The BudgetOptimizer.allocate_budget() doc string includes the argument minimize_kwargs which is described as consisting of "Extra kwargs for scipy.optimize.minimize". The "extra" is because the BudgetOptimizer.allocate_budget() argument budget_bounds gets passed to scipy.optimize.minimize (though it is processed first. No mention of this in the docstring). But the x0 argument is also passed into scipy.optimize.minimize but there is no mention of this. As a result, the following fails:

BudgetOptimizer.allocate_budget(total_budget=100_000, x0=[50_000, 50_000])` 

with the following error:

TypeError: scipy.optimizer_minimize.minimize() got multiple values for keyword argument `x0`

Docstring should be updated or BudgetOptimizer.allocate_budget() should handle x0 as an argument.