smt_optim.acquisition_functions

Acquisition Functions

expected_improvement(mu: float, s2: float, f_min: float) → float

Expected Improvement acquisition function.

Parameters:

• mu (float) – Mean prediction.

• s2 (float) – Variance prediction.

• f_min (float) – Best minimum objective value in training data.

Returns:

Expected Improvement value.

Return type:

float

fidelity_correlation(covariance: ndarray, li_var: ndarray, lj_var: ndarray) → ndarray

Compute the posterior correlation between two fidelity levels.

This function evaluates the pointwise Pearson correlation coefficient between the predictive distributions of two fidelity levels based on their posterior covariance and variances. The resulting correlation is clipped to the interval [0, 1].

Parameters:

• covariance (ndarray of shape (n_eval,)) – Posterior covariance between the predictions at fidelity levels \(i\) and \(j\) for each evaluation point.

• li_var (ndarray of shape (n_eval,)) – Posterior predictive variance at fidelity level \(i\).

• lj_var (ndarray of shape (n_eval,)) – Posterior predictive variance at fidelity level \(j\).

Returns:

Absolute value of the correlation coefficient between fidelity levels \(i\) and \(j\), clipped to the interval [0, 1].

Return type:

ndarray of shape (n_eval,)

init_ehvi_2o(state) → Callable

Initialize the Expected Hypervolume Improvement (EHVI) acquisition function for bi-objective Bayesian optimization.

The returned callable evaluates EHVI using the objective surrogate models stored in state. The Pareto front is extracted from the scaled dataset and augmented with reference points required by the two-objective EHVI computation (ehvi_2o).

Parameters:

state (State) –

Optimization state containing the problem definition, scaled dataset, and trained objective surrogate models.

The problem must have exactly two objectives. The scaled dataset must contain objective evaluations from which the current Pareto front can be computed.

Returns:

EHVI acquisition function with signature:

acquisition(x) -> float

where x is a candidate point and the returned value is the expected hypervolume improvement at x.

Return type:

Callable

Raises:

ValueError – If the optimization problem is not bi-objective.

Notes

The Pareto front is augmented with two additional points and sorted by the second objective before being passed to the EHVI computation.

Follows the implementation discussed in: Yang, K., Emmerich, M., Deutz, A., & Bäck, T. (2019). Multi-objective Bayesian global optimization using expected hypervolume improvement gradient. Swarm and evolutionary computation, 44, 945-956.

init_mpi(state) → Callable

Initialize the Minimum Probability of Improvement (MPoI / MPI) multi-objective acquisition function. DOI: https://doi.org/10.1145/3071178.3071276

Parameters:

• state (State)

• Returns – Callable acquisition function

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Notes

Uses the scaled dataset

integrated_variance_reduction(model, points: ndarray, integration_points: ndarray = None, inv_block: bool = True) → ndarray

Integrated Variance Reduction (IVR) acquisition function. Evaluates IVR for one or multiple candidate points. Returns the absolute reduction in IMSE: |IMSE_current - IMSE_new|.

Parameters:

• model (Surrogate) – The smt-optim surrogate model representing the objective function.

• points (np.ndarray) – Candidate points for enrichment, shape (num_points, num_dim).

• integration_points (np.ndarray, optional) – Monte-Carlo points for integration over the domain, shape (N_mc, num_dim). If None, a 500-point LHS grid is automatically generated (requires model to have xlimits or infers from training points).

• inv_block (bool, optional) – Whether to use block matrix inversion (faster). Default True.

Returns:

The IVR values of shape (num_points, 1) (higher is better).

Return type:

np.ndarray

log_ei(mu: float, s2: float, f_min: float) → float

Log Expected Improvement acquisition function.

LogEI is more numerically stable that the EI acquisition function especially when the GP’s variance is small. From: https://arxiv.org/abs/2310.20708.

Parameters:

• mu (float) – Mean prediction

• s2 (float) – Variance prediction

• f_min (float) – Best minimum objective value in training data.

Return type:

float

log_pi(mu: float, s2: float, f_min: float) → float

Log Probability of Improvement acquisition function.

LogPI is more numerically stable that the pi acquisition function especially when the GP’s variance is small. From: https://arxiv.org/abs/2310.20708.

Parameters:

• mu (float) – Mean prediction

• s2 (float) – Variance prediction

• f_min (float) – Best minimum objective value in training data.

Return type:

float

probability_of_improvement(mu: float, s2: float, f_min: float) → float

Probability of Improvement (PI) acquisition function.

Parameters:

• mu (float) – Mean prediction.

• s2 (float) – Variance prediction.

• f_min (float) – Best minimum objective value in training data.

Returns:

Probability of Improvement value.

Return type:

float

vec_expected_improvement(mu: ndarray, s2: ndarray, f_min: float) → ndarray

Vectorized Expected Improvement acquisition function.

Parameters:

• mu (np.ndarray) – Mean prediction of shape (num_points, 1).

• s2 (np.ndarray) – Variance prediction of shape (num_points, 1).

• f_min (float) – Best minimum objective value in training data.

Returns:

Expected Improvement values of shape (num_points, 1).

Return type:

np.ndarray

vec_probability_of_improvement(mu: ndarray, s2: ndarray, f_min: float) → ndarray

Probability of Improvement (PI) acquisition function.

Parameters:

• mu (float) – Mean prediction.

• s2 (float) – Variance prediction.

• f_min (float) – Best minimum objective value in training data.

Returns:

Probability of Improvement value.

Return type:

float