GPareto (version 1.1.8)

GParetoptim: Sequential multi-objective Expected Improvement maximization and model re-estimation, with a number of iterations fixed in advance by the user

Description

Executes nsteps iterations of multi-objective EGO methods to objects of class km. At each step, kriging models are 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 one of the four multi-objective Expected Improvement criteria available. Handles noiseless and noisy objective functions.

Usage

GParetoptim(
  model,
  fn,
  ...,
  cheapfn = NULL,
  crit = "SMS",
  nsteps,
  lower,
  upper,
  type = "UK",
  cov.reestim = TRUE,
  critcontrol = NULL,
  noise.var = NULL,
  reinterpolation = NULL,
  optimcontrol = list(method = "genoud", trace = 1),
  ncores = 1
)

Value

A list with components:

• par: a data frame representing the additional points visited during the algorithm,

• values: a data frame representing the response values at the points given in par,

• nsteps: an integer representing the desired number of iterations (given in argument),

• lastmodel: a list of objects of class km corresponding to the last kriging models fitted.

• observations.denoised: if noise.var!=NULL, a matrix representing the mean values of the km models at observation points. If a problem occurs during either model updates or criterion maximization, the last working model and corresponding values are returned.

Arguments

model

list of objects of class km, one for each objective functions,

fn

the multi-objective function to be minimized (vectorial output), found by a call to match.fun,

...

additional parameters to be given to the objective fn.

cheapfn

optional additional fast-to-evaluate objective function (handled next with class fastfun), which does not need a kriging model, handled by a call to match.fun,

crit

choice of multi-objective improvement function: "SMS", "EHI", "EMI" or "SUR", see details below,

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,

type

"SK" or "UK" (by default), depending whether uncertainty related to trend estimation has to be taken into account, see km

cov.reestim

optional boolean specifying if the kriging hyperparameters should be re-estimated at each iteration,

critcontrol

optional list of parameters for criterion crit, see details,

noise.var

noise variance (of the objective functions). Either NULL (noiseless objectives), a scalar (constant noise, identical for all objectives), a vector (constant noise, different for each objective) or a function (type closure) with vectorial output (variable noise, different for each objective). Alternatively, set noise.var="given_by_fn", see details. If not provided but km models are based on noisy observations, noise.var is taken as the average of [e-mail].

reinterpolation

Boolean: for noisy problems, indicates whether a reinterpolation model is used, see details,

optimcontrol

an optional list of control parameters for optimization of the selected infill criterion: "method" can be set to "discrete", "pso", "genoud" or a user defined method name (passed to match.fun). For "discrete", a matrix candidate.points must be given. For "pso" and "genoud", specific parameters to the chosen method can also be specified (see genoud and psoptim). A user defined method must have arguments like the default optim method, i.e. par, fn, lower, upper, ... and eventually control.
A trace trace argument is available, it can be set to 0 to suppress all messages, to 1 (default) for displaying the optimization progresses, and >1 for the highest level of details.

ncores

number of CPU available (> 1 makes mean parallel TRUE). Only used with discrete optimization for now.

Details

Extension of the function EGO.nsteps for multi-objective optimization.
Available infill criteria with crit are:

• Expected Hypervolume Improvement (EHI) crit_EHI,

• SMS criterion (SMS) crit_SMS,

• Expected Maximin Improvement (EMI) crit_EMI,

• Stepwise Uncertainty Reduction of the excursion volume (SUR) crit_SUR.

Depending on the selected criterion, parameters such as reference point for SMS and EHI or arguments for integration_design_optim with SUR can be given with critcontrol. Also options for checkPredict are available. More precisions are given in the corresponding help pages.

The reinterpolation=TRUE setting can be used to handle noisy objective functions. It works with all criteria and is the recommended option. If reinterpolation=FALSE and noise.var!=NULL, the criteria are used based on a "denoised" Pareto front.

If noise.var="given_by_fn", fn must return a list of two vectors, the first being the objective functions and the second the corresponding noise variances (see examples).

Display of results and various post-processings are available with plotGPareto.

References

M. T. Emmerich, A. H. Deutz, J. W. Klinkenberg (2011), Hypervolume-based expected improvement: Monotonicity properties and exact computation, Evolutionary Computation (CEC), 2147-2154.

V. Picheny (2014), Multiobjective optimization using Gaussian process emulators via stepwise uncertainty reduction, Statistics and Computing, 25(6), 1265-1280

T. Wagner, M. Emmerich, A. Deutz, W. Ponweiser (2010), On expected-improvement criteria for model-based multi-objective optimization. Parallel Problem Solving from Nature, 718-727, Springer, Berlin.

J. D. Svenson (2011), Computer Experiments: Multiobjective Optimization and Sensitivity Analysis, Ohio State university, PhD thesis. V. Picheny and D. Ginsbourger (2013), Noisy kriging-based optimization methods: A unified implementation within the DiceOptim package, Computational Statistics & Data Analysis, 71: 1035-1053.

Examples

set.seed(25468)
library(DiceDesign)

################################################
# NOISELESS PROBLEMS
################################################
d <- 2 
fname <- ZDT3
n.grid <- 21
test.grid <- expand.grid(seq(0, 1, length.out = n.grid), seq(0, 1, length.out = n.grid))
nappr <- 15 
design.grid <- maximinESE_LHS(lhsDesign(nappr, d, seed = 42)$design)$design
response.grid <- t(apply(design.grid, 1, fname))
Front_Pareto <- t(nondominated_points(t(response.grid)))

mf1 <- km(~1, design = design.grid, response = response.grid[, 1], lower=c(.1,.1))
mf2 <- km(~., design = design.grid, response = response.grid[, 2], lower=c(.1,.1))
model <- list(mf1, mf2)

nsteps <- 2
lower <- rep(0, d)
upper <- rep(1, d)

# Optimization 1: EHI with pso
optimcontrol <- list(method = "pso", maxit = 20)
critcontrol <- list(refPoint = c(1, 10))
omEGO1 <- GParetoptim(model = model, fn = fname, crit = "EHI", nsteps = nsteps,
                     lower = lower, upper = upper, critcontrol = critcontrol,
                     optimcontrol = optimcontrol)
print(omEGO1$par)
print(omEGO1$values)

if (FALSE) {
nsteps <- 10
# Optimization 2: SMS with discrete search
optimcontrol <- list(method = "discrete", candidate.points = test.grid)
critcontrol <- list(refPoint = c(1, 10))
omEGO2 <- GParetoptim(model = model, fn = fname, crit = "SMS", nsteps = nsteps,
                     lower = lower, upper = upper, critcontrol = critcontrol,
                     optimcontrol = optimcontrol)
print(omEGO2$par)
print(omEGO2$values)

# Optimization 3: SUR with genoud
optimcontrol <- list(method = "genoud", pop.size = 20, max.generations = 10)
critcontrol <- list(distrib = "SUR", n.points = 100)
omEGO3 <- GParetoptim(model = model, fn = fname, crit = "SUR", nsteps = nsteps,
                     lower = lower, upper = upper, critcontrol = critcontrol,
                     optimcontrol = optimcontrol)
print(omEGO3$par)
print(omEGO3$values)

# Optimization 4: EMI with pso
optimcontrol <- list(method = "pso", maxit = 20)
critcontrol <- list(nbsamp = 200)
omEGO4 <- GParetoptim(model = model, fn = fname, crit = "EMI", nsteps = nsteps,
                     lower = lower, upper = upper, optimcontrol = optimcontrol)
print(omEGO4$par)
print(omEGO4$values)

# graphics
sol.grid <- apply(expand.grid(seq(0, 1, length.out = 100),
                              seq(0, 1, length.out = 100)), 1, fname)
plot(t(sol.grid), pch = 20, col = rgb(0, 0, 0, 0.05), xlim = c(0, 1),
     ylim = c(-2, 10), xlab = expression(f[1]), ylab = expression(f[2]))
plotGPareto(res = omEGO1, add = TRUE,
            control = list(pch = 20, col = "blue", PF.pch = 17,
                           PF.points.col = "blue", PF.line.col = "blue"))
text(omEGO1$values[,1], omEGO1$values[,2], labels = 1:nsteps, pos = 3, col = "blue")
plotGPareto(res = omEGO2, add = TRUE,
            control = list(pch = 20, col = "green", PF.pch = 17,
                           PF.points.col = "green", PF.line.col = "green"))
text(omEGO2$values[,1], omEGO2$values[,2], labels = 1:nsteps, pos = 3, col = "green")
plotGPareto(res = omEGO3, add = TRUE,
            control = list(pch = 20, col = "red", PF.pch = 17,
                           PF.points.col = "red", PF.line.col = "red"))
text(omEGO3$values[,1], omEGO3$values[,2], labels = 1:nsteps, pos = 3, col = "red") 
plotGPareto(res = omEGO4, add = TRUE,
            control = list(pch = 20, col = "orange", PF.pch = 17,
                           PF.points.col = "orange", PF.line.col = "orange"))
text(omEGO4$values[,1], omEGO4$values[,2], labels = 1:nsteps, pos = 3, col = "orange")
points(response.grid[,1], response.grid[,2], col = "black", pch = 20)
legend("topright", c("EHI", "SMS", "SUR", "EMI"), col = c("blue", "green", "red", "orange"),
 pch = rep(17,4))
 
 
# Post-processing
plotGPareto(res = omEGO1, UQ_PF = TRUE, UQ_PS = TRUE, UQ_dens = TRUE)

################################################
# NOISY PROBLEMS
################################################
set.seed(25468)
library(DiceDesign)
d <- 2 
nsteps <- 3
lower <- rep(0, d)
upper <- rep(1, d)
optimcontrol <- list(method = "pso", maxit = 20)
critcontrol <- list(refPoint = c(1, 10))

n.grid <- 21
test.grid <- expand.grid(seq(0, 1, length.out = n.grid), seq(0, 1, length.out = n.grid))
n.init <- 30
design <- maximinESE_LHS(lhsDesign(n.init, d, seed = 42)$design)$design

fit.models <- function(u) km(~., design = design, response = response[, u],
                             noise.var=design.noise.var[,u])

# Test 1: EHI, constant noise.var
noise.var <- c(0.1, 0.2)
funnoise1 <- function(x) {ZDT3(x) + sqrt(noise.var)*rnorm(n=d)}
response <- t(apply(design, 1, funnoise1))
design.noise.var <- matrix(rep(noise.var, n.init), ncol=d, byrow=TRUE)
model <- lapply(1:d, fit.models)

omEGO1 <- GParetoptim(model = model, fn = funnoise1, crit = "EHI", nsteps = nsteps,
                      lower = lower, upper = upper, critcontrol = critcontrol,
                      reinterpolation=TRUE, noise.var=noise.var, optimcontrol = optimcontrol)
plotGPareto(omEGO1)

# Test 2: EMI, noise.var given by fn
funnoise2 <- function(x) {list(ZDT3(x) + sqrt(0.05 + abs(0.1*x))*rnorm(n=d), 0.05 + abs(0.1*x))}
temp <- funnoise2(design)
response <- temp[[1]]
design.noise.var <- temp[[2]]
model <- lapply(1:d, fit.models)

omEGO2 <- GParetoptim(model = model, fn = funnoise2, crit = "EMI", nsteps = nsteps,
                      lower = lower, upper = upper, critcontrol = critcontrol,
                      reinterpolation=TRUE, noise.var="given_by_fn", optimcontrol = optimcontrol)
plotGPareto(omEGO2)

# Test 3: SMS, functional noise.var
funnoise3 <- function(x) {ZDT3(x) + sqrt(0.025 + abs(0.05*x))*rnorm(n=d)}
noise.var <- function(x) return(0.025 + abs(0.05*x))
response <- t(apply(design, 1, funnoise3))
design.noise.var <- t(apply(design, 1, noise.var))
model <- lapply(1:d, fit.models)

omEGO3 <- GParetoptim(model = model, fn = funnoise3, crit = "SMS", nsteps = nsteps,
                           lower = lower, upper = upper, critcontrol = critcontrol,
                           reinterpolation=TRUE, noise.var=noise.var, optimcontrol = optimcontrol)
plotGPareto(omEGO3)

# Test 4: SUR, fastfun, constant noise.var
noise.var <- 0.1
funnoise4 <- function(x) {ZDT3(x)[1] + sqrt(noise.var)*rnorm(1)}
cheapfn <- function(x) ZDT3(x)[2]
response <- apply(design, 1, funnoise4)
design.noise.var <- rep(noise.var, n.init)
model <- list(km(~., design = design, response = response, noise.var=design.noise.var))

omEGO4 <- GParetoptim(model = model, fn = funnoise4, cheapfn = cheapfn, crit = "SUR", 
                      nsteps = nsteps, lower = lower, upper = upper, critcontrol = critcontrol,
                      reinterpolation=TRUE, noise.var=noise.var, optimcontrol = optimcontrol)
 plotGPareto(omEGO4)
                            
 # Test 5: EMI, fastfun, noise.var given by fn
 funnoise5 <- function(x) {
   if (is.null(dim(x))) x <- matrix(x, nrow=1)
   list(apply(x, 1, ZDT3)[1,] + sqrt(abs(0.05*x[,1]))*rnorm(nrow(x)), abs(0.05*x[,1]))
 }
 
 cheapfn <- function(x) {
   if (is.null(dim(x))) x <- matrix(x, nrow=1)
   apply(x, 1, ZDT3)[2,]
 }
 
 temp <- funnoise5(design)
 response <- temp[[1]]
 design.noise.var <- temp[[2]]
 model <- list(km(~., design = design, response = response, noise.var=design.noise.var))
 
 omEGO5 <- GParetoptim(model = model, fn = funnoise5, cheapfn = cheapfn, crit = "EMI", 
                       nsteps = nsteps, lower = lower, upper = upper, critcontrol = critcontrol,
                       reinterpolation=TRUE, noise.var="given_by_fn", optimcontrol = optimcontrol)
 plotGPareto(omEGO5)               
 
 # Test 6: EHI, fastfun, functional noise.var
 noise.var <- 0.1
 funnoise6 <- function(x) {ZDT3(x)[1] + sqrt(abs(0.1*x[1]))*rnorm(1)}
 noise.var <- function(x) return(abs(0.1*x[1]))
 cheapfn <- function(x) ZDT3(x)[2]
 response <- apply(design, 1, funnoise6)
 design.noise.var <- t(apply(design, 1, noise.var))
 model <- list(km(~., design = design, response = response, noise.var=design.noise.var))
 
 omEGO6 <- GParetoptim(model = model, fn = funnoise6, cheapfn = cheapfn, crit = "EMI", 
                       nsteps = nsteps, lower = lower, upper = upper, critcontrol = critcontrol,
                       reinterpolation=TRUE, noise.var=noise.var, optimcontrol = optimcontrol)
 plotGPareto(omEGO6)    
}