EHVI: EHVI: Constrained Expected Hypervolume Improvement
In moko: Multi-Objective Kriging Optimization

Description Usage Arguments Details Value References Examples

Multi-objective Expected Hypervolume Improvement with respect to the current Pareto front. It's based on the crit_EHI function of the GPareto-package package. However, the present implementation accounts for inequality constrains embedded into the mkm model.

1	EHVI(x, model, control = NULL)

x

a vector representing the input for which one wishes to calculate EHI, alternatively a matrix with one point per row,

model

An object of class mkm.

control

An optional list of control parameters, some of them passed to the crit_EHI function. One can control:

minimization: logical indicating if the EHVI is minimizing all objectives (TRUE, by default) or maximizing all objectives (FALSE). Mixed optimization is not currently accepted, if the user needs it, the cost functions should be modified prior Kriging modeling (i.e. inverting or multiplying the output by -1).
paretoFront: object of class ps containing the actual Pareto set. If not provided a Pareto set is built based on the current feasible observations (model@response[model@feasible,]).
nb.samp: number of random samples from the posterior distribution (with more than two objectives), default to 50, increasing gives more reliable results at the cost of longer computation time
seed: seed used for the random samples (with more than two objectives);
refPoint: reference point for Hypervolume Expected Improvement. If not provided, it is set to the maximum or minimum of each objective.

The way that the constraints are handled are based on the probability of feasibility. The strong assumption here is that the cost functions and the constraints are uncorrelated. With that assumption in mind, a simple closed-form solution can be derived that consists in the product of the probability that each constraint will be met and the expected improvement of the objective.

The constrained expected hypervolume improvement at x.

Forrester, A., Sobester, A., & Keane, A. (2008). Engineering design via surrogate modeling: a practical guide. John Wiley & Sons.

# ------------------------
# The Nowacki Beam
# ------------------------
n <- 10
d <- 2
doe <- replicate(d,sample(0:n,n))/n
res <- t(apply(doe, 1, nowacki_beam, box = data.frame(b = c(10, 50), h = c(50, 250))))
model <- mkm(doe, res, modelcontrol = list(objective = 1:2, lower=rep(0.1,d)))
grid <- expand.grid(seq(0, 1, , 10),seq(0, 1, , 10))
ehvi <- apply(grid, 1, EHVI, model)
contour(matrix(ehvi, 20))
points(model@design, col=ifelse(model@feasible,'blue','red'))
points(grid[which.max(ehvi),], col='green', pch=19)