| max_EI {DiceOptim} | R Documentation |
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 |
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 |
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.
Value
A list with components:
par |
The best set of parameters found. |
value |
The value of expected improvement at par. |
Author(s)
David Ginsbourger
Olivier Roustant
Victor Picheny
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
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")
## End(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")
## End(Not run)