`trieste.objectives.single_objectives`#

This module contains toy objective functions, useful for experimentation. A number of them have been taken from this Virtual Library of Simulation Experiments <https://web.archive.org/web/20211015101644/https://www.sfu.ca/~ssurjano/> (:cite:`ssurjano2021)`_.

Module Contents#

branin(x: trieste.types.TensorType) → trieste.types.TensorType [source]#

The Branin-Hoo function over \([0, 1]^2\). See [PWG13] for details.

Parameters: x – The points at which to evaluate the function, with shape […, 2].
Returns: The function values at x, with shape […, 1].
Raises: ValueError (or InvalidArgumentError) – If x has an invalid shape.

scaled_branin(x: trieste.types.TensorType) → trieste.types.TensorType [source]#

The Branin-Hoo function, rescaled to have zero mean and unit variance over \([0, 1]^2\). See [PWG13] for details.

Parameters: x – The points at which to evaluate the function, with shape […, 2].
Returns: The function values at x, with shape […, 1].
Raises: ValueError (or InvalidArgumentError) – If x has an invalid shape.

BRANIN_MINIMIZERS[source]#: The three global minimizers of the branin() function over \([0, 1]^2\), with shape [3, 2] and dtype float64.

BRANIN_MINIMUM[source]#: The global minimum of the branin() function, with shape [1] and dtype float64.

SCALED_BRANIN_MINIMUM[source]#: The global minimum of the branin() function, with shape [1] and dtype float64.

BRANIN_SEARCH_SPACE[source]#: The search space for the branin() function.

simple_quadratic(x: trieste.types.TensorType) → trieste.types.TensorType [source]#

A trivial quadratic function over \([0, 1]^2\). Useful for quick testing.

Parameters: x – The points at which to evaluate the function, with shape […, 2].
Returns: The function values at x, with shape […, 1].
Raises: ValueError (or InvalidArgumentError) – If x has an invalid shape.

SIMPLE_QUADRATIC_MINIMIZER[source]#: The global minimizer of the simple_quadratic() function over \([0, 1]^2\), with shape [1, 2] and dtype float64.

SIMPLE_QUADRATIC_MINIMUM[source]#: The global minimum of the simple_quadratic() function over \([0, 1]^2\), with shape [1] and dtype float64.

SIMPLE_QUADRATIC_SEARCH_SPACE[source]#: The search space for the simple_quadratic() function.

gramacy_lee(x: trieste.types.TensorType) → trieste.types.TensorType [source]#

The Gramacy & Lee function, typically evaluated over \([0.5, 2.5]\). See [GL12] for details.

Parameters: x – Where to evaluate the function, with shape […, 1].
Returns: The function values, with shape […, 1].
Raises: ValueError (or InvalidArgumentError) – If x has an invalid shape.

GRAMACY_LEE_MINIMIZER[source]#: The global minimizer of the gramacy_lee() function over \([0.5, 2.5]\), with shape [1, 1] and dtype float64.

GRAMACY_LEE_MINIMUM[source]#: The global minimum of the gramacy_lee() function over \([0.5, 2.5]\), with shape [1] and dtype float64.

GRAMACY_LEE_SEARCH_SPACE[source]#: The search space for the gramacy_lee() function.

logarithmic_goldstein_price(x: trieste.types.TensorType) → trieste.types.TensorType [source]#

A logarithmic form of the Goldstein-Price function, with zero mean and unit variance over \([0, 1]^2\). See [PWG13] for details.

Parameters: x – The points at which to evaluate the function, with shape […, 2].
Returns: The function values at x, with shape […, 1].
Raises: ValueError (or InvalidArgumentError) – If x has an invalid shape.

LOGARITHMIC_GOLDSTEIN_PRICE_MINIMIZER[source]#: The global minimizer for the logarithmic_goldstein_price() function, with shape [1, 2] and dtype float64.

LOGARITHMIC_GOLDSTEIN_PRICE_MINIMUM[source]#: The global minimum for the logarithmic_goldstein_price() function, with shape [1] and dtype float64.

LOGARITHMIC_GOLDSTEIN_PRICE_SEARCH_SPACE[source]#: The search space for the logarithmic_goldstein_price() function.

hartmann_3(x: trieste.types.TensorType) → trieste.types.TensorType [source]#

The Hartmann 3 test function over \([0, 1]^3\). This function has 3 local and one global minima. See https://www.sfu.ca/~ssurjano/hart3.html for details.

Parameters: x – The points at which to evaluate the function, with shape […, 3].
Returns: The function values at x, with shape […, 1].
Raises: ValueError (or InvalidArgumentError) – If x has an invalid shape.

HARTMANN_3_MINIMIZER[source]#: The global minimizer for the hartmann_3() function, with shape [1, 3] and dtype float64.

HARTMANN_3_MINIMUM[source]#: The global minimum for the hartmann_3() function, with shape [1] and dtype float64.

HARTMANN_3_SEARCH_SPACE[source]#: The search space for the hartmann_3() function.

shekel_4(x: trieste.types.TensorType) → trieste.types.TensorType [source]#

The Shekel test function over \([0, 1]^4\). This function has ten local minima and a single global minimum. See https://www.sfu.ca/~ssurjano/shekel.html for details. Note that we rescale the original problem, which is typically defined over [0, 10]^4.

Parameters: x – The points at which to evaluate the function, with shape […, 4].
Returns: The function values at x, with shape […, 1].
Raises: ValueError (or InvalidArgumentError) – If x has an invalid shape.

SHEKEL_4_MINIMIZER[source]#: The global minimizer for the shekel_4() function, with shape [1, 4] and dtype float64.

SHEKEL_4_MINIMUM[source]#: The global minimum for the shekel_4() function, with shape [1] and dtype float64.

SHEKEL_4_SEARCH_SPACE[source]#: The search space for the shekel_4() function.

rosenbrock_4(x: trieste.types.TensorType) → trieste.types.TensorType [source]#

The Rosenbrock function, rescaled to have zero mean and unit variance over \([0, 1]^4. See :cite:\) for details. This function (also known as the Banana function) is unimodal, however the minima lies in a narrow valley.

Parameters: x – The points at which to evaluate the function, with shape […, 4].
Returns: The function values at x, with shape […, 1].
Raises: ValueError (or InvalidArgumentError) – If x has an invalid shape.

ROSENBROCK_4_MINIMIZER[source]#: The global minimizer for the rosenbrock_4() function, with shape [1, 4] and dtype float64.

ROSENBROCK_4_MINIMUM[source]#: The global minimum for the rosenbrock_4() function, with shape [1] and dtype float64.

ROSENBROCK_4_SEARCH_SPACE[source]#: The search space for the rosenbrock_4() function.

ackley_5(x: trieste.types.TensorType) → trieste.types.TensorType [source]#

The Ackley test function over \([0, 1]^5\). This function has many local minima and a global minima. See https://www.sfu.ca/~ssurjano/ackley.html for details. Note that we rescale the original problem, which is typically defined over [-32.768, 32.768].

Parameters: x – The points at which to evaluate the function, with shape […, 5].
Returns: The function values at x, with shape […, 1].
Raises: ValueError (or InvalidArgumentError) – If x has an invalid shape.

ACKLEY_5_MINIMIZER[source]#: The global minimizer for the ackley_5() function, with shape [1, 5] and dtype float64.

ACKLEY_5_MINIMUM[source]#: The global minimum for the ackley_5() function, with shape [1] and dtype float64.

ACKLEY_5_SEARCH_SPACE[source]#: The search space for the ackley_5() function.

hartmann_6(x: trieste.types.TensorType) → trieste.types.TensorType [source]#

The Hartmann 6 test function over \([0, 1]^6\). This function has 6 local and one global minima. See https://www.sfu.ca/~ssurjano/hart6.html for details.

Parameters: x – The points at which to evaluate the function, with shape […, 6].
Returns: The function values at x, with shape […, 1].
Raises: ValueError (or InvalidArgumentError) – If x has an invalid shape.

HARTMANN_6_MINIMIZER[source]#: The global minimizer for the hartmann_6() function, with shape [1, 6] and dtype float64.

HARTMANN_6_MINIMUM[source]#: The global minimum for the hartmann_6() function, with shape [1] and dtype float64.

HARTMANN_6_SEARCH_SPACE[source]#: The search space for the hartmann_6() function.

michalewicz(x: trieste.types.TensorType, d: int = 2, m: int = 10) → trieste.types.TensorType [source]#: The Michalewicz function over \([0, \pi]\) for all i=1,…,d. Dimensionality is determined by the parameter d and it features steep ridges and drops. It has \(d!\) local minima, and it is multimodal. The parameter m defines the steepness of they valleys and ridges; a larger m leads to a more difficult search. The recommended value of m is 10. See https://www.sfu.ca/~ssurjano/egg.html for details. :param x: The points at which to evaluate the function, with shape […, d]. :param d: The dimension of the function. :param m: The steepness of the valleys/ridges. :return: The function values at x, with shape […, 1]. :raise ValueError (or InvalidArgumentError): If x has an invalid shape.

michalewicz_2(x: trieste.types.TensorType) → trieste.types.TensorType [source]#: Convenience function for the 2-dimensional michalewicz() function with steepness 10. :param x: The points at which to evaluate the function, with shape […, 2]. :return: The function values at x, with shape […, 1].

michalewicz_5(x: trieste.types.TensorType) → trieste.types.TensorType [source]#: Convenience function for the 5-dimensional michalewicz() function with steepness 10. :param x: The points at which to evaluate the function, with shape […, 5]. :return: The function values at x, with shape […, 1].

michalewicz_10(x: trieste.types.TensorType) → trieste.types.TensorType [source]#: Convenience function for the 10-dimensional michalewicz() function with steepness 10. :param x: The points at which to evaluate the function, with shape […, 10]. :return: The function values at x, with shape […, 1].

MICHALEWICZ_2_MINIMIZER[source]#: The global minimizer of the michalewicz() function over \([0, \pi]^2\), with shape [1, 2] and dtype float64. Taken from https://arxiv.org/abs/2003.09867

MICHALEWICZ_5_MINIMIZER[source]#: The global minimizer of the michalewicz() function over \([0, \pi]^5\), with shape [1, 5] and dtype float64. Taken from https://arxiv.org/abs/2003.09867

MICHALEWICZ_10_MINIMIZER[source]#: The global minimizer of the michalewicz() function over \([0, \pi]^10\), with shape [1, 10] and dtype float64. Taken from https://arxiv.org/abs/2003.09867

MICHALEWICZ_2_MINIMUM[source]#: The global minimum of the 2-dimensional michalewicz() function, with shape [1] and dtype float64. Taken from https://arxiv.org/abs/2003.09867

MICHALEWICZ_5_MINIMUM[source]#: The global minimum of the 5-dimensional michalewicz() function, with shape [1] and dtype float64. Taken from https://arxiv.org/abs/2003.09867

MICHALEWICZ_10_MINIMUM[source]#: The global minimum of the 10-dimensional michalewicz() function, with shape [1] and dtype float64. Taken from https://arxiv.org/abs/2003.09867

MICHALEWICZ_2_SEARCH_SPACE[source]#: The search space for the 2-dimensional michalewicz() function.

MICHALEWICZ_5_SEARCH_SPACE[source]#: The search space for the 5-dimensional michalewicz() function.

MICHALEWICZ_10_SEARCH_SPACE[source]#: The search space for the 10-dimensional michalewicz() function.

trid(x: trieste.types.TensorType, d: int = 10) → trieste.types.TensorType [source]#

The Trid function over \([-d^2, d^2]\) for all i=1,…,d. Dimensionality is determined by the parameter d and it has a global minimum. This function has large variation in output which makes it challenging for Bayesian optimisation with vanilla Gaussian processes with non-stationary kernels. Models that can deal with non-stationarities, such as deep Gaussian processes, can be useful for modelling these functions. See [HBB+19] and https://www.sfu.ca/~ssurjano/trid.html for details.

Parameters

x – The points at which to evaluate the function, with shape […, d].
d – Dimensionality.

Returns

The function values at x, with shape […, 1].

Raises

ValueError (or InvalidArgumentError) – If x has an invalid shape.

trid_10(x: trieste.types.TensorType) → trieste.types.TensorType [source]#

The Trid function with dimension 10.

Parameters: x – The points at which to evaluate the function, with shape […, 10].
Returns: The function values at x, with shape […, 1].
Raises: ValueError (or InvalidArgumentError) – If x has an invalid shape.

TRID_10_MINIMIZER[source]#: The global minimizer of trid() function is defined as \(x_i=i(d+1-i)\) for all i=1,…,d. Here, we define it specifically for the 10-dimensional variant, with shape [1, 10] and dtype float64.

TRID_10_MINIMUM[source]#: The global minimum of trid() function is defined as \(d(d+4)(d-1)/6\) for dimensionality d. Here, we define it specifically for the 10-dimensional variant, with shape [1] and dtype float64.

TRID_10_SEARCH_SPACE[source]#: The search space for trid() function is defined over \([-d^2, d^2]\) for all i=1,…,d. Here, we define it specifically for the 10-dimensional variant.

trieste.objectives.single_objectives#

Module Contents#

`trieste.objectives.single_objectives`#