Efficient Global Optimization

Description

The Efficient Global Optimization (EGO) algorithms is a global optimization working with kriging surrogate, by maximization of the expected improvement. Several iterations of this maximization of expected information enrichment leads to fill the space design at interesting locations to achieve optimization. The following parameters are available for fine tuning of the algorithm:

The resulting analysis is the following:

Sources

References

D. Ginsbourger (2009), Multiples metamodeles pour l'approximation et l'optimisation de fonctions numeriques multivariables, Ph.D. thesis, Ecole Nationale Superieure des Mines de Saint-Etienne, 2009.
D.R. Jones, M. Schonlau, and W.J. Welch (1998), Efficient global optimization of expensive black-box functions, _Journal of Global Optimization_, 13, 455-492.
W.R. Jr. Mebane and J.S. Sekhon (2009), in press, Genetic optimization using derivatives: The rgenoud package for R, _Journal of Statistical Software_.

Implementation

This EGO algorithm is wrapped from the [R] packages DiceOptim and DiceKriging from DICE consortium.This basic designer is implemented in following R file (EGO.R):

#help=Efficient Global Optimization (EGO) algorithm
#type=Optimization
#output=Optimum
#options=initBatchSize=10,batchSize=10,iterations=10,nugget.estim=false,nugget=0.0,trend=y~1,covtype=matern5_2,liar=min,Xbounds=false,search_min=true
#require=lhs,DiceKriging,DiceOptim,DiceView
 
# iteration index
iEGO <<- 0
 
#' constructor and initializer of R session
#' @returnType NULL
#' @return null
#' @author richet
init <- function() {
	library(lhs)
	library(MASS)
	library(rgenoud)
	library(DiceKriging)
	library(DiceOptim)
	library(DiceView)
 
# all parameters are initialy strings, so you have to put as global non-string values
	initBatchSize <<- as.integer(initBatchSize)
	batchSize <<- as.integer(batchSize)
	iterations <<- as.integer(iterations)
	#covtype
	nugget.estim <<- as.logical(nugget.estim)
	nugget <<- as.numeric(nugget)
	if (nugget == 0) {
		nugget <<- NULL
	}
	trend <<- as.formula(trend)
	if (liar == "min") {
		liar <<- as.function(min)
	}
	if (liar == "max") {
		liar <<- as.function(max)
	}
	noise.var <<- NULL
	search_min <<- as.logical(search_min)
	Xbounds <<- as.logical(Xbounds)
	iEGO <<- 0
}
 
#' first design building. All variables are set in [0,1]. d is the dimension, or number of variables
#' @param d number of variables
#' @returnType matrix
#' @return next design of experiments
#' @author richet
initDesign <- function(d) {
	set.seed(1)
	lhs <- maximinLHS(n=initBatchSize,k=d)
	if (Xbounds) {
		e=c(0,1)
		id=1
		while(id<d){
			e=rbind(cbind(e,0),cbind(e,1))
			id=id+1
		}
		Xinit=rbind(as.matrix(lhs),as.matrix(e))
	} else {
		Xinit=as.matrix(lhs)
	}
	return(Xinit)
}
 
#' iterated design building.
#' @param X data frame of current doe variables (in [0,1])
#' @param Y data frame of current results
#' @returnType data frame or matrix
#' @return  next doe step
#' @author richet
nextDesign <- function(X,Y) {
	if (iEGO > iterations) return();
 
	d = dim(X)[2]
	if (dim(Y)[2] == 2) {
		noise.var <<- as.array(Y[,2])
	} else {
		noise.var <<- NULL
	}
 
	if (search_min) {y=Y[,1]} else {y=-Y[,1]}
	kmi <- km(control=list(trace=FALSE),trend,optim.method='gen',penalty = NULL,covtype=covtype,nugget.estim = nugget.estim, nugget = nugget, noise.var = noise.var,design=X,response=y)
 
	EGOi <- max_qEI.CL(model=kmi,npoints=batchSize,L=liar(as.array(Y[,1])),lower=rep(0,d),upper=rep(1,d),control=list(trace=FALSE))
 
	Xnext <- EGOi$par
 
	iEGO <<- iEGO + 1
	return(as.matrix(Xnext))
}
 
#' final analysis. All variables are set in [0,1]. Return HTML string
#' @param X data frame of doe variables (in [0,1])
#'q @param Y data frame of results
#' @returnType String
#' @return HTML string of analysis
#' @author richet
analyseDesign <- function(X,Y) {
	analyse.files <<- paste("sectionview_",iEGO-1,".png",sep="")
	resolution <- 600
	if (dim(Y)[2] == 2) {
		noise.var <<- as.array(Y[,2])
	} else {
		noise.var <<- NULL
	}
	if (search_min) {
		m = min(Y)
		x = as.matrix(X)[Y==m,]
		html=paste(sep="<br/>",paste("<HTML>minimum is ",m),paste(sep="","found at ",paste(collapse="= ",capture.output(x)),"<br/><img src='",analyse.files,"' width='",resolution,"' height='",resolution,"'/></HTML>"))
		plot=paste("<Plot1D name='min'>",m,"</Plot1D>")
 
		d = dim(X)[2]
		if (d == 1) {
			plotx=paste("<Plot1D name='argmin'>",paste(x),"</Plot1D>")
		} else if (d == 2) {
			plotx=paste("<Plot2D name='argmin'>(",paste(collapse=",",x),")</Plot2D>")
		} else {
			plotx=paste("<PlotnD name='argmin'>(",paste(collapse=",",x),")</PlotnD>")
		}
	} else {
		m = max(Y)
		x = as.matrix(X)[Y==m,]
		html=paste(sep="<br/>",paste("<HTML>maximum is ",m),paste(sep="","found at ",paste(collapse="= ",capture.output(x)),"<br/><img src='",analyse.files,"' width='",resolution,"' height='",resolution,"'/></HTML>"))
		plot=paste("<Plot1D name='max'>",m,"</Plot1D>")
 
		d = dim(X)[2]
		if (d == 1) {
			plotx=paste("<Plot1D name='argmax'>",paste(x),"</Plot1D>")
		} else if (d == 2) {
			plotx=paste("<Plot2D name='argmax'>(",paste(collapse=",",x),")</Plot2D>")
		} else {
			plotx=paste("<PlotnD name='argmax'>(",paste(collapse=",",x),")</PlotnD>")
		}
	}
 
	model <- km(control=list(trace=FALSE),trend,optim.method='gen',penalty = NULL,covtype=covtype,nugget.estim = nugget.estim, nugget = nugget, noise.var = noise.var,design=X,response=Y)
 
	png(file=analyse.files,bg="transparent",height=resolution,width = resolution)
	sectionview(model=model,center=x)
	dev.off()
 
	return(paste(html,plot,plotx))
}
 
#' temporary analysis. All variables are set in [0,1]. Return HTML string
#' @param X data frame of doe variables (in [0,1])
#' @param Y data frame of  results
#' @returnType String
#' @return HTML string of analysis
#' @author richet
analyseDesignTmp <- function(X,Y) {
	analyseDesign(X,Y)
}

Installation

copy
EGO.R
in “Promethee_Install_Path/plugins/doe” directory,
restart Promethee GUI.