ExpectedImprovement returns NaN for zero predictive variance at the improvement threshold #621

On master af8b928, ExpectedImprovement.base_acq computes a/std unconditionally. At std=0 and mean=y_max+xi it returns NaN, although a deterministic predictive distribution has expected improvement max(mean-y_max-xi,0). Other zero-variance points also emit divide-by-zero warnings.

import numpy as np
from bayes_opt.acquisition import ExpectedImprovement
from sklearn.gaussian_process import GaussianProcessRegressor
from sklearn.gaussian_process.kernels import ConstantKernel
a = ExpectedImprovement(xi=0)
a.y_max = 1.0
gp = GaussianProcessRegressor(
    kernel=ConstantKernel(1.0, constant_value_bounds='fixed'),
    alpha=0, optimizer=None,
).fit([[0.]], [1.])
mean, std = gp.predict([[0.], [1.]], return_std=True)
print(mean, std)              # [1. 1.] [0. 0.]
print(a.base_acq(mean, std))  # [nan nan], expected [0. 0.]

Reproduced in Python 3.12 / NumPy 2.5.3 / sklearn 1.9.1. The older #516 reported the warning without a complete reproduction and was closed pending information; the current API and an actual GP reproduction are shown here. No open overlapping PR found.

Proposed fix: avoid division by zero and use the deterministic positive-part expectation at zero variance, preserving the formula for positive std, broadcasting and scalar calls. Add a focused mixed-variance regression checked against a numerical Gaussian expectation and a real zero-variance GP control. ProbabilityOfImprovement's threshold/tie convention is outside this issue.

AI assistance: Codex autonomously reproduced this report and is preparing the contribution; no independent human review has occurred.