GPareto (version 1.1.8)

crit_EMI: Expected Maximin Improvement with m objectives

Description

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.

Usage

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

Value

The Expected Maximin Improvement at x.

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.

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.

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.

See Also

EI from package DiceOptim, crit_EHI, crit_SUR, crit_SMS.

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")
                            }
              )