DiceOptim (version 2.1.1)

max_EI: Maximization of the Expected Improvement criterion

Description

Given an object of class km and a set of tuning parameters (lower,upper,parinit, and control), max_EI performs the maximization of the Expected Improvement criterion and delivers the next point to be visited in an EGO-like procedure.

Usage

max_EI(
  model,
  plugin = NULL,
  type = "UK",
  lower,
  upper,
  parinit = NULL,
  minimization = TRUE,
  control = NULL
)

Arguments

model

an object of class km ,

plugin

optional scalar: if provided, it replaces the minimum of the current observations,

type

Kriging type: "SK" or "UK"

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,

minimization

logical specifying if EI is used in minimiziation or in maximization,

control

optional list of control parameters for optimization. One can control "pop.size" (default : [N=3*2^dim for dim<6 and N=32*dim otherwise]), "max.generations" (12), "wait.generations" (2) and "BFGSburnin" (2) of function "genoud" (see genoud). Numbers into brackets are the default values

Value

A list with components:

par

The best set of parameters found.

value

The value of expected improvement at par.

Details

The latter maximization relies on a genetic algorithm using derivatives, genoud. This function plays a central role in the package since it is in constant use in the proposed algorithms. It is important to remark that the information needed about the objective function reduces here to the vector of response values embedded in model (no call to the objective function or simulator).

The current minimum of the observations can be replaced by an arbitrary value (plugin), which is usefull in particular in noisy frameworks.

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.

Examples

# NOT RUN {
set.seed(123)
##########################################################
### "ONE-SHOT" EI-MAXIMIZATION OF THE BRANIN FUNCTION ####
### 	KNOWN AT 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 one step
library(rgenoud)
lower <- rep(0,d) 
upper <- rep(1,d)     # domain for Branin function
oEGO <- max_EI(fitted.model1, lower=lower, upper=upper, 
control=list(pop.size=20, BFGSburnin=2))
print(oEGO)

# graphics
n.grid <- 20
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$par[1], oEGO$par[2], pch=19, col="red")


#############################################################
### "ONE-SHOT" EI-MAXIMIZATION OF THE CAMELBACK FUNCTION ####
###	KNOWN AT A 16-POINTS FACTORIAL DESIGN            ####
#############################################################
# }
# NOT RUN {
# a 16-points factorial design, and the corresponding response
d <- 2 
n <- 16
design.fact <- expand.grid(seq(0,1,length=4), seq(0,1,length=4)) 
names(design.fact)<-c("x1", "x2")
design.fact <- data.frame(design.fact) 
names(design.fact) <- c("x1", "x2")
response.camelback <- apply(design.fact, 1, camelback)
response.camelback <- data.frame(response.camelback) 
names(response.camelback) <- "y" 

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

# EI maximization
library(rgenoud)
lower <- rep(0,d) 
upper <- rep(1,d)   
oEGO <- max_EI(fitted.model1, lower=lower, upper=upper, 
control=list(pop.size=20, BFGSburnin=2))
print(oEGO)

# graphics
n.grid <- 20
x.grid <- y.grid <- seq(0,1,length=n.grid)
design.grid <- expand.grid(x.grid, y.grid)
response.grid <- apply(design.grid, 1, camelback)
z.grid <- matrix(response.grid, n.grid, n.grid)
contour(x.grid,y.grid,z.grid,40)
title("Camelback Function")
points(design.fact[,1], design.fact[,2], pch=17, col="blue")
points(oEGO$par[1], oEGO$par[2], pch=19, col="red")
# }
# NOT RUN {
####################################################################
### "ONE-SHOT" EI-MAXIMIZATION OF THE GOLDSTEIN-PRICE FUNCTION #####
### 	     KNOWN AT 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="gen")

# EI maximization
library(rgenoud)
lower <- rep(0,d); upper <- rep(1,d);     # domain for Branin function
oEGO <- max_EI(fitted.model1, lower=lower, upper=upper, control
=list(pop.size=50, max.generations=50, wait.generations=5, BFGSburnin=10))
print(oEGO)

# graphics
n.grid <- 20
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$par[1], oEGO$par[2], pch=19, col="red")
# }
# NOT RUN {
# }