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: Module, ABC

Base class for test functions.

Base constructor for test functions.

Parameters:

noise_std (Union[None, float, List[float]]) – Standard deviation of the observation noise. If a list is provided, specifies separate noise standard deviations for each objective in a multiobjective problem.
negate (bool) – If True, negate the function.

dim: int¶

forward(X, noise=True)[source]¶

Evaluate the function on a set of points.

Parameters:

X (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:

Tensor

abstract evaluate_true(X)[source]¶

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

Parameters:: X (Tensor) –
Return type:: Tensor

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

Bases: BaseTestProblem, 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 (Union[None, float, List[float]]) – Standard deviation of the observation noise. If a list is provided, specifies separate noise standard deviations for each objective in a multiobjective problem.
negate (bool) – If True, negate the function.

num_constraints: int¶

constraint_noise_std: None | float | List[float] = None¶

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 (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:

Tensor

is_feasible(X, noise=True)[source]¶

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

Parameters:

X (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:

Tensor

abstract evaluate_slack_true(X)[source]¶

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

Parameters:

X (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:

Tensor

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

Bases: 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 (Union[None, float, List[float]]) – Standard deviation of the observation noise. If a list is provided, specifies separate noise standard deviations for each objective.
negate (bool) – If True, negate the objectives.

num_objectives: int¶

property max_hv: float¶

gen_pareto_front(n)[source]¶

Generate n pareto optimal points.

Parameters:: n (int) –
Return type:: Tensor

Synthetic Test Functions¶

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] (1,2)

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.

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

Bases: BaseTestProblem

Base class for synthetic test functions.

Parameters:

noise_std (Union[None, float, List[float]]) – Standard deviation of the observation noise. If a list is provided, specifies separate noise standard deviations for each objective in a multiobjective problem.
negate (bool) – If True, negate the function.
bounds (Optional[List[Tuple[float, float]]]) – Custom bounds for the function specified as (lower, upper) pairs.

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, bounds=None)[source]¶

Bases: 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.

Parameters:

dim (int) – The (input) dimension.
noise_std (Optional[float]) – Standard deviation of the observation noise.
negate (bool) – If True, negate the function.
bounds (Optional[List[Tuple[float, float]]]) – Custom bounds for the function specified as (lower, upper) pairs.

evaluate_true(X)[source]¶

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

Parameters:: X (Tensor) –
Return type:: Tensor

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

Bases: SyntheticTestFunction

Parameters:

noise_std (Union[None, float, List[float]]) – Standard deviation of the observation noise. If a list is provided, specifies separate noise standard deviations for each objective in a multiobjective problem.
negate (bool) – If True, negate the function.
bounds (Optional[List[Tuple[float, float]]]) – Custom bounds for the function specified as (lower, upper) pairs.

dim: int = 2¶

evaluate_true(X)[source]¶

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

Parameters:: X (Tensor) –
Return type:: Tensor

training: bool¶

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

Bases: 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.

Parameters:

noise_std (Union[None, float, List[float]]) – Standard deviation of the observation noise. If a list is provided, specifies separate noise standard deviations for each objective in a multiobjective problem.
negate (bool) – If True, negate the function.
bounds (Optional[List[Tuple[float, float]]]) – Custom bounds for the function specified as (lower, upper) pairs.

dim: int = 2¶

evaluate_true(X)[source]¶

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

Parameters:: X (Tensor) –
Return type:: Tensor

training: bool¶

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

Bases: SyntheticTestFunction

Parameters:

noise_std (Union[None, float, List[float]]) – Standard deviation of the observation noise. If a list is provided, specifies separate noise standard deviations for each objective in a multiobjective problem.
negate (bool) – If True, negate the function.
bounds (Optional[List[Tuple[float, float]]]) – Custom bounds for the function specified as (lower, upper) pairs.

dim: int = 2¶

evaluate_true(X)[source]¶

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

Parameters:: X (Tensor) –
Return type:: Tensor

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

Bases: 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

Parameters:

noise_std (Union[None, float, List[float]]) – Standard deviation of the observation noise. If a list is provided, specifies separate noise standard deviations for each objective in a multiobjective problem.
negate (bool) – If True, negate the function.
bounds (Optional[List[Tuple[float, float]]]) – Custom bounds for the function specified as (lower, upper) pairs.

dim: int = 8¶

evaluate_true(X)[source]¶

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

Parameters:: X (Tensor) –
Return type:: Tensor

training: bool¶

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

Bases: SyntheticTestFunction

Parameters:

noise_std (Union[None, float, List[float]]) – Standard deviation of the observation noise. If a list is provided, specifies separate noise standard deviations for each objective in a multiobjective problem.
negate (bool) – If True, negate the function.
bounds (Optional[List[Tuple[float, float]]]) – Custom bounds for the function specified as (lower, upper) pairs.

dim: int = 2¶

evaluate_true(X)[source]¶

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

Parameters:: X (Tensor) –
Return type:: Tensor

training: bool¶

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

Bases: SyntheticTestFunction

Parameters:

dim – The (input) dimension.
noise_std (Optional[float]) – Standard deviation of the observation noise.
negate (bool) – If True, negate the function.
bounds (Optional[List[Tuple[float, float]]]) –

dim: int¶

evaluate_true(X)[source]¶

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

Parameters:: X (Tensor) –
Return type:: Tensor

training: bool¶

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

Bases: 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)|).

Parameters:

noise_std (Union[None, float, List[float]]) – Standard deviation of the observation noise. If a list is provided, specifies separate noise standard deviations for each objective in a multiobjective problem.
negate (bool) – If True, negate the function.
bounds (Optional[List[Tuple[float, float]]]) – Custom bounds for the function specified as (lower, upper) pairs.

dim: int = 2¶

evaluate_true(X)[source]¶

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

Parameters:: X (Tensor) –
Return type:: Tensor

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

Bases: SyntheticTestFunction

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.

Parameters:

dim – The (input) dimension.
noise_std (Optional[float]) – Standard deviation of the observation noise.
negate (bool) – If True, negate the function.
bounds (Optional[List[Tuple[float, float]]]) – Custom bounds for the function specified as (lower, upper) pairs.

dim: int¶

evaluate_true(X)[source]¶

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

Parameters:: X (Tensor) –
Return type:: Tensor

training: bool¶

class botorch.test_functions.synthetic.Hartmann(dim=6, noise_std=None, negate=False, bounds=None)[source]¶

Bases: 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.

Parameters:

dim – The (input) dimension.
noise_std (Optional[float]) – Standard deviation of the observation noise.
negate (bool) – If True, negate the function.
bounds (Optional[List[Tuple[float, float]]]) – Custom bounds for the function specified as (lower, upper) pairs.

dim: int¶

property optimal_value: float¶: The global minimum (maximum if negate=True) of the function.

property optimizers: Tensor¶

evaluate_true(X)[source]¶

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

Parameters:: X (Tensor) –
Return type:: Tensor

training: bool¶

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

Bases: 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)

Parameters:

noise_std (Union[None, float, List[float]]) – Standard deviation of the observation noise. If a list is provided, specifies separate noise standard deviations for each objective in a multiobjective problem.
negate (bool) – If True, negate the function.
bounds (Optional[List[Tuple[float, float]]]) – Custom bounds for the function specified as (lower, upper) pairs.

dim: int = 2¶

evaluate_true(X)[source]¶

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

Parameters:: X (Tensor) –
Return type:: Tensor

training: bool¶

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

Bases: 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.

Parameters:

dim – The (input) dimension.
noise_std (Optional[float]) – Standard deviation of the observation noise.
negate (bool) – If True, negate the function.
bounds (Optional[List[Tuple[float, float]]]) – Custom bounds for the function specified as (lower, upper) pairs.

dim: int¶

evaluate_true(X)[source]¶

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

Parameters:: X (Tensor) –
Return type:: Tensor

training: bool¶

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

Bases: 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)

Parameters:

dim – The (input) dimension.
noise_std (Optional[float]) – Standard deviation of the observation noise.
negate (bool) – If True, negate the function.
bounds (Optional[List[Tuple[float, float]]]) – Custom bounds for the function specified as (lower, upper) pairs.

dim: int¶

property optimizers: Tensor¶

evaluate_true(X)[source]¶

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

Parameters:: X (Tensor) –
Return type:: Tensor

training: bool¶

class botorch.test_functions.synthetic.Powell(dim=4, noise_std=None, negate=False, bounds=None)[source]¶

Bases: SyntheticTestFunction

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.

Parameters:

dim – The (input) dimension.
noise_std (Optional[float]) – Standard deviation of the observation noise.
negate (bool) – If True, negate the function.
bounds (Optional[List[Tuple[float, float]]]) – Custom bounds for the function specified as (lower, upper) pairs.

dim: int¶

evaluate_true(X)[source]¶

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

Parameters:: X (Tensor) –
Return type:: Tensor

training: bool¶

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

Bases: SyntheticTestFunction

Parameters:

dim – The (input) dimension.
noise_std (Optional[float]) – Standard deviation of the observation noise.
negate (bool) – If True, negate the function.
bounds (Optional[List[Tuple[float, float]]]) – Custom bounds for the function specified as (lower, upper) pairs.

dim: int¶

evaluate_true(X)[source]¶

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

Parameters:: X (Tensor) –
Return type:: Tensor

training: bool¶

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

Bases: 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.

Parameters:

dim – The (input) dimension.
noise_std (Optional[float]) – Standard deviation of the observation noise.
negate (bool) – If True, negate the function.
bounds (Optional[List[Tuple[float, float]]]) – Custom bounds for the function specified as (lower, upper) pairs.

dim: int¶

evaluate_true(X)[source]¶

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

Parameters:: X (Tensor) –
Return type:: Tensor

training: bool¶

class botorch.test_functions.synthetic.Shekel(m=10, noise_std=None, negate=False, bounds=None)[source]¶

Bases: 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.

Parameters:

m (int) – Defaults to 10.
noise_std (Optional[float]) – Standard deviation of the observation noise.
negate (bool) – If True, negate the function.
bounds (Optional[List[Tuple[float, float]]]) – Custom bounds for the function specified as (lower, upper) pairs.

dim: int = 4¶

evaluate_true(X)[source]¶

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

Parameters:: X (Tensor) –
Return type:: Tensor

training: bool¶

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

Bases: SyntheticTestFunction

Parameters:

noise_std (Union[None, float, List[float]]) – Standard deviation of the observation noise. If a list is provided, specifies separate noise standard deviations for each objective in a multiobjective problem.
negate (bool) – If True, negate the function.
bounds (Optional[List[Tuple[float, float]]]) – Custom bounds for the function specified as (lower, upper) pairs.

dim: int = 2¶

evaluate_true(X)[source]¶

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

Parameters:: X (Tensor) –
Return type:: Tensor

training: bool¶

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

Bases: 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

Parameters:

dim – The (input) dimension.
noise_std (Optional[float]) – Standard deviation of the observation noise.
negate (bool) – If True, negate the function.
bounds (Optional[List[Tuple[float, float]]]) – Custom bounds for the function specified as (lower, upper) pairs.

dim: int¶

evaluate_true(X)[source]¶

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

Parameters:: X (Tensor) –
Return type:: Tensor

training: bool¶

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

Bases: SyntheticTestFunction

Parameters:

noise_std (Union[None, float, List[float]]) – Standard deviation of the observation noise. If a list is provided, specifies separate noise standard deviations for each objective in a multiobjective problem.
negate (bool) – If True, negate the function.
bounds (Optional[List[Tuple[float, float]]]) – Custom bounds for the function specified as (lower, upper) pairs.

dim: int = 2¶

evaluate_true(X)[source]¶

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

Parameters:: X (Tensor) –
Return type:: Tensor

training: bool¶

class botorch.test_functions.synthetic.ConstrainedSyntheticTestFunction(noise_std=None, constraint_noise_std=None, negate=False, bounds=None)[source]¶

Bases: ConstrainedBaseTestProblem, SyntheticTestFunction

Base class for constrained synthetic test functions.

Parameters:

noise_std (Union[None, float, List[float]]) – 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 (Union[None, float, List[float]]) – Standard deviation of the constraint noise. If a list is provided, specifies separate noise standard deviations for each constraint.
negate (bool) – If True, negate the function.
bounds (Optional[List[Tuple[float, float]]]) – Custom bounds for the function specified as (lower, upper) pairs.

num_constraints: int¶

dim: int¶

training: bool¶

class botorch.test_functions.synthetic.ConstrainedGramacy(noise_std=None, constraint_noise_std=None, negate=False, bounds=None)[source]¶

Bases: ConstrainedSyntheticTestFunction

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.

Parameters:

noise_std (Union[None, float, List[float]]) – 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 (Union[None, float, List[float]]) – Standard deviation of the constraint noise. If a list is provided, specifies separate noise standard deviations for each constraint.
negate (bool) – If True, negate the function.
bounds (Optional[List[Tuple[float, float]]]) – Custom bounds for the function specified as (lower, upper) pairs.

num_objectives: int = 1¶

num_constraints: int = 2¶

dim: int = 2¶

evaluate_true(X)[source]¶

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

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

evaluate_slack_true(X)[source]¶

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

Parameters:

X (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:

Tensor

training: bool¶

class botorch.test_functions.synthetic.ConstrainedHartmann(dim=6, noise_std=None, constraint_noise_std=None, negate=False, bounds=None)[source]¶

Bases: Hartmann, ConstrainedSyntheticTestFunction

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].

Parameters:

dim (int) – The (input) dimension.
noise_std (Union[None, float]) – Standard deviation of the observation noise.
constraint_noise_std (Union[None, float, List[float]]) – Standard deviation of the constraint noise. If a list is provided, specifies separate noise standard deviations for each constraint.
negate (bool) – If True, negate the function.
bounds (Optional[List[Tuple[float, float]]]) – Custom bounds for the function specified as (lower, upper) pairs.

num_constraints: int = 1¶

evaluate_slack_true(X)[source]¶

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

Parameters:

X (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:

Tensor

dim: int¶

training: bool¶

class botorch.test_functions.synthetic.ConstrainedHartmannSmooth(dim=6, noise_std=None, constraint_noise_std=None, negate=False, bounds=None)[source]¶

Bases: Hartmann, ConstrainedSyntheticTestFunction

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.

Parameters:

dim (int) – The (input) dimension.
noise_std (Union[None, float]) – Standard deviation of the observation noise.
constraint_noise_std (Union[None, float, List[float]]) – Standard deviation of the constraint noise. If a list is provided, specifies separate noise standard deviations for each constraint.
negate (bool) – If True, negate the function.
bounds (Optional[List[Tuple[float, float]]]) – Custom bounds for the function specified as (lower, upper) pairs.

num_constraints: int = 1¶

evaluate_slack_true(X)[source]¶

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

Parameters:

X (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:

Tensor

dim: int¶

training: bool¶

class botorch.test_functions.synthetic.PressureVessel(noise_std=None, constraint_noise_std=None, negate=False, bounds=None)[source]¶

Bases: ConstrainedSyntheticTestFunction

Pressure vessel design problem with constraints.

The four-dimensional pressure vessel design problem with four black-box constraints from [CoelloCoello2002constraint].

Parameters:

noise_std (Union[None, float, List[float]]) – 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 (Union[None, float, List[float]]) – Standard deviation of the constraint noise. If a list is provided, specifies separate noise standard deviations for each constraint.
negate (bool) – If True, negate the function.
bounds (Optional[List[Tuple[float, float]]]) – Custom bounds for the function specified as (lower, upper) pairs.

dim: int = 4¶

num_constraints: int = 4¶

evaluate_true(X)[source]¶

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

Parameters:: X (Tensor) –
Return type:: Tensor

evaluate_slack_true(X)[source]¶

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

Parameters:

X (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:

Tensor

training: bool¶

class botorch.test_functions.synthetic.WeldedBeamSO(noise_std=None, constraint_noise_std=None, negate=False, bounds=None)[source]¶

Bases: ConstrainedSyntheticTestFunction

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.

Parameters:

noise_std (Union[None, float, List[float]]) – 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 (Union[None, float, List[float]]) – Standard deviation of the constraint noise. If a list is provided, specifies separate noise standard deviations for each constraint.
negate (bool) – If True, negate the function.
bounds (Optional[List[Tuple[float, float]]]) – Custom bounds for the function specified as (lower, upper) pairs.

dim: int = 4¶

num_constraints: int = 6¶

evaluate_true(X)[source]¶

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

Parameters:: X (Tensor) –
Return type:: Tensor

evaluate_slack_true(X)[source]¶

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

Parameters:

X (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:

Tensor

training: bool¶

class botorch.test_functions.synthetic.TensionCompressionString(noise_std=None, constraint_noise_std=None, negate=False, bounds=None)[source]¶

Bases: ConstrainedSyntheticTestFunction

Tension compression string optimization problem with constraints.

The three-dimensional tension compression string optimization problem with four black-box constraints from [Hedar2006derivfree].

Parameters:

noise_std (Union[None, float, List[float]]) – 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 (Union[None, float, List[float]]) – Standard deviation of the constraint noise. If a list is provided, specifies separate noise standard deviations for each constraint.
negate (bool) – If True, negate the function.
bounds (Optional[List[Tuple[float, float]]]) – Custom bounds for the function specified as (lower, upper) pairs.

dim: int = 3¶

num_constraints: int = 4¶

evaluate_true(X)[source]¶

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

Parameters:: X (Tensor) –
Return type:: Tensor

evaluate_slack_true(X)[source]¶

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

Parameters:

X (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:

Tensor

training: bool¶

class botorch.test_functions.synthetic.SpeedReducer(noise_std=None, constraint_noise_std=None, negate=False, bounds=None)[source]¶

Bases: ConstrainedSyntheticTestFunction

Speed Reducer design problem with constraints.

The seven-dimensional speed reducer design problem with eleven black-box constraints from [Lemonge2010constrained].

Parameters:

noise_std (Union[None, float, List[float]]) – 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 (Union[None, float, List[float]]) – Standard deviation of the constraint noise. If a list is provided, specifies separate noise standard deviations for each constraint.
negate (bool) – If True, negate the function.
bounds (Optional[List[Tuple[float, float]]]) – Custom bounds for the function specified as (lower, upper) pairs.

dim: int = 7¶

num_constraints: int = 11¶

evaluate_true(X)[source]¶

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

Parameters:: X (Tensor) –
Return type:: Tensor

evaluate_slack_true(X)[source]¶

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

Parameters:

X (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:

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, bounds=None)[source]¶

Bases: 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

Parameters:

noise_std (Union[None, float, List[float]]) – Standard deviation of the observation noise. If a list is provided, specifies separate noise standard deviations for each objective in a multiobjective problem.
negate (bool) – If True, negate the function.
bounds (Optional[List[Tuple[float, float]]]) – Custom bounds for the function specified as (lower, upper) pairs.

dim: int = 3¶

evaluate_true(X)[source]¶

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

Parameters:: X (Tensor) –
Return type:: Tensor

training: bool¶

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

Bases: 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

Parameters:

noise_std (Optional[float]) – Standard deviation of the observation noise.
negate (bool) – If True, negate the function.

dim: int = 7¶

evaluate_true(X)[source]¶

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

Parameters:: X (Tensor) –
Return type:: Tensor

training: bool¶

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

Bases: 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.

Parameters:

dim – The (input) dimension. Must be at least 3.
noise_std (Optional[float]) – Standard deviation of the observation noise.
negate (bool) – If True, negate the function.

dim: int¶

evaluate_true(X)[source]¶

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

Parameters:: X (Tensor) –
Return type:: Tensor

training: bool¶

Multi-Objective Synthetic Test Functions¶

Multi-objective optimization benchmark problems.

References

[Daulton2022] (1,2)

S. Daulton, S. Cakmak, M. Balandat, M. A. Osborne, E. Zhou, and E. Bakshy. Robust Multi-Objective Bayesian Optimization Under Input Noise. Proceedings of the 39th International Conference on Machine Learning, 2022.

[Deb2005dtlz]

K. Deb, L. Thiele, M. Laumanns, E. Zitzler, A. Abraham, L. Jain, and R. Goldberg. Scalable test problems for evolutionary multi-objective optimization. Evolutionary Multiobjective Optimization, Springer-Verlag, pp. 105-145, 2005.

[Deb2005robust] (1,2)

K. Deb and H. Gupta. Searching for Robust Pareto-Optimal Solutions in Multi-objective Optimization. Evolutionary Multi-Criterion Optimization, Springer-Berlin, pp. 150-164, 2005.

[Frohlich2020]

L. Frohlich, E. Klenske, J. Vinogradska, C. Daniel, and M. Zeilinger. Noisy-Input Entropy Search for Efficient Robust Bayesian Optimization. Proceedings of the Twenty Third International Conference on Artificial Intelligence and Statistics, PMLR 108:2262-2272, 2020.

[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.

[Liang2021]

Q. Liang and L. Lai, Scalable Bayesian Optimization Accelerates Process Optimization of Penicillin Production. NeurIPS 2021 AI for Science Workshop, 2021.

[Ma2019]

Z. Ma and Y. Wang. Evolutionary Constrained Multiobjective Optimization: Test Suite Construction and Performance Comparisons. IEEE Transactions on Evolutionary Computation, 23(6):972–986, December 2019.

[Oszycka1995]

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

[Tanabe2020] (1,2,3,4,5,6,7)

Ryoji Tanabe and 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. Multi-Objective Bayesian Global Optimization using expected hypervolume improvement gradient. Swarm and evolutionary computation 44, pp. 945–956, 2019.

[Zitzler2000]

E. Zitzler, K. Deb, and L. Thiele. Comparison of multiobjective evolutionary algorithms: Empirical results. Evolutionary Computation, vol. 8, no. 2,pp. 173–195, 2000.

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

Bases: 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

Parameters:

noise_std (Union[None, float, List[float]]) – Standard deviation of the observation noise.
negate (bool) – If True, negate the objectives.

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 (Tensor) –
Return type:: Tensor

training: bool¶

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

Bases: MultiObjectiveTestProblem, ABC

Base class for DH problems for robust multi-objective optimization.

In their paper, [Deb2005robust] consider these problems under a mean-robustness setting, and use uniformly distributed input perturbations from the box with edge lengths delta_0 = delta, delta_i = 2 * delta, i > 0, with delta ranging up to 0.01 for DH1 and DH2, and delta = 0.03 for DH3 and DH4.

These are d-dimensional problems with two objectives:

f_0(x) = x_0 f_1(x) = h(x) + g(x) * S(x) for DH1 and DH2 f_1(x) = h(x) * (g(x) + S(x)) for DH3 and DH4

The goal is to minimize both objectives. See [Deb2005robust] for more details on DH. The reference points were set using infer_reference_point.

Parameters:

dim (int) – The (input) dimension.
noise_std (Union[None, float, List[float]]) – Standard deviation of the observation noise.
negate (bool) – If True, negate the function.

num_objectives: int = 2¶

class botorch.test_functions.multi_objective.DH1(dim, noise_std=None, negate=False)[source]¶

Bases: DH

DH1 test problem.

d-dimensional problem evaluated on [0, 1] x [-1, 1]^{d-1}:

f_0(x) = x_0 f_1(x) = h(x_0) + g(x) * S(x_0) h(x_0) = 1 - x_0^2 g(x) = sum_{i=1}^{d-1} (10 + x_i^2 - 10 * cos(4 * pi * x_i)) S(x_0) = alpha / (0.2 + x_0) + beta * x_0^2

where alpha = 1 and beta = 1.

The Pareto front corresponds to the equation f_1 = 1 - f_0^2, and it is found at x_i = 0 for i > 0 and any value of x_0 in (0, 1].

Parameters:

dim (int) – The (input) dimension.
noise_std (Union[None, float, List[float]]) – Standard deviation of the observation noise.
negate (bool) – If True, negate the function.

alpha = 1.0¶

beta = 1.0¶

evaluate_true(X)[source]¶

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

Parameters:: X (Tensor) –
Return type:: Tensor

dim: int¶

training: bool¶

class botorch.test_functions.multi_objective.DH2(dim, noise_std=None, negate=False)[source]¶

Bases: DH1

DH2 test problem.

This is identical to DH1 except for having beta = 10.0.

Parameters:

dim (int) – The (input) dimension.
noise_std (Union[None, float, List[float]]) – Standard deviation of the observation noise.
negate (bool) – If True, negate the function.

beta = 10.0¶

dim: int¶

training: bool¶

class botorch.test_functions.multi_objective.DH3(dim, noise_std=None, negate=False)[source]¶

Bases: DH

DH3 test problem.

d-dimensional problem evaluated on [0, 1]^2 x [-1, 1]^{d-2}:

f_0(x) = x_0 f_1(x) = h(x_1) * (g(x) + S(x_0)) h(x_1) = 2 - 0.8 * exp(-((x_1 - 0.35) / 0.25)^2) - exp(-((x_1 - 0.85) / 0.03)^2) g(x) = sum_{i=2}^{d-1} (50 * x_i^2) S(x_0) = 1 - sqrt(x_0)

The Pareto front is found at x_i = 0 for i > 1. There’s a local and a global Pareto front, which are found at x_1 = 0.35 and x_1 = 0.85, respectively. The approximate relationships between the objectives at local and global Pareto fronts are given by f_1 = 1.2 (1 - sqrt(f_0)) and f_1 = 1 - f_0, respectively. The specific values on the Pareto fronts can be found by varying x_0.

Parameters:

dim (int) – The (input) dimension.
noise_std (Union[None, float, List[float]]) – Standard deviation of the observation noise.
negate (bool) – If True, negate the function.

evaluate_true(X)[source]¶

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

Parameters:: X (Tensor) –
Return type:: Tensor

dim: int¶

training: bool¶

class botorch.test_functions.multi_objective.DH4(dim, noise_std=None, negate=False)[source]¶

Bases: DH3

DH4 test problem.

This is similar to DH3 except that it is evaluated on [0, 1] x [-0.15, 1] x [-1, 1]^{d-2} and:

h(x_0, x_1) = 2 - x_0 - 0.8 * exp(-((x_0 + x_1 - 0.35) / 0.25)^2) - exp(-((x_0 + x_1 - 0.85) / 0.03)^2)

The Pareto front is found at x_i = 0 for i > 2, with the local one being near x_0 + x_1 = 0.35 and the global one near x_0 + x_1 = 0.85.

Parameters:

dim (int) – The (input) dimension.
noise_std (Union[None, float, List[float]]) – Standard deviation of the observation noise.
negate (bool) – If True, negate the function.

dim: int¶

training: bool¶

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

Bases: MultiObjectiveTestProblem

Base class for DTLZ problems.

See [Deb2005dtlz] for more details on DTLZ.

Parameters:

dim (int) – The (input) dimension of the function.
num_objectives (int) – Must be less than dim.
noise_std (Union[None, float, List[float]]) – Standard deviation of the observation noise.
negate (bool) – If True, negate the function.

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: 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}^{d-1} ( k + (x_i - 0.5)^2 - cos(20 * pi * (x_i - 0.5)) )

where k = d - 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].

Parameters:

dim (int) – The (input) dimension of the function.
num_objectives (int) – Must be less than dim.
noise_std (Union[None, float, List[float]]) – Standard deviation of the observation noise.
negate (bool) – If True, negate the function.

evaluate_true(X)[source]¶

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

Parameters:: X (Tensor) –
Return type:: 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:: 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: 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}^{d-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.

Parameters:

dim (int) – The (input) dimension of the function.
num_objectives (int) – Must be less than dim.
noise_std (Union[None, float, List[float]]) – Standard deviation of the observation noise.
negate (bool) – If True, negate the function.

evaluate_true(X)[source]¶

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

Parameters:: X (Tensor) –
Return type:: 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:: Tensor

num_objectives: int¶

dim: int¶

training: bool¶

class botorch.test_functions.multi_objective.DTLZ3(dim, num_objectives=2, noise_std=None, negate=False)[source]¶

Bases: DTLZ2

DTLZ3 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) = 100 * [k + sum_{i=m}^{n-1} (x_i - 0.5)^2 - cos(20 * pi * (x_i - 0.5))]

g(x) introduces (3k−1) local Pareto fronts that are parallel to the one global Pareto-optimal front.

The global Pareto-optimal front corresponds to x_i = 0.5 for x_i in X_m.

Parameters:

dim (int) – The (input) dimension of the function.
num_objectives (int) – Must be less than dim.
noise_std (Union[None, float, List[float]]) – Standard deviation of the observation noise.
negate (bool) – If True, negate the function.

evaluate_true(X)[source]¶

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

Parameters:: X (Tensor) –
Return type:: Tensor

num_objectives: int¶

dim: int¶

training: bool¶

class botorch.test_functions.multi_objective.DTLZ4(dim, num_objectives=2, noise_std=None, negate=False)[source]¶

Bases: DTLZ2

DTLZ4 test problem.

This is the same as DTLZ2, but with alpha=100 as the exponent, resulting in dense solutions near the f_M-f_1 plane.

The global Pareto-optimal front corresponds to x_i = 0.5 for x_i in X_m.

Parameters:

dim (int) – The (input) dimension of the function.
num_objectives (int) – Must be less than dim.
noise_std (Union[None, float, List[float]]) – Standard deviation of the observation noise.
negate (bool) – If True, negate the function.

num_objectives: int¶

dim: int¶

training: bool¶

class botorch.test_functions.multi_objective.DTLZ5(dim, num_objectives=2, noise_std=None, negate=False)[source]¶

Bases: DTLZ

DTLZ5 test problem.

d-dimensional problem evaluated on [0, 1]^d:

f_0(x) = (1 + g(x)) * cos(theta_0 * pi / 2) f_1(x) = (1 + g(x)) * sin(theta_0 * pi / 2) theta_i = pi / (4 * (1 + g(X_m)) * (1 + 2 * g(X_m) * x_i)) for i = 1, … , M-2 g(x) = sum_{i=m}^{d-1} (x_i - 0.5)^2

The global Pareto-optimal front corresponds to x_i = 0.5 for x_i in X_m.

Parameters:

dim (int) – The (input) dimension of the function.
num_objectives (int) – Must be less than dim.
noise_std (Union[None, float, List[float]]) – Standard deviation of the observation noise.
negate (bool) – If True, negate the function.

evaluate_true(X)[source]¶

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

Parameters:: X (Tensor) –
Return type:: Tensor

num_objectives: int¶

dim: int¶

training: bool¶

class botorch.test_functions.multi_objective.DTLZ7(dim, num_objectives=2, noise_std=None, negate=False)[source]¶

Bases: DTLZ

DTLZ7 test problem.

d-dimensional problem evaluated on [0, 1]^d:: f_0(x) = x_0 f_1(x) = x_1 … f_{M-1}(x) = (1 + g(X_m)) * h(f_0, f_1, …, f_{M-2}, g, x) h(f_0, f_1, …, f_{M-2}, g, x) = M - sum_{i=0}^{M-2} f_i(x)/(1+g(x)) * (1 + sin(3 * pi * f_i(x)))

This test problem has 2M-1 disconnected Pareto-optimal regions in the search space.

The pareto frontier corresponds to X_m = 0.

Parameters:

dim (int) – The (input) dimension of the function.
num_objectives (int) – Must be less than dim.
noise_std (Union[None, float, List[float]]) – Standard deviation of the observation noise.
negate (bool) – If True, negate the function.

evaluate_true(X)[source]¶: Evaluate the function (w/o observation noise) on a set of points.

num_objectives: int¶

dim: int¶

training: bool¶

class botorch.test_functions.multi_objective.GMM(noise_std=None, negate=False, num_objectives=2)[source]¶

Bases: MultiObjectiveTestProblem

A test problem where each objective is a Gaussian mixture model.

This implementation is adapted from the single objective version (proposed by [Frohlich2020]) at https://github.com/boschresearch/NoisyInputEntropySearch/blob/master/ core/util/objectives.py.

See [Daulton2022] for details on this multi-objective problem.

Parameters:

noise_std (Union[None, float, List[float]]) – Standard deviation of the observation noise.
negate (bool) – If True, negate the objectives.
num_objectives (int) – The number of objectives.

dim: int = 2¶

num_objectives: int¶

evaluate_true(X)[source]¶

Evaluate the GMMs.

Parameters:: X (Tensor) –
Return type:: Tensor

training: bool¶

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

Bases: MultiObjectiveTestProblem

A penicillin production simulator from [Liang2021].

This implementation is adapted from https://github.com/HarryQL/TuRBO-Penicillin.

The goal is to maximize the penicillin yield while minimizing time to ferment and the CO2 byproduct.

The function is defined for minimization of all objectives.

The reference point was set using the infer_reference_point heuristic on the Pareto frontier over a large discrete set of random designs.

Base constructor for multi-objective test functions.

Parameters:

noise_std (Union[None, float, List[float]]) – Standard deviation of the observation noise. If a list is provided, specifies separate noise standard deviations for each objective.
negate (bool) – If True, negate the objectives.

dim: int = 7¶

num_objectives: int = 3¶

Y_xs = 0.45¶

Y_ps = 0.9¶

K_1 = 1e-10¶

K_2 = 7.000000000000001e-05¶

m_X = 0.014¶

alpha_1 = 0.143¶

alpha_2 = 4e-07¶

alpha_3 = 0.0001¶

mu_X = 0.092¶

K_X = 0.15¶

mu_p = 0.005¶

K_p = 0.0002¶

K_I = 0.1¶

K = 0.04¶

k_g = 7000.0¶

E_g = 5100.0¶

k_d = 1e+33¶

E_d = 50000.0¶

lambd = 0.00025¶

T_v = 273.0¶

T_o = 373.0¶

R = 1.9872¶

V_max = 180.0¶

classmethod penicillin_vectorized(X_input)[source]¶

Penicillin simulator, simplified and vectorized.

The 7 input parameters are (in order): culture volume, biomass concentration, temperature, glucose concentration, substrate feed rate, substrate feed concentration, and H+ concentration.

Parameters:: X_input (Tensor) – A n x 7-dim tensor of inputs.
Returns:: An n x 3-dim tensor of (negative) penicillin yield, CO2 and time.
Return type:: Tensor

evaluate_true(X)[source]¶

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

Parameters:: X (Tensor) –
Return type:: Tensor

training: bool¶

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

Bases: MultiObjectiveTestProblem

A 1D problem where the Pareto frontier is sensitive to input noise.

Specifically, the pareto frontier over the nominal objectives is sensitive to input noise. The first objective is a mixture of a linear function and a sinusoidal function, and the second objective is a modified Levy function, where the second parameter is fixed.

This function comes from [Daulton2022].

The reference point was set using the infer_reference_point heuristic on the Pareto frontier over a large discrete set of random designs.

Base constructor for multi-objective test functions.

Parameters:

noise_std (Union[None, float, List[float]]) – Standard deviation of the observation noise. If a list is provided, specifies separate noise standard deviations for each objective.
negate (bool) – If True, negate the objectives.

dim: int = 1¶

num_objectives: int = 2¶

levy = Levy()¶

f_1(X)[source]¶

Parameters:: X (Tensor) –
Return type:: Tensor

f_2(X)[source]¶

Parameters:: X (Tensor) –
Return type:: Tensor

evaluate_true(X)[source]¶

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

Parameters:: X (Tensor) –
Return type:: Tensor

training: bool¶

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

Bases: 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 (Union[None, float, List[float]]) – Standard deviation of the observation noise. If a list is provided, specifies separate noise standard deviations for each objective.
negate (bool) – If True, negate the objectives.

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 (Tensor) –
Return type:: Tensor

training: bool¶

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

Bases: MultiObjectiveTestProblem

Base class for ZDT problems.

See [Zitzler2000] for more details on ZDT.

Parameters:

dim (int) – The (input) dimension of the function.
num_objectives (int) – Number of objectives. Must not be larger than dim.
noise_std (Union[None, float, List[float]]) – Standard deviation of the observation noise.
negate (bool) – If True, negate the function.

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: 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.

Parameters:

dim (int) – The (input) dimension of the function.
num_objectives (int) – Number of objectives. Must not be larger than dim.
noise_std (Union[None, float, List[float]]) – Standard deviation of the observation noise.
negate (bool) – If True, negate the function.

evaluate_true(X)[source]¶

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

Parameters:: X (Tensor) –
Return type:: Tensor

gen_pareto_front(n)[source]¶

Generate n pareto optimal points.

Parameters:: n (int) –
Return type:: 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: 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.

Parameters:

dim (int) – The (input) dimension of the function.
num_objectives (int) – Number of objectives. Must not be larger than dim.
noise_std (Union[None, float, List[float]]) – Standard deviation of the observation noise.
negate (bool) – If True, negate the function.

evaluate_true(X)[source]¶

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

Parameters:: X (Tensor) –
Return type:: Tensor

gen_pareto_front(n)[source]¶

Generate n pareto optimal points.

Parameters:: n (int) –
Return type:: 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: 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.

Parameters:

dim (int) – The (input) dimension of the function.
num_objectives (int) – Number of objectives. Must not be larger than dim.
noise_std (Union[None, float, List[float]]) – Standard deviation of the observation noise.
negate (bool) – If True, negate the function.

evaluate_true(X)[source]¶

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

Parameters:: X (Tensor) –
Return type:: Tensor

gen_pareto_front(n)[source]¶

Generate n pareto optimal points.

Parameters:: n (int) –
Return type:: Tensor

num_objectives: int¶

dim: int¶

training: bool¶

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

Bases: MultiObjectiveTestProblem

Car side impact problem.

See [Tanabe2020] for details.

The reference point is nadir + 0.1 * (ideal - nadir) where the ideal and nadir points come from the approximate Pareto frontier from [Tanabe2020]. The max_hv was computed based on the approximate Pareto frontier from [Tanabe2020].

Base constructor for multi-objective test functions.

Parameters:

noise_std (Union[None, float, List[float]]) – Standard deviation of the observation noise. If a list is provided, specifies separate noise standard deviations for each objective.
negate (bool) – If True, negate the objectives.

num_objectives: int = 4¶

dim: int = 7¶

evaluate_true(X)[source]¶

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

Parameters:: X (Tensor) –
Return type:: Tensor

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

Bases: MultiObjectiveTestProblem, 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 (Union[None, float, List[float]]) – Standard deviation of the observation noise. If a list is provided, specifies separate noise standard deviations for each objective.
negate (bool) – If True, negate the objectives.

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 (Tensor) –
Return type:: Tensor

evaluate_slack_true(X)[source]¶

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

Parameters:

X (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:

Tensor

training: bool¶

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

Bases: MultiObjectiveTestProblem, 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 (Union[None, float, List[float]]) – Standard deviation of the observation noise. If a list is provided, specifies separate noise standard deviations for each objective.
negate (bool) – If True, negate the objectives.

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 (Tensor) –
Return type:: Tensor

evaluate_slack_true(X)[source]¶

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

Parameters:

X (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:

Tensor

training: bool¶

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

Bases: BraninCurrin, ConstrainedBaseTestProblem

Constrained Branin Currin Function.

This uses the disk constraint from [Gelbart2014].

Parameters:

noise_std (Union[None, float, List[float]]) – Standard deviation of the observation noise of the objectives.
constraint_noise_std (Union[None, float, List[float]]) – Standard deviation of the observation noise of the constraint.
negate (bool) – If True, negate the function.

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 (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:

Tensor

training: bool¶

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

Bases: DTLZ2, ConstrainedBaseTestProblem

Parameters:

dim (int) – The (input) dimension of the function.
num_objectives (int) – Must be less than dim.
noise_std (Union[None, float, List[float]]) – Standard deviation of the observation noise.
negate (bool) – If True, negate the function.

num_constraints: int = 1¶

evaluate_slack_true(X)[source]¶

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

Parameters:

X (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:

Tensor

num_objectives: int¶

dim: int¶

training: bool¶

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

Bases: MultiObjectiveTestProblem, ConstrainedBaseTestProblem

The Disc Brake problem.

There are 2 objectives and 4 constraints.

Both objectives should be minimized.

See [Tanabe2020] for details.

The reference point was set using the infer_reference_point heuristic on the Pareto frontier over a large discrete set of random designs.

Base constructor for multi-objective test functions.

Parameters:

noise_std (Union[None, float, List[float]]) – Standard deviation of the observation noise. If a list is provided, specifies separate noise standard deviations for each objective.
negate (bool) – If True, negate the objectives.

dim: int = 4¶

num_objectives: int = 2¶

num_constraints: int = 4¶

evaluate_true(X)[source]¶

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

Parameters:: X (Tensor) –
Return type:: Tensor

evaluate_slack_true(X)[source]¶

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

Parameters:

X (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:

Tensor

training: bool¶

class botorch.test_functions.multi_objective.MW7(dim, noise_std=None, constraint_noise_std=None, negate=False)[source]¶

Bases: MultiObjectiveTestProblem, ConstrainedBaseTestProblem

The MW7 problem.

This problem has 2 objectives, 2 constraints, and a disconnected Pareto frontier. It supports arbitrary input dimension > 1. See [Ma2019] for details.

This implementation is adapted from: https://github.com/anyoptimization/pymoo/blob/master/pymoo/problems/multi/mw.py

Parameters:

dim (int) – The (input) dimension of the function. Must be at least 2.
noise_std (Union[None, float, List[float]]) – Standard deviation of the observation noise of the objectives.
constraint_noise_std (Union[None, float, List[float]]) – Standard deviation of the observation noise of the constraints.
negate (bool) – If True, negate the function.

num_constraints: int = 2¶

num_objectives: int = 2¶

dim: int¶

LA2(A, B, C, D, theta)[source]¶

evaluate_true(X)[source]¶

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

Parameters:: X (Tensor) –
Return type:: Tensor

evaluate_slack_true(X)[source]¶

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

Parameters:

X (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:

Tensor

training: bool¶

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

Bases: MultiObjectiveTestProblem, 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 (Union[None, float, List[float]]) – Standard deviation of the observation noise. If a list is provided, specifies separate noise standard deviations for each objective.
negate (bool) – If True, negate the objectives.

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 (Tensor) –
Return type:: Tensor

evaluate_slack_true(X)[source]¶

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

Parameters:

X (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:

Tensor

training: bool¶

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

Bases: MultiObjectiveTestProblem, 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 (Union[None, float, List[float]]) – Standard deviation of the observation noise. If a list is provided, specifies separate noise standard deviations for each objective.
negate (bool) – If True, negate the objectives.

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 (Tensor) –
Return type:: Tensor

evaluate_slack_true(X)[source]¶

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

Parameters:

X (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:

Tensor

training: bool¶

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

Bases: MultiObjectiveTestProblem, ConstrainedBaseTestProblem

The Welded Beam multi-objective test problem. Similar to WeldedBeamSO in botorch.test_function.synthetic, but with an additional output, somewhat modified constraints, and a different domain.

Implementation from https://github.com/msu-coinlab/pymoo/blob/master/pymoo/problems/multi/welded_beam.py Note that this implementation assumes minimization, so please choose negate=True.

Base constructor for multi-objective test functions.

Parameters:

noise_std (Union[None, float, List[float]]) – Standard deviation of the observation noise. If a list is provided, specifies separate noise standard deviations for each objective.
negate (bool) – If True, negate the objectives.

dim: int = 4¶

num_constraints: int = 4¶

num_objectives: int = 2¶

evaluate_true(X)[source]¶

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

Parameters:: X (Tensor) –
Return type:: Tensor

evaluate_slack_true(X)[source]¶

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

Parameters:

X (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:

Tensor

training: bool¶

Multi-Objective Multi-Fidelity Synthetic Test Functions¶

Multi-objective multi-fidelity optimization benchmark problems.

References

[Irshad2021] (1,2)

F. Irshad, S. Karsch, and A. Döpp. Expected hypervolume improvement for simultaneous multi-objective and multi-fidelity optimization. arXiv preprint arXiv:2112.13901, 2021.

class botorch.test_functions.multi_objective_multi_fidelity.MOMFBraninCurrin(noise_std=None, negate=False)[source]¶

Bases: MultiObjectiveTestProblem

Branin-Currin problem for multi-objective-multi-fidelity optimization.

(2+1)-dimensional function with domain [0,1]^3 where the last dimension is the fidelity parameter s. Both functions assume minimization. See [Irshad2021] for more details.

Modified Branin function:

B(x,s) = 21-(( 15*x_2 - b(s) * (15 * x_1 - 5) ** 2 + c(s) * (15 * x_1 - 5) - 6 ) ** 2 + 10 * (1 - t(s)) * cos(15 * x_1 - 5)+10)/22

Here b, c, r and t are constants and s is the fidelity parameter:: where b = 5.1 / (4 * math.pi ** 2) - 0.01(1-s), c = 5 / math.pi - 0.1*(1 - s), r = 6, t = 1 / (8 * math.pi) + 0.05*(1-s)

Modified Currin function:

C(x) = 14-((1 - 0.1(1-s)exp(-1 / (2 * x_2))) * ( 2300 * x_1 ** 3 + 1900 * x_1 ** 2 + 2092 * x_1 + 60 ) / 100 * x_1 ** 3 + 500 * x_1 ** 2 + 4 * x_2 + 20)/15

Base constructor for multi-objective test functions.

Parameters:

noise_std (Union[None, float, List[float]]) – Standard deviation of the observation noise. If a list is provided, specifies separate noise standard deviations for each objective.
negate (bool) – If True, negate the objectives.

dim: int = 3¶

num_objectives: int = 2¶

evaluate_true(X)[source]¶

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

Parameters:: X (Tensor) –
Return type:: Tensor

training: bool¶

class botorch.test_functions.multi_objective_multi_fidelity.MOMFPark(noise_std=None, negate=False)[source]¶

Bases: MultiObjectiveTestProblem

Modified Park test functions for multi-objective multi-fidelity optimization.

(4+1)-dimensional function with domain [0,1]^5 where the last dimension is the fidelity parameter s. See [Irshad2021] for more details.

The first modified Park function is

P1(x, s)=A*(T1(x,s)+T2(x,s)-B)/22-0.8

The second modified Park function is

P2(x,s)=A*(5-2/3*exp(x1+x2)-x4*sin(x3)*A+x3-B)/4 - 0.7

Here

T_1(x,s) = (x1+0.001*(1-s))/2*sqrt(1+(x2+x3**2)*x4/(x1**2))

T_2(x, s) = (x1+3*x4)*exp(1+sin(x3))

and A(s)=(0.9+0.1*s), B(s)=0.1*(1-s).

Base constructor for multi-objective test functions.

Parameters:

noise_std (Union[None, float, List[float]]) – Standard deviation of the observation noise. If a list is provided, specifies separate noise standard deviations for each objective.
negate (bool) – If True, negate the objectives.

dim: int = 5¶

num_objectives: int = 2¶

evaluate_true(X)[source]¶

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

Parameters:: X (Tensor) –
Return type:: Tensor

training: bool¶

Sensitivity Analysis Test Functions¶

class botorch.test_functions.sensitivity_analysis.Ishigami(b=0.1, noise_std=None, negate=False)[source]¶

Bases: SyntheticTestFunction

Ishigami test function.

three-dimensional function (usually evaluated on [-pi, pi]^3):

f(x) = sin(x_1) + a sin(x_2)^2 + b x_3^4 sin(x_1)

Here a and b are constants where a=7 and b=0.1 or b=0.05 Proposed to test sensitivity analysis methods because it exhibits strong nonlinearity and nonmonotonicity and a peculiar dependence on x_3.

Parameters:

b (float) – the b constant, should be 0.1 or 0.05.
noise_std (float | None) – Standard deviation of the observation noise.
negative – If True, negative the objective.
negate (bool) –

dim: int¶

compute_dgsm(X)[source]¶

Compute derivative global sensitivity measures.

This function can be called separately to estimate the dgsm measure The exact global integrals of these values are already added under as attributes dgsm_gradient, dgsm_gradient_bas, and dgsm_gradient_square.

Parameters:: X (Tensor) – Set of points at which to compute derivative measures.
Return type:: Tuple[List[float], List[float], List[float]]

Returns: The average gradient, absolute gradient, and square gradients.

evaluate_true(X)[source]¶

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

Parameters:: X (Tensor) –
Return type:: Tensor

training: bool¶

class botorch.test_functions.sensitivity_analysis.Gsobol(dim, a=None, noise_std=None, negate=False)[source]¶

Bases: SyntheticTestFunction

Gsobol test function.

d-dimensional function (usually evaluated on [0, 1]^d):

f(x) = Prod_{i=1}^{d} ((|4x_i-2|+a_i)/(1+a_i)), a_i >=0

common combinations of dimension and a vector:

dim=8, a= [0, 1, 4.5, 9, 99, 99, 99, 99] dim=6, a=[0, 0.5, 3, 9, 99, 99] dim = 15, a= [1, 2, 5, 10, 20, 50, 100, 500, 1000, …, 1000]

Proposed to test sensitivity analysis methods First order Sobol indices have closed form expression S_i=V_i/V with :

V_i= 1/(3(1+a_i)^2) V= Prod_{i=1}^{d} (1+V_i) - 1

Parameters:

dim (int) – Dimensionality of the problem. If 6, 8, or 15, will use standard a.
a (List) – a parameter, unless dim is 6, 8, or 15.
noise_std (float | None) – Standard deviation of observation noise.
negate (bool) – Return negatie of function.

dim: int¶

optimal_sobol_indicies()[source]¶

evaluate_true(X)[source]¶

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

Parameters:: X (Tensor) –
Return type:: Tensor

training: bool¶

class botorch.test_functions.sensitivity_analysis.Morris(noise_std=None, negate=False)[source]¶

Bases: SyntheticTestFunction

Morris test function.

20-dimensional function (usually evaluated on [0, 1]^20):

f(x) = sum_{i=1}^20 beta_i w_i + sum_{i<j}^20 beta_ij w_i w_j + sum_{i<j<l}^20 beta_ijl w_i w_j w_l + 5w_1 w_2 w_3 w_4

Proposed to test sensitivity analysis methods

Parameters:

noise_std (float | None) – Standard deviation of observation noise.
negate (bool) – Return negative of function.

training: bool¶

dim: int¶

evaluate_true(X)[source]¶

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

Parameters:: X (Tensor) –
Return type:: Tensor

Utilities For Test Functions¶

botorch.test_functions.utils.round_nearest(X, increment, bounds)[source]¶

Rounds the input tensor to the nearest multiple of increment.

Parameters:

X (Tensor) – The input to be rounded.
increment (float) – The increment to round to.
bounds (Tuple[float, float] | None) – An optional tuple of two floats representing the lower and upper bounds on X. If provided, this will round to the nearest multiple of increment that lies within the bounds.

Returns:

The rounded input.

Return type:

Tensor

BoTorch

0.10.0

botorch.test_functions¶

Abstract Test Function API¶

Synthetic Test Functions¶

Multi-Fidelity Synthetic Test Functions¶

Multi-Objective Synthetic Test Functions¶

Multi-Objective Multi-Fidelity Synthetic Test Functions¶

Sensitivity Analysis Test Functions¶

Utilities For Test Functions¶

BoTorch

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