crit_optimizer {GPareto}

Maximization of multiobjective infill criterion

Description

Given a list of objects of class km and a set of tuning parameters (lower, upper and critcontrol), crit_optimizer performs the maximization of an infill criterion and delivers the next point to be visited in a multi-objective EGO-like procedure.

The latter maximization relies either on a genetic algorithm using derivatives, genoud, particle swarm algorithm pso-package, exhaustive search at pre-specified points or on a user defined method. It is important to remark that the information needed about the objective function reduces here to the vector of response values embedded in the models (no call to the objective functions or simulators (except with cheapfn)).

Usage

crit_optimizer(
  crit = "SMS",
  model,
  lower,
  upper,
  cheapfn = NULL,
  type = "UK",
  paretoFront = NULL,
  critcontrol = NULL,
  optimcontrol = NULL,
  ncores = 1
)

Arguments

crit

sampling criterion. Four choices are available : "SMS", "EHI", "EMI" and "SUR",

model

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

lower

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

upper

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

cheapfn

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

type

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

paretoFront

(optional) matrix corresponding to the Pareto front of size [n.pareto x n.obj], or any reference set of observations,

critcontrol

optional list of control parameters for criterion crit, see details. Options for the checkPredict function: threshold (1e-4) and distance (covdist) are used to avoid numerical issues occuring when adding points too close to the existing ones.

optimcontrol

optional list of control parameters for optimization of the selected infill criterion. "method" set the optimization method; one can choose between "discrete", "pso" and "genoud" or a user defined method name (passed to match.fun). For each method, further parameters can be set.
For "discrete", one has to provide the argument "candidate.points".
For "pso", one can control the maximum number of iterations "maxit" (400) and the population size "s" (default : max(20, floor(10+2*sqrt(dim))) (see psoptim).
For "genoud", one can control, among others, "pop.size" (default : [N = 3*2^dim for dim < 6 and N = 32*dim otherwise]), "max.generations" (12), "wait.generations" (2), "BFGSburnin" (2), BFGSmaxit (N) and solution.tolerance (1e-21) of function "genoud" (see genoud). Numbers into brackets are the default values.
For a user defined method, it must have arguments like the default optim method, i.e. par, fn, lower, upper, ... and possibly control, and return a list with par and value. 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 max_EI 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 a 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.

Value

A list with components:

• par: The best set of parameters found,

• value: The value of expected improvement at par.

References

W.R. Jr. Mebane and J.S. Sekhon (2011), Genetic optimization using derivatives: The rgenoud package for R, Journal of Statistical Software, 42(11), 1-26 doi:10.18637/jss.v042.i11

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.

Examples

## Not run: 
#---------------------------------------------------------------------------
# EHI surface associated with the "P1" problem at a 15 points design
#---------------------------------------------------------------------------

set.seed(25468)
library(DiceDesign)

d <- 2
n.obj <- 2 
fname <- "P1" 
n.grid <- 51
test.grid <- expand.grid(seq(0, 1, length.out = n.grid), seq(0, 1, length.out = n.grid))
nappr <- 15 
design.grid <- round(maximinESE_LHS(lhsDesign(nappr, d, seed = 42)$design)$design, 1)
response.grid <- t(apply(design.grid, 1, fname))
paretoFront <- t(nondominated_points(t(response.grid)))
mf1 <- km(~., design = design.grid, response = response.grid[,1])
mf2 <- km(~., design = design.grid, response = response.grid[,2])
model <- list(mf1, mf2)

EHI_grid <- apply(test.grid, 1, crit_EHI, model = list(mf1, mf2),
                  critcontrol = list(refPoint = c(300, 0)))

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

omEGO <- crit_optimizer(crit = "EHI", model = model,  lower = lower, upper = upper, 
                 optimcontrol = list(method = "genoud", pop.size = 200, BFGSburnin = 2),
                 critcontrol = list(refPoint = c(300, 0)))
                 
print(omEGO)
 
filled.contour(seq(0, 1, length.out = n.grid), seq(0, 1, length.out = n.grid), nlevels = 50,
               matrix(EHI_grid, nrow = n.grid), main = "Expected Hypervolume Improvement", 
               xlab = expression(x[1]), ylab = expression(x[2]), color = terrain.colors,
               plot.axes = {axis(1); axis(2);
                            points(design.grid[, 1], design.grid[, 2], pch = 21, bg = "white");
                            points(omEGO$par, col = "red", pch = 4)
                            }
              )
              
#---------------------------------------------------------------------------
# SMS surface associated with the "P1" problem at a 15 points design
#---------------------------------------------------------------------------


SMS_grid <- apply(test.grid, 1, crit_SMS, model = model,
                  critcontrol = list(refPoint = c(300, 0)))

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

omEGO2 <- crit_optimizer(crit = "SMS", model = model,  lower = lower, upper = upper, 
                 optimcontrol = list(method="genoud", pop.size = 200, BFGSburnin = 2),
                 critcontrol = list(refPoint = c(300, 0)))
                 
print(omEGO2)
 
filled.contour(seq(0, 1, length.out = n.grid), seq(0, 1, length.out = n.grid), nlevels = 50,
               matrix(pmax(0,SMS_grid), nrow = n.grid), main = "SMS Criterion (>0)",
               xlab = expression(x[1]), ylab = expression(x[2]), color = terrain.colors,
               plot.axes = {axis(1); axis(2);
                            points(design.grid[, 1], design.grid[, 2], pch = 21, bg = "white");
                            points(omEGO2$par, col = "red", pch = 4)
                            }
              )
#---------------------------------------------------------------------------
# Maximin Improvement surface associated with the "P1" problem at a 15 points design
#---------------------------------------------------------------------------


EMI_grid <- apply(test.grid, 1, crit_EMI, model = model,
                  critcontrol = list(nb_samp = 20, type ="EMI"))

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

omEGO3 <- crit_optimizer(crit = "EMI", model = model,  lower = lower, upper = upper, 
                 optimcontrol = list(method = "genoud", pop.size = 200, BFGSburnin = 2))
                 
print(omEGO3)
 
filled.contour(seq(0, 1, length.out = n.grid), seq(0, 1, length.out = n.grid), nlevels = 50,
               matrix(EMI_grid, nrow = n.grid), main = "Expected Maximin Improvement",
               xlab = expression(x[1]), ylab = expression(x[2]), color = terrain.colors, 
               plot.axes = {axis(1);axis(2);
                            points(design.grid[, 1], design.grid[, 2], pch = 21, bg = "white");
                            points(omEGO3$par, col = "red", pch = 4)
                            }
              )
#---------------------------------------------------------------------------
# crit_SUR surface associated with the "P1" problem at a 15 points design
#---------------------------------------------------------------------------
library(KrigInv)

integration.param <- integration_design_optim(lower = c(0, 0), upper = c(1, 1), model = model)
integration.points <- as.matrix(integration.param$integration.points)
integration.weights <- integration.param$integration.weights

precalc.data <- list()
mn.X <- sn.X <- matrix(0, n.obj, nrow(integration.points))

for (i in 1:n.obj){
  p.tst.all <- predict(model[[i]], newdata = integration.points, type = "UK",
                       checkNames = FALSE)
  mn.X[i,] <- p.tst.all$mean
  sn.X[i,]   <- p.tst.all$sd
  precalc.data[[i]] <- precomputeUpdateData(model[[i]], integration.points)
}
critcontrol <- list(integration.points = integration.points,
                    integration.weights = integration.weights,
                    mn.X = mn.X, sn.X = sn.X, precalc.data = precalc.data)
EEV_grid <- apply(test.grid, 1, crit_SUR, model=model, paretoFront = paretoFront,
                  critcontrol = critcontrol)
lower <- rep(0, d)
upper <- rep(1, d)
                  
omEGO4 <- crit_optimizer(crit = "SUR", model = model,  lower = lower, upper = upper, 
                 optimcontrol = list(method = "genoud", pop.size = 200, BFGSburnin = 2))
print(omEGO4)

filled.contour(seq(0, 1, length.out = n.grid), seq(0, 1, length.out = n.grid),
               matrix(pmax(0,EEV_grid), n.grid), main = "EEV criterion", nlevels = 50,
               xlab = expression(x[1]), ylab = expression(x[2]), color = terrain.colors,
               plot.axes = {axis(1); axis(2);
                            points(design.grid[,1], design.grid[,2], pch = 21, bg = "white")
                            points(omEGO4$par, col = "red", pch = 4)
                            }
              )

# example using user defined optimizer, here L-BFGS-B from base optim
userOptim <- function(par, fn, lower, upper, control, ...){
  return(optim(par = par, fn = fn, method = "L-BFGS-B", lower = lower, upper = upper,
         control = control, ...))
}
omEGO4bis <- crit_optimizer(crit = "SUR", model = model,  lower = lower, upper = upper, 
                 optimcontrol = list(method = "userOptim"))
print(omEGO4bis)


#---------------------------------------------------------------------------
# crit_SMS surface with problem "P1" with 15 design points, using cheapfn
#---------------------------------------------------------------------------

# Optimization with fastfun: SMS with discrete search
# Separation of the problem P1 in two objectives: 
# the first one to be kriged, the second one with fastobj

# Definition of the fastfun
f2 <-   function(x){
  return(P1(x)[2])
}
 
SMS_grid_cheap <- apply(test.grid, 1, crit_SMS, model = list(mf1, fastfun(f2, design.grid)),
                        paretoFront = paretoFront, critcontrol = list(refPoint = c(300, 0)))


optimcontrol <- list(method = "pso")
model2 <- list(mf1)
omEGO5 <- crit_optimizer(crit = "SMS", model = model2,  lower = lower, upper = upper,
                         cheapfn = f2, critcontrol = list(refPoint = c(300, 0)),
                         optimcontrol = list(method = "genoud", pop.size = 200, BFGSburnin = 2))
print(omEGO5)

filled.contour(seq(0, 1, length.out = n.grid), seq(0, 1, length.out = n.grid), 
               matrix(pmax(0, SMS_grid_cheap), nrow = n.grid), nlevels = 50,
               main = "SMS criterion with cheap 2nd objective (>0)", xlab = expression(x[1]),
               ylab = expression(x[2]), color = terrain.colors,
               plot.axes = {axis(1); axis(2);
                            points(design.grid[,1], design.grid[,2], pch = 21, bg = "white")
                            points(omEGO5$par, col = "red", pch = 4)
                            }
              )

## End(Not run)