DiceOptim (version 2.0)

EGO.nsteps: Sequential EI maximization and model re-estimation, with a number of iterations fixed in advance by the user

Description

Executes nsteps iterations of the EGO method to an object of class km. At each step, a kriging model is re-estimated (including covariance parameters re-estimation) based on the initial design points plus the points visited during all previous iterations; then a new point is obtained by maximizing the Expected Improvement criterion (EI).

Usage

EGO.nsteps(model, fun, nsteps, lower, upper, parinit = NULL, control = NULL, kmcontrol = NULL)

Arguments

model

an object of class km ,

fun

the objective function to be minimized,

nsteps

an integer representing the desired number of iterations,

lower

vector of lower bounds for the variables to be optimized over,

upper

vector of upper bounds for the variables to be optimized over,

parinit

optional vector of initial values for the variables to be optimized over,

control

an optional list of control parameters for optimization. One can control

"pop.size" (default : [4+3*log(nb of variables)]),

"max.generations" (default :5),

"wait.generations" (default :2),

"BFGSburnin" (default :0),

of the function genoud.

kmcontrol

an optional list representing the control variables for the re-estimation of the kriging model. The items are the same as in km :

penalty, optim.method, parinit, control.

The default values are those contained in model, typically corresponding to the variables used in km to estimate a kriging model from the initial design points.

Value

A list with components:

References

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.

J. Mockus (1988), Bayesian Approach to Global Optimization. Kluwer academic publishers.

T.J. Santner, B.J. Williams, and W.J. Notz (2003), The design and analysis of computer experiments, Springer.

M. Schonlau (1997), Computer experiments and global optimization, Ph.D. thesis, University of Waterloo.

See Also

EI, max_EI, EI.grad

Examples

set.seed(123)
###############################################################
### 	10 ITERATIONS OF EGO ON THE BRANIN FUNCTION, 	   ####
###	 STARTING FROM A 9-POINTS FACTORIAL DESIGN         ####
###############################################################

# a 9-points factorial design, and the corresponding response
d <- 2
n <- 9
design.fact <- expand.grid(seq(0,1,length=3), seq(0,1,length=3))
names(design.fact)<-c("x1", "x2")
design.fact <- data.frame(design.fact)
names(design.fact)<-c("x1", "x2")
response.branin <- apply(design.fact, 1, branin)
response.branin <- data.frame(response.branin)
names(response.branin) <- "y"

# model identification
fitted.model1 <- km(~1, design=design.fact, response=response.branin,
covtype="gauss", control=list(pop.size=50,trace=FALSE), parinit=c(0.5, 0.5))

# EGO n steps
library(rgenoud)
nsteps <- 5 # Was 10, reduced to 5 for speeding up compilation
lower <- rep(0,d)
upper <- rep(1,d)
oEGO <- EGO.nsteps(model=fitted.model1, fun=branin, nsteps=nsteps,
lower=lower, upper=upper, control=list(pop.size=20, BFGSburnin=2))
print(oEGO$par)
print(oEGO$value)

# graphics
n.grid <- 15 # Was 20, reduced to 15 for speeding up compilation
x.grid <- y.grid <- seq(0,1,length=n.grid)
design.grid <- expand.grid(x.grid, y.grid)
response.grid <- apply(design.grid, 1, branin)
z.grid <- matrix(response.grid, n.grid, n.grid)
contour(x.grid, y.grid, z.grid, 40)
title("Branin function")
points(design.fact[,1], design.fact[,2], pch=17, col="blue")
points(oEGO$par, pch=19, col="red")
text(oEGO$par[,1], oEGO$par[,2], labels=1:nsteps, pos=3)


###############################################################
### 	20 ITERATIONS OF EGO ON THE GOLDSTEIN-PRICE,  	   ####
###	 STARTING FROM A 9-POINTS FACTORIAL DESIGN   	   ####
###############################################################
## Not run: 
# # a 9-points factorial design, and the corresponding response
# d <- 2
# n <- 9
# design.fact <- expand.grid(seq(0,1,length=3), seq(0,1,length=3))
# names(design.fact)<-c("x1", "x2")
# design.fact <- data.frame(design.fact)
# names(design.fact)<-c("x1", "x2")
# response.goldsteinPrice <- apply(design.fact, 1, goldsteinPrice)
# response.goldsteinPrice <- data.frame(response.goldsteinPrice)
# names(response.goldsteinPrice) <- "y"
# 
# # model identification
# fitted.model1 <- km(~1, design=design.fact, response=response.goldsteinPrice,
# covtype="gauss", control=list(pop.size=50, max.generations=50,
# wait.generations=5, BFGSburnin=10,trace=FALSE), parinit=c(0.5, 0.5), optim.method="BFGS")
# 
# # EGO n steps
# library(rgenoud)
# nsteps <- 10 # Was 20, reduced to 10 for speeding up compilation
# lower <- rep(0,d)
# upper <- rep(1,d)
# oEGO <- EGO.nsteps(model=fitted.model1, fun=goldsteinPrice, nsteps=nsteps,
# lower, upper, control=list(pop.size=20, BFGSburnin=2))
# print(oEGO$par)
# print(oEGO$value)
# 
# # graphics
# n.grid <- 15 # Was 20, reduced to 15 for speeding up compilation
# x.grid <- y.grid <- seq(0,1,length=n.grid)
# design.grid <- expand.grid(x.grid, y.grid)
# response.grid <- apply(design.grid, 1, goldsteinPrice)
# z.grid <- matrix(response.grid, n.grid, n.grid)
# contour(x.grid, y.grid, z.grid, 40)
# title("Goldstein-Price Function")
# points(design.fact[,1], design.fact[,2], pch=17, col="blue")
# points(oEGO$par, pch=19, col="red")
# text(oEGO$par[,1], oEGO$par[,2], labels=1:nsteps, pos=3)
# ## End(Not run)

#######################################################################
### 	nsteps ITERATIONS OF EGO ON THE HARTMAN6 FUNCTION,  	   ####
###	  STARTING FROM A 10-POINTS UNIFORM DESIGN      	   ####
#######################################################################

## Not run: 
# fonction<-hartman6
# data(mydata)
# a <- mydata
# nb<-10
# nsteps <- 3 # Maybe be changed to a larger value
# x1<-a[[1]][1:nb];x2<-a[[2]][1:nb];x3<-a[[3]][1:nb]
# x4<-a[[4]][1:nb];x5<-a[[5]][1:nb];x6<-a[[6]][1:nb]
# design <- data.frame(cbind(x1,x2,x3,x4,x5,x6))
# names(design)<-c("x1", "x2","x3","x4","x5","x6")
# n <- nrow(design)
# 
# response <- data.frame(q=apply(design,1,fonction))
# names(response) <- "y"
# fitted.model1 <- km(~1, design=design, response=response, covtype="gauss",
# control=list(pop.size=50, max.generations=20, wait.generations=5, BFGSburnin=5,
# trace=FALSE), optim.method="gen", parinit=rep(0.8,6))
# 
# res.nsteps <- EGO.nsteps(model=fitted.model1, fun=fonction, nsteps=nsteps,
# lower=rep(0,6), upper=rep(1,6), parinit=rep(0.5,6), control=list(pop.size=50,
# max.generations=20, wait.generations=5, BFGSburnin=5), kmcontrol=NULL)
# print(res.nsteps)
# plot(res.nsteps$value,type="l")
# ## End(Not run)