crit_EFI {DiceOptim}

Expected Feasible Improvement

Description

Computes the Expected Feasible Improvement at current location. The current feasible minimum of the observations can be replaced by an arbitrary value (plugin), which is usefull in particular in noisy frameworks.

Usage

crit_EFI(
  x,
  model.fun,
  model.constraint,
  equality = FALSE,
  critcontrol = NULL,
  plugin = NULL,
  type = "UK"
)

Arguments

x

a vector representing the input for which one wishes to calculate EFI,

model.fun

object of class km corresponding to the objective function, or, if the objective function is fast-to-evaluate, a fastfun object,

model.constraint

either one or a list of objects of class km, one for each constraint function,

equality

either FALSE if all constraints are for inequalities, else a vector of boolean indicating which are equalities,

critcontrol

optional list with argument tolConstraints, an optional vector giving a tolerance (> 0) for each of the constraints (equality or inequality). It is highly recommended to use it when there are equality constraints since the default tolerance of 0.05 in such case might not be suited.

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.

plugin

optional scalar: if provided, it replaces the feasible minimum of the current observations. If set to Inf, e.g. when their is no feasible solution, then the criterion is equal to the probability of feasibility,

type

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

Value

The Expected Feasible Improvement at x.

Author(s)

Victor Picheny

Mickael Binois

References

M. Schonlau, W.J. Welch, and D.R. Jones (1998), Global versus local search in constrained optimization of computer models, Lecture Notes-Monograph Series, 11-25.

M.J. Sasena, P. Papalambros, and P.Goovaerts (2002), Exploration of metamodeling sampling criteria for constrained global optimization, Engineering optimization, 34, 263-278.

See Also

EI from package DiceOptim, crit_AL, crit_SUR_cst.

Examples

#---------------------------------------------------------------------------
# Expected Feasible Improvement surface with one inequality constraint
#---------------------------------------------------------------------------

set.seed(25468)
library(DiceDesign)

n_var <- 2 
fun.obj <- goldsteinprice
fun.cst <- function(x){return(-branin(x) + 25)}
n.grid <- 51
test.grid <- expand.grid(X1 = seq(0, 1, length.out = n.grid), X2 = seq(0, 1, length.out = n.grid))
obj.grid <- apply(test.grid, 1, fun.obj)
cst.grid <- apply(test.grid, 1, fun.cst)
n.init <- 15 
design.grid <- round(maximinESE_LHS(lhsDesign(n.init, n_var, seed = 42)$design)$design, 1)
obj.init <- apply(design.grid, 1, fun.obj)
cst.init <- apply(design.grid, 1, fun.cst)
model.fun <- km(~., design = design.grid, response = obj.init)
model.constraint <- km(~., design = design.grid, response = cst.init)

EFI_grid <- apply(test.grid, 1, crit_EFI, model.fun = model.fun,
                  model.constraint = model.constraint)

filled.contour(seq(0, 1, length.out = n.grid), seq(0, 1, length.out = n.grid), nlevels = 50,
               matrix(EFI_grid, n.grid), main = "Expected Feasible 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")
                            contour(seq(0, 1, length.out = n.grid), seq(0, 1, length.out = n.grid), 
                            matrix(obj.grid, n.grid), nlevels = 10,
                                   add=TRUE,drawlabels=TRUE, col = "black")
                            contour(seq(0, 1, length.out = n.grid), seq(0, 1, length.out = n.grid), 
                            matrix(cst.grid, n.grid), level = 0, add=TRUE,
                                   drawlabels=FALSE,lwd=1.5, col = "red")
                            }
              )

#---------------------------------------------------------------------------
# Expected Feasible Improvement surface with one inequality and one equality constraint
#---------------------------------------------------------------------------

set.seed(25468)
library(DiceDesign)

n_var <- 2 
fun.obj <- goldsteinprice
fun.cstineq <- function(x){return(3/2 - x[1] - 2*x[2] - .5*sin(2*pi*(x[1]^2 - 2*x[2])))}
fun.csteq <- function(x){return(branin(x) - 25)}
n.grid <- 51
test.grid <- expand.grid(X1 = seq(0, 1, length.out = n.grid), X2 = seq(0, 1, length.out = n.grid))
obj.grid <- apply(test.grid, 1, fun.obj)
cstineq.grid <- apply(test.grid, 1, fun.cstineq)
csteq.grid <- apply(test.grid, 1, fun.csteq)
n.init <- 25 
design.grid <- round(maximinESE_LHS(lhsDesign(n.init, n_var, seed = 42)$design)$design, 1)
obj.init <- apply(design.grid, 1, fun.obj)
cstineq.init <- apply(design.grid, 1, fun.cstineq)
csteq.init <- apply(design.grid, 1, fun.csteq)
model.fun <- km(~., design = design.grid, response = obj.init)
model.constraintineq <- km(~., design = design.grid, response = cstineq.init)
model.constrainteq <- km(~., design = design.grid, response = csteq.init)

models.cst <- list(model.constraintineq, model.constrainteq)
 
EFI_grid <- apply(test.grid, 1, crit_EFI, model.fun = model.fun, model.constraint = models.cst,
                  equality = c(FALSE, TRUE), critcontrol = list(tolConstraints = c(0.05, 3)))

filled.contour(seq(0, 1, length.out = n.grid), seq(0, 1, length.out = n.grid), nlevels = 50,
               matrix(EFI_grid, n.grid), main = "Expected Feasible 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")
                            contour(seq(0, 1, length.out = n.grid), seq(0, 1, length.out = n.grid), 
                            matrix(obj.grid, n.grid), nlevels = 10,
                                   add=TRUE,drawlabels=TRUE, col = "black")
                            contour(seq(0, 1, length.out = n.grid), seq(0, 1, length.out = n.grid), 
                            matrix(cstineq.grid, n.grid), level = 0, add=TRUE,
                                   drawlabels=FALSE,lwd=1.5, col = "red")
                            contour(seq(0, 1, length.out = n.grid), seq(0, 1, length.out = n.grid),
                            matrix(csteq.grid, n.grid), level = 0, add=TRUE,
                                   drawlabels=FALSE,lwd=1.5, col = "orange")
                            }
              )