crit_EMI() R function from [GPareto]

Expected Maximin Improvement with m objectives

Expected Maximin Improvement with respect to the current Pareto front with Sample Average Approximation. The semi-analytical formula is used in the bi-objective scale if the Pareto front is in [-2,2]^2, for numerical stability reasons. To avoid numerical instabilities, the new point is penalized if it is too close to an existing observation.


crit_EMI(
  x,
  model,
  paretoFront = NULL,
  critcontrol = list(nb.samp = 50, seed = 42),
  type = "UK"
)

Arguments

x: a vector representing the input for which one wishes to calculate EMI,
model: list of objects of class km, one for each objective functions,
paretoFront: (optional) matrix corresponding to the Pareto front of size [n.pareto x n.obj], or any reference set of observations,
critcontrol: optional list with arguments (for more than 2 objectives only):
- nb.samp number of random samples from the posterior distribution, default to 50, increasing gives more reliable results at the cost of longer computation time;
- seed seed used for the random samples.
Options for the checkPredict function: threshold (1e-4) and distance (covdist) are used to avoid numerical issues occuring when adding points too close to the existing ones.
type: "SK" or "UK" (by default), depending whether uncertainty related to trend estimation has to be taken into account.

Returns

The Expected Maximin Improvement at x.

Details

It is recommanded to scale objectives, e.g. to [0,1]. If the Pareto front does not belong to [-2,2]^2, then SAA is used.

Examples


#---------------------------------------------------------------------------
# Expected Maximin Improvement surface associated with the "P1" problem at a 15 points design
#---------------------------------------------------------------------------
set.seed(25468)
library(DiceDesign)

n_var <- 2 
f_name <- "P1" 
n.grid <- 21
test.grid <- expand.grid(seq(0, 1, length.out = n.grid), seq(0, 1, length.out = n.grid))
n_appr <- 15 
design.grid <- round(maximinESE_LHS(lhsDesign(n_appr, n_var, seed = 42)$design)$design, 1)
response.grid <- t(apply(design.grid, 1, f_name))
Front_Pareto <- t(nondominated_points(t(response.grid)))
mf1 <- km(~., design = design.grid, response = response.grid[,1])
mf2 <- km(~., design = design.grid, response = response.grid[,2])

EMI_grid <- apply(test.grid, 1, crit_EMI, model = list(mf1, mf2), paretoFront = Front_Pareto,
                     critcontrol = list(nb_samp = 20))

filled.contour(seq(0, 1, length.out = n.grid), seq(0, 1, length.out = n.grid), nlevels = 50,
               matrix(EMI_grid, nrow = n.grid), main = "Expected Maximin Improvement",
               xlab = expression(x[1]), ylab = expression(x[2]), color = terrain.colors,
               plot.axes = {axis(1); axis(2);
                            points(design.grid[,1], design.grid[,2], pch = 21, bg = "white")
                            }
              )

References

J. D. Svenson & T. J. Santner (2010), Multiobjective Optimization of Expensive Black-Box Functions via Expected Maximin Improvement, Technical Report.

J. D. Svenson (2011), Computer Experiments: Multiobjective Optimization and Sensitivity Analysis, Ohio State University, PhD thesis.

crit_EMI function