Bayesian Optimization: Maximize MaxPro Criterion

[#1]

Hi dear Community and JMP experts,

I'm confused by the denomination of one of the option in the batch customizer for Bayesian Optimization. One of the option is called Maximize MaxPro criterion in Bayesian Optimization platform. The Help in this section mentions: "This option is a model-free exploration of the factor space that avoids replication of any of the factor settings in both the training data and the current batch. Use this option for any batch size when one or more of the models is not fitting well."

Regarding the definition of this MaxPro criterion in the Fast Flexible Filling Design Details help section, "The MaxPro criterion maximizes the product of the distances between potential design points in a way that involves all factors. This supports the goal of providing good space-filling properties on projections of factors". It is also mentioned that the MaxPro criterion "strives to find points in the clusters that minimize the following criterion" (before looking at the formula):

As the product of the squared distances is at the denominator in the formula, minimizing this criterion does maximize the distances between potential design points. So maximizing the MaxPro criterion may actually lead to smaller distances between points, causing a higher risk of runs that may be clustered or "stacked" when looking at smaller subspaces of the experimental space. Looking at the Bayesian Optimization Profiler, it seems the formula used behind this acquisition function does not use (squared) distances at the denominator (like in the original MaxPro criterion), but at the numerator, since we can see concave desirability profile matching a -X² type of formula involving distances between original design points. MaxPro acquisition function is at the maximum where the distances between the new point and original design points are the highest:

So can someone clarify this confusion between the "minimize MaxPro criterion" of Space Filling designs and the "maximize MaxPro criterion" of Bayesian Optimization ? Does the MaxPro criterion in the Bayesian Optimization platform use the same formula as the one in the Space Filling platform ? What is the formula displayed in the MaxPro Space Filling Criterion acquisition function ?

Thanks in advance !