Table 1 Pseudocode for the Bayesian optimization algorithm and outline for computing the acquisition function used in the above algorithm

 1. Data set: D_1:n = {x_i,y_i}i = 1 to n

 2. Build Gaussian process regression model: y = f_n(x)~GP(m(x),k(x,x^'))

 3. Bayesian optimization () {

   for t = 1 to t_max

   a. Find next x_t by optimizing the acquisition function u over GP

   x_t = argmax_xu(x|D_1:n)

   b. Compute the value y_t for this new x_t

   c. Augment (x_t,y_t) into data set D_1:n = {x_i,y_i}

   d. Update the Gaussian Process Regression model

 }

Acquisition function u

 1. Find x such that Expected Improvement (\({\Bbb E}\)) is maximum

 \(X = {\it{argmax}}_{\it{x}}EI\left( x \right) = argmax_x{\Bbb E}\left( {\max \left\{ {0,f_n\left( x \right) - f^{max}} \right\}\left| {D_{1:n}} \right.} \right)\)

   where, f_n(x) is a Gaussian process regression model made from D_1:n and f^max is the maximum value of this function.

 2. Equations to compute Expected Improvement (𝔼)

 \(EI\left( x \right) = \left\{ {\begin{array}{*{20}{c}} {\left( {\mu \left( x \right) - f^{max}} \right)\Phi \left( Z \right) + \sigma \left( x \right)\phi \left( z \right)} & {if\,\sigma \left( x \right) > 0} \\ 0 & {0\,if\,\sigma \left( x \right) = 0} \end{array}} \right.\)

 \(Z = \frac{{\mu \left( x \right) - f^{max}}}{{\sigma (x)}}\)

   where μ(x) and σ(x) are predicted mean and standard deviation values for x by Gaussian process regression model f_n(x).

   φ(Z) and Φ(Z)are probability density function (PDF) and cumulative density function (CDF) of standard normal distribution

 3. Return x