include center points/replicates to design

[jsdiazpo]

The method find_local_max_ipopt could include the possibility of including center points and/or replicates in the design.

[Osburg]

Can you further explain what you mean? I think there is already a functionality to manually define a number of experiments that will definitely be part of the design. These experiments are excluded from the optimization process (they are still considered for optimizing all other experiments, but will remain constant during optimization). You can make use of this with the fixed_experiments argument.

[jsdiazpo]

The purpose of replicates (an experiment reproduced a given number, say two, of times) to quantify the experimental uncertainty. Center points are a few experiments (can be the same as the replicates) somewhere in near the middle of the design hypervolume that can help assessing nonlinearity in the modeling. These experiments must be excluded from the optimization procedure, they just need to be added at the end.

[Osburg]

Okok, got you. When I implemented this feature my thought was that it is advantageous for the optimizer to use the information that there are certain fixed experiments. For example when you manually set two replicants in one corner of the design space (for whatever reason you want) the optimizer will know this and will not place a third point at the same corner. Would you disagree with this?
In this case: we could add another optional argument for fixed experiments that will be taken out of the optimization process completely or change the implementation for the currently existing argument accordingly.

[jsdiazpo]

I think that fixed_experiments can be very useful for another type of feature that is absent: augmented designs. Sometimes a user has done a few experiments already and would like to extend or augment the design space and fixed_experiments can avoid the repetition of existing experiments. Replicates/center points are different (and simpler), they just need to be added after the optimization at the end of the design matrix.

[Osburg]

Okay, I will add such an argument (if no one else does it before) soon :)

[DavidWalz]

@jsdiazpo a flag for adding center points / replicates to a design is a good idea.
For linear and linear + interaction models the center points also allows to check for non-linearities. For linear + quadratic and fully quadratic models the center point will anyway result from the optimization. Should a we add 3 replicate center points when this is set to true?

@Osburg You're right that the fixed_experiments you implemented already supports this functionality as well as augmented designs.

I don't see the benefit of adding center points only after the optimization. On the contrary, for quadratic models as well es for purely linear models with non-symmetric design spaces (constraints!) considering the center points during the optimization will lead to better designs.