Motivation

This PR introduces a refactoring of batched_lbfgsb and its related functions, making the interface of the batched_lbfgsb compatible with the scipy.optimize module. The goal is to address two pieces of technical debt that could lead to future bugs and make the API less intuitive.

1. Resolve Variable Capture to Prevent Bugs
   The core issue is that the nested function negative_acqf_with_grad was implicitly using the normalized_params variable from its enclosing scope (_gradient_ascent_batched). While this may not be causing an active bug, it is a form of technical debt that makes the code harder to reason about and could easily lead to errors in the future. This PR resolves the capture.

2. Improve API Flexibility for func_and_grad
   The current implementation of batched_lbfgsb implicitly requires that the func_and_grad callable accepts a second argument (batch_indices), even if the function itself does not use it. This design is inflexible and counter-intuitive.

For example, a simple single-argument function is currently incompatible:

# This function is not used, but is required as a placeholder argument
def myfunc1(x: np.ndarray, batched_indices: np.ndarray) -> tuple[np.ndarray, np.ndarray]:
    value = -np.sum(x**2)
    grad = -2 * x
    return value, grad

# This more natural, single-argument function should work, but currently throws an error
def myfunc2(x: np.ndarray) -> tuple[np.ndarray, np.ndarray]:
    value = -np.sum(x**2)
    grad = -2 * x
    return value, grad

# Works correctly
batched_lbfgsb(myfunc1, ...)

# Fails with a TypeError because the second argument is missing
batched_lbfgsb(myfunc2, ...)

This change allows batched_lbfgsb to correctly handle callables with and without the second argument, making the API more flexible and user-friendly.

Description of the changes

• The function signature for batched_lbfgsb has been updated.
• The batch_indices argument has been removed fromfunc_and_grad. This information is now passed via args_list instead.