GPareto (version 1.1.8)

crit_EHI: Expected Hypervolume Improvement with m objectives

Description

Multi-objective Expected Hypervolume Improvement with respect to the current Pareto front. With two objectives the analytical formula is used, while Sample Average Approximation (SAA) is used with more objectives. To avoid numerical instabilities, the new point is penalized if it is too close to an existing observation.

Usage

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

Value

The Expected Hypervolume Improvement at x.

Arguments

x

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

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:

• 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;

• extendper if no reference point refPoint is provided, for each objective it is fixed to the maximum over the Pareto front plus extendper times the range, Default value to 0.2, corresponding to 1.1 for a scaled objective with a Pareto front in [0,1]^n.obj.

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

The computation of the analytical formula with two objectives is adapted from the Matlab source code by Michael Emmerich and Andre Deutz, LIACS, Leiden University, 2010 available here : http://liacs.leidenuniv.nl/~csmoda/code/HV_based_expected_improvement.zip.

References

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

M. T. Emmerich, A. H. Deutz, J. W. Klinkenberg (2011), Hypervolume-based expected improvement: Monotonicity properties and exact computation, Evolutionary Computation (CEC), 2147-2154.

See Also

EI from package DiceOptim, crit_EMI, crit_SUR, crit_SMS.

Examples

#---------------------------------------------------------------------------
# Expected Hypervolume 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 <- 26
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])

EHI_grid <- crit_EHI(x = as.matrix(test.grid), model = list(mf1, mf2),
         critcontrol = list(refPoint = c(300,0)))

filled.contour(seq(0, 1, length.out = n.grid), seq(0, 1, length.out = n.grid), nlevels = 50,
               matrix(EHI_grid, n.grid), main = "Expected Hypervolume 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")
                            }
              )