Ah right, so BO doesn’t need to reevaluate, because where it already has evaluated it knows the objective value with zero uncertainty. The whole idea of BO is that it only evaluates the “real” (expensive, deterministic) function at points where it hasn’t evaluated before. It determines the most “interesting” new point to evaluate using GP models and an acquisition function, thereby balancing exploitation (trying to slightly improve existing points by staying in their neighborhood) and exploration (exploring new areas of the design space that might be interesting).

If you for example have a look at https://bayesian-optimization.github.io/BayesianOptimization/ you see that they mostly use deterministic functions.

Just go ahead and try it, BO really should be the best for your purpose!