Batch Expected Hypervolume Improvement with m objectives
Parallel Multi-objective Expected Hypervolume Improvement with respect to the current Pareto front, via Sample Average Approximation (SAA).
crit_qEHI( x, model, paretoFront = NULL, critcontrol = list(nb.samp = 50, seed = 42), type = "UK" )
x: a matrix representing the inputs (one per row) for which one wishes to calculate EHI,
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.type: "SK" or "UK" (by default), depending whether uncertainty related to trend estimation has to be taken into account.
The Expected Hypervolume Improvement at x.
The batch EHI is computed by simulated nb.samp conditional simulation of the objectives on the candidate points, before averaging the corresponding hypervolume improvement of each set of m simulations.
#--------------------------------------------------------------------------- # 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)) test.grid <- as.matrix(test.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]) refPoint <- c(300, 0) EHI_grid <- crit_EHI(x = test.grid, model = list(mf1, mf2), critcontrol = list(refPoint = refPoint)) ## Create potential batches q <- 3 ncb <- 500 Xbcands <- array(NA, dim = c(ncb, q, n_var)) for(i in 1:ncb) Xbcands[i,,] <- test.grid[sample(1:nrow(test.grid), q, prob = pmax(0,EHI_grid)),] qEHI_grid <- apply(Xbcands, 1, crit_qEHI, model = list(mf1, mf2), critcontrol = list(refPoint = refPoint)) Xq <- Xbcands[which.max(qEHI_grid),,] ## For further optimization (gradient may not be reliable) # sol <- optim(as.vector(Xq), function(x, ...) crit_qEHI(matrix(x, q), ...), # model = list(mf1, mf2), lower = c(0,0), upper = c(1,1), method = "L-BFGS-B", # control = list(fnscale = -1), critcontrol = list(refPoint = refPoint, nb.samp = 10000)) # sol <- psoptim(as.vector(Xq), function(x, ...) crit_qEHI(matrix(x, q), ...), # model = list(mf1, mf2), lower = c(0,0), upper = c(1,1), # critcontrol = list(refPoint = refPoint, nb.samp = 100), # control = list(fnscale = -1)) # Plot EHI surface and selected designs for parallel evaluation 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 surface and best 3-EHI designs", 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") points(Xq, pch = 20, col = 2) # points(matrix(sol$par, q), col = 4) } )
J. D. Svenson (2011), Computer Experiments: Multiobjective Optimization and Sensitivity Analysis, Ohio State University, PhD thesis.
EI from package DiceOptim, crit_EMI, crit_SUR, crit_SMS.