTLZ1(dim, num_objectives=2, noise_std=None, negate=False)[source]

Bases: botorch.test_functions.multi_objective.DTLZ

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].

Base constructor for multi-objective test functions.

Parameters
  • noise_std (Optional[float]) – Standard deviation of the observation noise.

  • negate (bool) – If True, negate the objectives.

  • dim (int) –

  • num_objectives (int) –

Return type

None

evaluate_true(X)[source]

Evaluate the function (w/o observation noise) on a set of points.

Parameters

X (torch.Tensor) –

Return type

torch.Tensor

gen_pareto_front(n)[source]

Generate n pareto optimal points.

The pareto points randomly sampled from the hyperplane sum_i f(x_i) = 0.5.

Parameters

n (int) –

Return type

torch.Tensor

num_objectives: int
dim: int
training: bool
class botorch.test_functions.multi_objective.DTLZ2(dim, num_objectives=2, noise_std=None, negate=False)[source]

Bases: botorch.test_functions.multi_objective.DTLZ

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.

Base constructor for multi-objective test functions.

Parameters
  • noise_std (Optional[float]) – Standard deviation of the observation noise.

  • negate (bool) – If True, negate the objectives.

  • dim (int) –

  • num_objectives (int) –

Return type

None

evaluate_true(X)[source]

Evaluate the function (w/o observation noise) on a set of points.

Parameters

X (torch.Tensor) –

Return type

torch.Tensor

gen_pareto_front(n)[source]

Generate n pareto optimal points.

The pareto points are randomly sampled from the hypersphere’s positive section.

Parameters

n (int) –

Return type

torch.Tensor

num_objectives: int
dim: int
training: bool
class botorch.test_functions.multi_objective.VehicleSafety(noise_std=None, negate=False)[source]

Bases: botorch.test_functions.base.MultiObjectiveTestProblem

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].

Base constructor for multi-objective test functions.

Parameters
  • noise_std (Optional[float]) – Standard deviation of the observation noise.

  • negate (bool) – If True, negate the objectives.

Return type

None

dim: int = 5
num_objectives: int = 3
evaluate_true(X)[source]

Evaluate the function (w/o observation noise) on a set of points.

Parameters

X (torch.Tensor) –

Return type

torch.Tensor

training: bool
class botorch.test_functions.multi_objective.ZDT(dim, num_objectives=2, noise_std=None, negate=False)[source]

Bases: botorch.test_functions.base.MultiObjectiveTestProblem

Base class for ZDT problems.

See [Zitzler2000] for more details on ZDT.

Base constructor for multi-objective test functions.

Parameters
  • noise_std (Optional[float]) – Standard deviation of the observation noise.

  • negate (bool) – If True, negate the objectives.

  • dim (int) –

  • num_objectives (int) –

Return type

None

num_objectives: int
dim: int
training: bool
class botorch.test_functions.multi_objective.ZDT1(dim, num_objectives=2, noise_std=None, negate=False)[source]

Bases: botorch.test_functions.multi_objective.ZDT

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.

Base constructor for multi-objective test functions.

Parameters
  • noise_std (Optional[float]) – Standard deviation of the observation noise.

  • negate (bool) – If True, negate the objectives.

  • dim (int) –

  • num_objectives (int) –

Return type

None

evaluate_true(X)[source]

Evaluate the function (w/o observation noise) on a set of points.

Parameters

X (torch.Tensor) –

Return type

torch.Tensor

gen_pareto_front(n)[source]

Generate n pareto optimal points.

Parameters

n (int) –

Return type

torch.Tensor

num_objectives: int
dim: int
training: bool
class botorch.test_functions.multi_objective.ZDT2(dim, num_objectives=2, noise_std=None, negate=False)[source]

Bases: botorch.test_functions.multi_objective.ZDT

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.

Base constructor for multi-objective test functions.

Parameters
  • noise_std (Optional[float]) – Standard deviation of the observation noise.

  • negate (bool) – If True, negate the objectives.

  • dim (int) –

  • num_objectives (int) –

Return type

None

evaluate_true(X)[source]

Evaluate the function (w/o observation noise) on a set of points.

Parameters

X (torch.Tensor) –

Return type

torch.Tensor

gen_pareto_front(n)[source]

Generate n pareto optimal points.

Parameters

n (int) –

Return type

torch.Tensor

num_objectives: int
dim: int
training: bool
class botorch.test_functions.multi_objective.ZDT3(dim, num_objectives=2, noise_std=None, negate=False)[source]

Bases: botorch.test_functions.multi_objective.ZDT

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.

Base constructor for multi-objective test functions.

Parameters
  • noise_std (Optional[float]) – Standard deviation of the observation noise.

  • negate (bool) – If True, negate the objectives.

  • dim (int) –

  • num_objectives (int) –

Return type

None

evaluate_true(X)[source]

Evaluate the function (w/o observation noise) on a set of points.

Parameters

X (torch.Tensor) –

Return type

torch.Tensor

gen_pareto_front(n)[source]

Generate n pareto optimal points.

Parameters

n (int) –

Return type

torch.Tensor

num_objectives: int
dim: int
training: bool
class botorch.test_functions.multi_objective.BNH(noise_std=None, negate=False)[source]

Bases: botorch.test_functions.base.MultiObjectiveTestProblem, botorch.test_functions.base.ConstrainedBaseTestProblem

The constrained BNH problem.

See [GarridoMerchan2020] for more details on this problem. Note that this is a minimization problem.

Base constructor for multi-objective test functions.

Parameters
  • noise_std (Optional[float]) – Standard deviation of the observation noise.

  • negate (bool) – If True, negate the objectives.

Return type

None

dim: int = 2
num_objectives: int = 2
num_constraints: int = 2
evaluate_true(X)[source]

Evaluate the function (w/o observation noise) on a set of points.

Parameters

X (torch.Tensor) –

Return type

torch.Tensor

evaluate_slack_true(X)[source]

Evaluate the constraint slack (w/o observation noise) on a set of points.

Parameters

X (torch.Tensor) – A batch_shape x d-dim tensor of point(s) at which to evaluate the constraint slacks: c_1(X), …., c_{n_c}(X).

Returns

A batch_shape x n_c-dim tensor of constraint slack (where positive slack

corresponds to the constraint being feasible).

Return type

torch.Tensor

training: bool
class botorch.test_functions.multi_objective.SRN(noise_std=None, negate=False)[source]

Bases: botorch.test_functions.base.MultiObjectiveTestProblem, botorch.test_functions.base.ConstrainedBaseTestProblem

The constrained SRN problem.

See [GarridoMerchan2020] for more details on this problem. Note that this is a minimization problem.

Base constructor for multi-objective test functions.

Parameters
  • noise_std (Optional[float]) – Standard deviation of the observation noise.

  • negate (bool) – If True, negate the objectives.

Return type

None

dim: int = 2
num_objectives: int = 2
num_constraints: int = 2
evaluate_true(X)[source]

Evaluate the function (w/o observation noise) on a set of points.

Parameters

X (torch.Tensor) –

Return type

torch.Tensor

evaluate_slack_true(X)[source]

Evaluate the constraint slack (w/o observation noise) on a set of points.

Parameters

X (torch.Tensor) – A batch_shape x d-dim tensor of point(s) at which to evaluate the constraint slacks: c_1(X), …., c_{n_c}(X).

Returns

A batch_shape x n_c-dim tensor of constraint slack (where positive slack

corresponds to the constraint being feasible).

Return type

torch.Tensor

training: bool
class botorch.test_functions.multi_objective.CONSTR(noise_std=None, negate=False)[source]

Bases: botorch.test_functions.base.MultiObjectiveTestProblem, botorch.test_functions.base.ConstrainedBaseTestProblem

The constrained CONSTR problem.

See [GarridoMerchan2020] for more details on this problem. Note that this is a minimization problem.

Base constructor for multi-objective test functions.

Parameters
  • noise_std (Optional[float]) – Standard deviation of the observation noise.

  • negate (bool) – If True, negate the objectives.

Return type

None

dim: int = 2
num_objectives: int = 2
num_constraints: int = 2
evaluate_true(X)[source]

Evaluate the function (w/o observation noise) on a set of points.

Parameters

X (torch.Tensor) –

Return type

torch.Tensor

evaluate_slack_true(X)[source]

Evaluate the constraint slack (w/o observation noise) on a set of points.

Parameters

X (torch.Tensor) – A batch_shape x d-dim tensor of point(s) at which to evaluate the constraint slacks: c_1(X), …., c_{n_c}(X).

Returns

A batch_shape x n_c-dim tensor of constraint slack (where positive slack

corresponds to the constraint being feasible).

Return type

torch.Tensor

training: bool
class botorch.test_functions.multi_objective.ConstrainedBraninCurrin(noise_std=None, negate=False)[source]

Bases: botorch.test_functions.multi_objective.BraninCurrin, botorch.test_functions.base.ConstrainedBaseTestProblem

Constrained Branin Currin Function.

This uses the disk constraint from [Gelbart2014].

Constructor for Branin-Currin.

Parameters
  • noise_std (Optional[float]) – Standard deviation of the observation noise.

  • negate (bool) – If True, negate the objectives.

Return type

None

dim: int = 2
num_objectives: int = 2
num_constraints: int = 1
evaluate_slack_true(X)[source]

Evaluate the constraint slack (w/o observation noise) on a set of points.

Parameters

X (torch.Tensor) – A batch_shape x d-dim tensor of point(s) at which to evaluate the constraint slacks: c_1(X), …., c_{n_c}(X).

Returns

A batch_shape x n_c-dim tensor of constraint slack (where positive slack

corresponds to the constraint being feasible).

Return type

torch.Tensor

training: bool
class botorch.test_functions.multi_objective.C2DTLZ2(dim, num_objectives=2, noise_std=None, negate=False)[source]

Bases: botorch.test_functions.multi_objective.DTLZ2, botorch.test_functions.base.ConstrainedBaseTestProblem

Base constructor for multi-objective test functions.

Parameters
  • noise_std (Optional[float]) – Standard deviation of the observation noise.

  • negate (bool) – If True, negate the objectives.

  • dim (int) –

  • num_objectives (int) –

Return type

None

num_constraints: int = 1
evaluate_slack_true(X)[source]

Evaluate the constraint slack (w/o observation noise) on a set of points.

Parameters

X (torch.Tensor) – A batch_shape x d-dim tensor of point(s) at which to evaluate the constraint slacks: c_1(X), …., c_{n_c}(X).

Returns

A batch_shape x n_c-dim tensor of constraint slack (where positive slack

corresponds to the constraint being feasible).

Return type

torch.Tensor

num_objectives: int
dim: int
training: bool
class botorch.test_functions.multi_objective.OSY(noise_std=None, negate=False)[source]

Bases: botorch.test_functions.base.MultiObjectiveTestProblem, botorch.test_functions.base.ConstrainedBaseTestProblem

The OSY test problem from [Oszycka1995]. Implementation from https://github.com/msu-coinlab/pymoo/blob/master/pymoo/problems/multi/osy.py Note that this implementation assumes minimization, so please choose negate=True.

Base constructor for multi-objective test functions.

Parameters
  • noise_std (Optional[float]) – Standard deviation of the observation noise.

  • negate (bool) – If True, negate the objectives.

Return type

None

dim: int = 6
num_constraints: int = 6
num_objectives: int = 2
evaluate_true(X)[source]

Evaluate the function (w/o observation noise) on a set of points.

Parameters

X (torch.Tensor) –

Return type

torch.Tensor

evaluate_slack_true(X)[source]

Evaluate the constraint slack (w/o observation noise) on a set of points.

Parameters

X (torch.Tensor) – A batch_shape x d-dim tensor of point(s) at which to evaluate the constraint slacks: c_1(X), …., c_{n_c}(X).

Returns

A batch_shape x n_c-dim tensor of constraint slack (where positive slack

corresponds to the constraint being feasible).

Return type

torch.Tensor

training: bool