MEGO: MEGO: Multi-Objective Efficient Global Optimization Algorithm...
In coldfir3/moko: Multi-Objective Kriging Optimization

Description Usage Arguments Details Value References Examples

Executes nsteps iterations of the MEGO method to an object of class mkm. At each step, a weighted kriging model is 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 the Constrained Expected Improvement criterion (EI).

1 2	MEGO(model, fun, nsteps, lower = rep(0, model@d), upper = rep(1, model@d), quiet = TRUE, control = NULL, optimcontrol = NULL)

`model`	An object of class `mkm`. This `model` must have a single objective (`model@m == 1`).
`fun`	The multi-objective and constraint cost function to be optimized. This function must return a vector with the size of `model@m + model@j` where `model@m` are the number of objectives and `model@j` the number of the constraints,
`nsteps`	An integer representing the desired number of iterations,
`lower`	Vector of lower bounds for the variables to be optimized over (default: 0 with length = `model@d`),
`upper`	Vector of upper bounds for the variables to be optimized over (default: 1 with length = `model@d`),
`quiet`	Logical indicating the verbosity of the routine,
`control`	An optional list of control parameters, some of them passed to the `EI` function. One can control: `minimization` logical specifying if EI is used in minimiziation or in maximization (default: `TRUE`) `plugin` optional scalar, if not provided, the minimum (or maximum) of the current feasible observations. If there isn't any feasible design plugin is set to `NA` and the algorithm returns the value of the probabilty of constraints be met. `envir` optional enviroment specifying where to assign intermediate values. Default: `NULL`.
`optimcontrol`	Optional list of control parameters passed to the `GenSA` function. Please, note that the values are passed as the `control` parameter inside the `GenSA` function (`genSA(control = optimcontrol)`).

Note that since MEGO is works by scalarizing a cost function, this technique is well suited for single objective problems with multiple constraints.

updated mkm model

Knowles, J. (2006). ParEGO: a hybrid algorithm with on-line landscape approximation for expensive multiobjective optimization problems. IEEE Transactions on Evolutionary Computation, 10(1), 50-66.

# ----------------
# The Nowacki Beam
# ----------------
n <- 20
d <- 2
nsteps <- 1 # value has been set to 1 to save compliation time, change this value to 40.
fun <- nowacki_beam
doe <- replicate(d,sample(0:n,n))/n
res <- t(apply(doe, 1, fun))
model <- mkm(doe, res, modelcontrol = list(objective = 1:2, lower = rep(0.1,d)))
model <- MEGO(model, fun, nsteps, quiet = FALSE, control = list(rho = 0.1))
plot(nowacki_beam_tps$set)
points(ps(model@response[which(model@feasible),model@objective])$set, col = 'green', pch = 19)

############################################
#### some single objective optimization ####
############################################
## Not run: 
## Those examples are flagged as "don't run" only to save compilation time. ##
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, DiceKriging::branin)
z.grid <- matrix(response.grid, n.grid, n.grid)

# -----------------------------------
# Branin-Hoo function (unconstrained)
# -----------------------------------
n <- 10
d <- 2
doe <- replicate(d,sample(0:n,n))/n
fun <- DiceKriging::branin
res <- apply(doe, 1, fun)
model <- mkm(doe, res, modelcontrol = list(lower=rep(0.1,d)))
model <- MEGO(model, fun, 10, quiet = FALSE)
contour(x.grid,y.grid,z.grid,40)
points(model@design, col=ifelse(model@feasible,'blue','red'))
# ---------------------------------------
# Branin-Hoo function (simple constraint)
# ---------------------------------------
n <- 10
d <- 2
doe <- replicate(d,sample(0:n,n))/n
fun_cost <- DiceKriging::branin
fun_cntr <- function(x) 0.2 - prod(x)
fun <- function(x) return(c(fun_cost(x),fun_cntr(x)))
res <- t(apply(doe, 1, fun))
model <- mkm(doe, res, modelcontrol = list(objective = 1, lower=rep(0.1,d)))
model <- MEGO(model, fun, 10, quiet = FALSE)
contour(x.grid,y.grid,z.grid,40)
points(model@design, col=ifelse(model@feasible,'blue','red'))
# ---------------------------------------
# Branin-Hoo function (narrow constraint)
# ---------------------------------------
n <- 10
d <- 2
doe <- replicate(d,sample(0:n,n))/n
fun_cost <- DiceKriging::branin
fun_cntr <- function(x){
 g1 <- 0.9 - sum(x)
 g2 <- sum(x) - 1.1
 g3 <- - x[1] + 0.75
 g4 <- x[2] - 0.25
 return(c(g1,g2,g3,g4))
}
fun <- function(x) return(c(fun_cost(x),fun_cntr(x)))
res <- t(apply(doe, 1, fun))
model <- mkm(doe, res, modelcontrol = list(objective = 1, lower=rep(0.1,d)))
model <- MEGO(model, fun, 10, quiet = FALSE)
contour(x.grid,y.grid,z.grid,40)
points(model@design, col=ifelse(model@feasible,'blue','red'))
# ---------------------------------------------
# Branin-Hoo function (disconnected constraint)
# ---------------------------------------------
n <- 10
d <- 2
doe <- replicate(d,sample(0:n,n))/n
Griewank <-  function(x) {
 ii <- c(1:length(x))
  sum <- sum(x^2/4000)
  prod <- prod(cos(x/sqrt(ii)))
  y <- sum - prod + 1
  return(y)
}
fun_cost <- DiceKriging::branin
fun_cntr <- function(x) 1.6 - Griewank(x*10-5)
fun <- function(x) return(c(fun_cost(x),fun_cntr(x)))
res <- t(apply(doe, 1, fun))
model <- mkm(doe, res, modelcontrol = list(objective = 1, lower=c(0.1,0.1)))
model <- MEGO(model, fun, 10, quiet = FALSE)
contour(x.grid,y.grid,z.grid,40)
points(model@design, col=ifelse(model@feasible,'blue','red'))

## End(Not run)