crit_SMS {GPareto}

Analytical expression of the SMS-EGO criterion with m>1 objectives

Description

Computes a slightly modified infill Criterion of the SMS-EGO. To avoid numerical instabilities, an additional penalty is added to the new point if it is too close to an existing observation.

Usage

crit_SMS(x, model, paretoFront = NULL, critcontrol = NULL, type = "UK")

Arguments

x

a vector representing the input for which one wishes to calculate the criterion,

model

a list of objects of class km (one for each objective),

paretoFront

(optional) matrix corresponding to the Pareto front of size [n.pareto x n.obj], or any reference set of observations,

critcontrol

list with arguments:

• currentHV current hypervolume;

• refPoint reference point for hypervolume computations;

• 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;

• epsilon optional value to use in additive epsilon dominance;

• gain optional gain factor for sigma.

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.

Value

Value of the criterion.

References

W. Ponweiser, T. Wagner, D. Biermann, M. Vincze (2008), Multiobjective Optimization on a Limited Budget of Evaluations Using Model-Assisted S-Metric Selection, Parallel Problem Solving from Nature, pp. 784-794. Springer, Berlin.

T. Wagner, M. Emmerich, A. Deutz, W. Ponweiser (2010), On expected-improvement criteria for model-based multi-objective optimization. Parallel Problem Solving from Nature, pp. 718-727. Springer, Berlin.

See Also

crit_EHI, crit_SUR, crit_EMI.

Examples

#---------------------------------------------------------------------------
# SMS-EGO 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))
PF <- t(nondominated_points(t(response.grid)))
mf1 <- km(~., design = design.grid, response = response.grid[,1])
mf2 <- km(~., design = design.grid, response = response.grid[,2])

model <- list(mf1, mf2)
critcontrol <- list(refPoint = c(300, 0), currentHV = dominated_hypervolume(t(PF), c(300, 0)))
SMSEGO_grid <- apply(test.grid, 1, crit_SMS, model = model,
                     paretoFront = PF, critcontrol = critcontrol)

filled.contour(seq(0, 1, length.out = n.grid), seq(0, 1, length.out = n.grid),
               matrix(pmax(0, SMSEGO_grid), nrow = n.grid), nlevels = 50,
               main = "SMS-EGO criterion (positive part)", 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")
                            }
              )