EGO.nsteps: Sequential EI maximization and model re-estimation, with a...

In DiceOptim: Kriging-Based Optimization for Computer Experiments

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:

par	a data frame representing the additional points visited during the algorithm,
value	a data frame representing the response values at the points given in par,
npoints	an integer representing the number of parallel computations (=1 here),
nsteps	an integer representing the desired number of iterations (given in argument),
lastmodel	an object of class km corresponding to the last kriging model fitted.

Note

Most EGO-like methods (EI algorithms) usually work with Ordinary Kriging (constant trend), by maximization of the expected improvement. Here, the EI maximization is also possible with any linear trend. However, note that the optimization may perform much faster and better when the trend is a constant since it is the only case where the analytical gradient is available.

For more details on kmcontrol, see the documentation of km.

Author(s)

David Ginsbourger

Olivier Roustant

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)

DiceOptim documentation built on Feb. 2, 2021, 1:06 a.m.