# How to analytically solve the probability of improvement acquisition function in Bayesian Optimization with Vector inputs?

I have been using the probability of improvement acquisition function in my Bayesian Optimization program, but I've run into a problem because I am not optimizing the acquisition function that quickly. Currently, I'm using a PSO algorithm to find points from the acquisition functions, but as the matrices get larger it quickly becomes computationally burdensome.

The Probability of Improvement Function is: PI(x) = P(f(x) ≥ f(x+)) = Φ (µ(x) - f(x+) / σ(x)) where f(x+) is the max value already found, µ(x) is the mean, σ(x) is the standard deviation, Φ () refers to the cumulative density function of a normal distribution.