Short Description

caRamel is a multiobjective evolutionary algorithm combining the MEAS algorithm and the NGSA-II algorithm.

Download the package from CRAN or GitHub and then install and load it.

library(caRamel)

## Loading required package: geometry

## Loading required package: parallel

## Package 'caRamel' version 1.4

Test functions

Schaffer

Schaffer test function has two objectives with one variable.

schaffer <- function(i) {
  if (x[i,1] <= 1) {
    s1 <- -x[i,1]
  } else if (x[i,1] <= 3) {
    s1 <- x[i,1] - 2
  } else if (x[i,1] <= 4) {
    s1 <- 4 - x[i,1]
  } else {
    s1 <- x[i,1] - 4
  }
  s2 <- (x[i,1] - 5) * (x[i,1] - 5)
  return(c(s1, s2))
}

Note that :

parameter i is mandatory for the management of parallelism.
the variable must be named x and is a matrix of size [npopulation, nvariables].

The variable lies in the range [-5, 10]:

nvar <- 1 # number of variables
bounds <- matrix(data = 1, nrow = nvar, ncol = 2) # upper and lower bounds
bounds[, 1] <- -5 * bounds[, 1]
bounds[, 2] <- 10 * bounds[, 2]

Both functions are to be minimized:

nobj <- 2 # number of objectives
minmax <- c(FALSE, FALSE) # min and min

Before calling caRamel in order to optimize the Schaffer’s problem, some algorithmic parameters need to be set:

popsize <- 100 # size of the genetic population
archsize <- 100 # size of the archive for the Pareto front
maxrun <- 1000 # maximum number of calls
prec <- matrix(1.e-3, nrow = 1, ncol = nobj) # accuracy for the convergence phase

Then the minimization problem can be launched:

results <-
  caRamel(nobj,
          nvar,
          minmax,
          bounds,
          schaffer,
          popsize,
          archsize,
          maxrun,
          prec,
          carallel=FALSE) # no parallelism

## Beginning of caRamel optimization <-- Mon Nov 11 07:26:39 2024

## Number of variables : 1

## Number of functions : 2

## Warning in delaunayn(obj): 1 points missing from triangulation.
## It is possible that setting the 'options' argument of delaunayn may help.
## For example:
## options = "Qt Qc Qz Qbb"
## options = "Qt Qc Qz QbB"
## If these options do not work, try shifting the centre of the points
## to the origin by subtracting the mean coordinates from every point.

## Warning in delaunayn(obj): 1 points missing from triangulation.
## It is possible that setting the 'options' argument of delaunayn may help.
## For example:
## options = "Qt Qc Qz Qbb"
## options = "Qt Qc Qz QbB"
## If these options do not work, try shifting the centre of the points
## to the origin by subtracting the mean coordinates from every point.

## Warning in delaunayn(obj): 1 points missing from triangulation.
## It is possible that setting the 'options' argument of delaunayn may help.
## For example:
## options = "Qt Qc Qz Qbb"
## options = "Qt Qc Qz QbB"
## If these options do not work, try shifting the centre of the points
## to the origin by subtracting the mean coordinates from every point.

## Warning in delaunayn(obj): 4 points missing from triangulation.
## It is possible that setting the 'options' argument of delaunayn may help.
## For example:
## options = "Qt Qc Qz Qbb"
## options = "Qt Qc Qz QbB"
## If these options do not work, try shifting the centre of the points
## to the origin by subtracting the mean coordinates from every point.

## Warning in delaunayn(obj): 1 points missing from triangulation.
## It is possible that setting the 'options' argument of delaunayn may help.
## For example:
## options = "Qt Qc Qz Qbb"
## options = "Qt Qc Qz QbB"
## If these options do not work, try shifting the centre of the points
## to the origin by subtracting the mean coordinates from every point.

## Warning in delaunayn(obj): 1 points missing from triangulation.
## It is possible that setting the 'options' argument of delaunayn may help.
## For example:
## options = "Qt Qc Qz Qbb"
## options = "Qt Qc Qz QbB"
## If these options do not work, try shifting the centre of the points
## to the origin by subtracting the mean coordinates from every point.

## Warning in delaunayn(obj): 1 points missing from triangulation.
## It is possible that setting the 'options' argument of delaunayn may help.
## For example:
## options = "Qt Qc Qz Qbb"
## options = "Qt Qc Qz QbB"
## If these options do not work, try shifting the centre of the points
## to the origin by subtracting the mean coordinates from every point.

## Warning in delaunayn(obj): 1 points missing from triangulation.
## It is possible that setting the 'options' argument of delaunayn may help.
## For example:
## options = "Qt Qc Qz Qbb"
## options = "Qt Qc Qz QbB"
## If these options do not work, try shifting the centre of the points
## to the origin by subtracting the mean coordinates from every point.

## Warning in delaunayn(obj): 1 points missing from triangulation.
## It is possible that setting the 'options' argument of delaunayn may help.
## For example:
## options = "Qt Qc Qz Qbb"
## options = "Qt Qc Qz QbB"
## If these options do not work, try shifting the centre of the points
## to the origin by subtracting the mean coordinates from every point.

## Warning in delaunayn(obj): 1 points missing from triangulation.
## It is possible that setting the 'options' argument of delaunayn may help.
## For example:
## options = "Qt Qc Qz Qbb"
## options = "Qt Qc Qz QbB"
## If these options do not work, try shifting the centre of the points
## to the origin by subtracting the mean coordinates from every point.

## Warning in delaunayn(obj): 1 points missing from triangulation.
## It is possible that setting the 'options' argument of delaunayn may help.
## For example:
## options = "Qt Qc Qz Qbb"
## options = "Qt Qc Qz QbB"
## If these options do not work, try shifting the centre of the points
## to the origin by subtracting the mean coordinates from every point.

## Warning in delaunayn(obj): 1 points missing from triangulation.
## It is possible that setting the 'options' argument of delaunayn may help.
## For example:
## options = "Qt Qc Qz Qbb"
## options = "Qt Qc Qz QbB"
## If these options do not work, try shifting the centre of the points
## to the origin by subtracting the mean coordinates from every point.

## Warning in delaunayn(obj): 1 points missing from triangulation.
## It is possible that setting the 'options' argument of delaunayn may help.
## For example:
## options = "Qt Qc Qz Qbb"
## options = "Qt Qc Qz QbB"
## If these options do not work, try shifting the centre of the points
## to the origin by subtracting the mean coordinates from every point.

## Warning in delaunayn(obj): 2 points missing from triangulation.
## It is possible that setting the 'options' argument of delaunayn may help.
## For example:
## options = "Qt Qc Qz Qbb"
## options = "Qt Qc Qz QbB"
## If these options do not work, try shifting the centre of the points
## to the origin by subtracting the mean coordinates from every point.

## Warning in delaunayn(obj): 1 points missing from triangulation.
## It is possible that setting the 'options' argument of delaunayn may help.
## For example:
## options = "Qt Qc Qz Qbb"
## options = "Qt Qc Qz QbB"
## If these options do not work, try shifting the centre of the points
## to the origin by subtracting the mean coordinates from every point.

## Warning in delaunayn(obj): 1 points missing from triangulation.
## It is possible that setting the 'options' argument of delaunayn may help.
## For example:
## options = "Qt Qc Qz Qbb"
## options = "Qt Qc Qz QbB"
## If these options do not work, try shifting the centre of the points
## to the origin by subtracting the mean coordinates from every point.

## Done in 3.01017665863037 secs --> Mon Nov 11 07:26:42 2024

## Size of the Pareto front : 100

## Number of calls : 1020

Test if the convergence is successful:

print(results$success==TRUE)

## [1] TRUE

Plot the Pareto front:

plot(results$objectives[,1], results$objectives[,2], main="Schaffer Pareto front", xlab="Objective #1", ylab="Objective #2")

plot(results$parameters, main="Corresponding values for X", xlab="Element of the archive", ylab="X Variable")

Kursawe

Kursawe test function has two objectives of three variables.

kursawe <- function(i) {
  k1 <- -10 * exp(-0.2 * sqrt(x[i,1] ^ 2 + x[i,2] ^ 2)) - 10 * exp(-0.2 * sqrt(x[i,2] ^2 + x[i,3] ^ 2))
  k2 <- abs(x[i,1]) ^ 0.8 + 5 * sin(x[i,1] ^ 3) + abs(x[i,2]) ^ 0.8 + 5 * sin(x[i,2] ^3) + abs(x[i,3]) ^ 0.8 + 5 * sin(x[i,3] ^ 3)
  return(c(k1, k2))
}

The variables lie in the range [-5, 5]:

nvar <- 3 # number of variables
bounds <- matrix(data = 1, nrow = nvar, ncol = 2) # upper and lower bounds
bounds[, 1] <- -5 * bounds[, 1]
bounds[, 2] <- 5 * bounds[, 2]

Both functions are to be minimized:

nobj <- 2 # number of objectives
minmax <- c(FALSE, FALSE) # min and min

Set algorithmic parameters and launch caRamel:

popsize <- 100 # size of the genetic population
archsize <- 100 # size of the archive for the Pareto front
maxrun <- 1000 # maximum number of calls
prec <- matrix(1.e-3, nrow = 1, ncol = nobj) # accuracy for the convergence phase

results <- 
  caRamel(nobj,
          nvar,
          minmax,
          bounds,
          kursawe,
          popsize,
          archsize,
          maxrun,
          prec,
          carallel=FALSE) # no parallelism

## Beginning of caRamel optimization <-- Mon Nov 11 07:26:45 2024

## Number of variables : 3

## Number of functions : 2

## Warning in delaunayn(obj): 4 points missing from triangulation.
## It is possible that setting the 'options' argument of delaunayn may help.
## For example:
## options = "Qt Qc Qz Qbb"
## options = "Qt Qc Qz QbB"
## If these options do not work, try shifting the centre of the points
## to the origin by subtracting the mean coordinates from every point.

## Warning in delaunayn(obj): 3 points missing from triangulation.
## It is possible that setting the 'options' argument of delaunayn may help.
## For example:
## options = "Qt Qc Qz Qbb"
## options = "Qt Qc Qz QbB"
## If these options do not work, try shifting the centre of the points
## to the origin by subtracting the mean coordinates from every point.

## Warning in delaunayn(obj): 4 points missing from triangulation.
## It is possible that setting the 'options' argument of delaunayn may help.
## For example:
## options = "Qt Qc Qz Qbb"
## options = "Qt Qc Qz QbB"
## If these options do not work, try shifting the centre of the points
## to the origin by subtracting the mean coordinates from every point.

## Warning in delaunayn(obj): 5 points missing from triangulation.
## It is possible that setting the 'options' argument of delaunayn may help.
## For example:
## options = "Qt Qc Qz Qbb"
## options = "Qt Qc Qz QbB"
## If these options do not work, try shifting the centre of the points
## to the origin by subtracting the mean coordinates from every point.

## Warning in delaunayn(obj): 2 points missing from triangulation.
## It is possible that setting the 'options' argument of delaunayn may help.
## For example:
## options = "Qt Qc Qz Qbb"
## options = "Qt Qc Qz QbB"
## If these options do not work, try shifting the centre of the points
## to the origin by subtracting the mean coordinates from every point.

## Warning in delaunayn(obj): 2 points missing from triangulation.
## It is possible that setting the 'options' argument of delaunayn may help.
## For example:
## options = "Qt Qc Qz Qbb"
## options = "Qt Qc Qz QbB"
## If these options do not work, try shifting the centre of the points
## to the origin by subtracting the mean coordinates from every point.

## Warning in delaunayn(obj): 3 points missing from triangulation.
## It is possible that setting the 'options' argument of delaunayn may help.
## For example:
## options = "Qt Qc Qz Qbb"
## options = "Qt Qc Qz QbB"
## If these options do not work, try shifting the centre of the points
## to the origin by subtracting the mean coordinates from every point.

## Warning in delaunayn(obj): 4 points missing from triangulation.
## It is possible that setting the 'options' argument of delaunayn may help.
## For example:
## options = "Qt Qc Qz Qbb"
## options = "Qt Qc Qz QbB"
## If these options do not work, try shifting the centre of the points
## to the origin by subtracting the mean coordinates from every point.

## Warning in delaunayn(obj): 1 points missing from triangulation.
## It is possible that setting the 'options' argument of delaunayn may help.
## For example:
## options = "Qt Qc Qz Qbb"
## options = "Qt Qc Qz QbB"
## If these options do not work, try shifting the centre of the points
## to the origin by subtracting the mean coordinates from every point.

## Warning in delaunayn(obj): 2 points missing from triangulation.
## It is possible that setting the 'options' argument of delaunayn may help.
## For example:
## options = "Qt Qc Qz Qbb"
## options = "Qt Qc Qz QbB"
## If these options do not work, try shifting the centre of the points
## to the origin by subtracting the mean coordinates from every point.

## Warning in delaunayn(obj): 5 points missing from triangulation.
## It is possible that setting the 'options' argument of delaunayn may help.
## For example:
## options = "Qt Qc Qz Qbb"
## options = "Qt Qc Qz QbB"
## If these options do not work, try shifting the centre of the points
## to the origin by subtracting the mean coordinates from every point.

## Warning in delaunayn(obj): 5 points missing from triangulation.
## It is possible that setting the 'options' argument of delaunayn may help.
## For example:
## options = "Qt Qc Qz Qbb"
## options = "Qt Qc Qz QbB"
## If these options do not work, try shifting the centre of the points
## to the origin by subtracting the mean coordinates from every point.

## Warning in delaunayn(obj): 1 points missing from triangulation.
## It is possible that setting the 'options' argument of delaunayn may help.
## For example:
## options = "Qt Qc Qz Qbb"
## options = "Qt Qc Qz QbB"
## If these options do not work, try shifting the centre of the points
## to the origin by subtracting the mean coordinates from every point.

## Warning in delaunayn(obj): 10 points missing from triangulation.
## It is possible that setting the 'options' argument of delaunayn may help.
## For example:
## options = "Qt Qc Qz Qbb"
## options = "Qt Qc Qz QbB"
## If these options do not work, try shifting the centre of the points
## to the origin by subtracting the mean coordinates from every point.

## Warning in delaunayn(obj): 1 points missing from triangulation.
## It is possible that setting the 'options' argument of delaunayn may help.
## For example:
## options = "Qt Qc Qz Qbb"
## options = "Qt Qc Qz QbB"
## If these options do not work, try shifting the centre of the points
## to the origin by subtracting the mean coordinates from every point.

## Warning in delaunayn(obj): 9 points missing from triangulation.
## It is possible that setting the 'options' argument of delaunayn may help.
## For example:
## options = "Qt Qc Qz Qbb"
## options = "Qt Qc Qz QbB"
## If these options do not work, try shifting the centre of the points
## to the origin by subtracting the mean coordinates from every point.

## Warning in delaunayn(obj): 1 points missing from triangulation.
## It is possible that setting the 'options' argument of delaunayn may help.
## For example:
## options = "Qt Qc Qz Qbb"
## options = "Qt Qc Qz QbB"
## If these options do not work, try shifting the centre of the points
## to the origin by subtracting the mean coordinates from every point.

## Warning in delaunayn(obj): 5 points missing from triangulation.
## It is possible that setting the 'options' argument of delaunayn may help.
## For example:
## options = "Qt Qc Qz Qbb"
## options = "Qt Qc Qz QbB"
## If these options do not work, try shifting the centre of the points
## to the origin by subtracting the mean coordinates from every point.

## Warning in delaunayn(obj): 5 points missing from triangulation.
## It is possible that setting the 'options' argument of delaunayn may help.
## For example:
## options = "Qt Qc Qz Qbb"
## options = "Qt Qc Qz QbB"
## If these options do not work, try shifting the centre of the points
## to the origin by subtracting the mean coordinates from every point.

## Warning in delaunayn(obj): 2 points missing from triangulation.
## It is possible that setting the 'options' argument of delaunayn may help.
## For example:
## options = "Qt Qc Qz Qbb"
## options = "Qt Qc Qz QbB"
## If these options do not work, try shifting the centre of the points
## to the origin by subtracting the mean coordinates from every point.

## Warning in delaunayn(obj): 7 points missing from triangulation.
## It is possible that setting the 'options' argument of delaunayn may help.
## For example:
## options = "Qt Qc Qz Qbb"
## options = "Qt Qc Qz QbB"
## If these options do not work, try shifting the centre of the points
## to the origin by subtracting the mean coordinates from every point.

## Warning in delaunayn(obj): 3 points missing from triangulation.
## It is possible that setting the 'options' argument of delaunayn may help.
## For example:
## options = "Qt Qc Qz Qbb"
## options = "Qt Qc Qz QbB"
## If these options do not work, try shifting the centre of the points
## to the origin by subtracting the mean coordinates from every point.

## Warning in delaunayn(obj): 5 points missing from triangulation.
## It is possible that setting the 'options' argument of delaunayn may help.
## For example:
## options = "Qt Qc Qz Qbb"
## options = "Qt Qc Qz QbB"
## If these options do not work, try shifting the centre of the points
## to the origin by subtracting the mean coordinates from every point.

## Warning in delaunayn(obj): 3 points missing from triangulation.
## It is possible that setting the 'options' argument of delaunayn may help.
## For example:
## options = "Qt Qc Qz Qbb"
## options = "Qt Qc Qz QbB"
## If these options do not work, try shifting the centre of the points
## to the origin by subtracting the mean coordinates from every point.

## Warning in delaunayn(obj): 1 points missing from triangulation.
## It is possible that setting the 'options' argument of delaunayn may help.
## For example:
## options = "Qt Qc Qz Qbb"
## options = "Qt Qc Qz QbB"
## If these options do not work, try shifting the centre of the points
## to the origin by subtracting the mean coordinates from every point.

## Warning in delaunayn(obj): 14 points missing from triangulation.
## It is possible that setting the 'options' argument of delaunayn may help.
## For example:
## options = "Qt Qc Qz Qbb"
## options = "Qt Qc Qz QbB"
## If these options do not work, try shifting the centre of the points
## to the origin by subtracting the mean coordinates from every point.

## Done in 2.45570254325867 secs --> Mon Nov 11 07:26:47 2024

## Size of the Pareto front : 76

## Number of calls : 1011

Test if the convergence is successful and plot the optimal front:

print(results$success==TRUE)

## [1] TRUE

plot(results$objectives[,1], results$objectives[,2], main="Kursawe Pareto front", xlab="Objective #1", ylab="Objective #2")

Finally plot the convergences of the objective functions:

matplot(results$save_crit[,1],cbind(results$save_crit[,2],results$save_crit[,3]),type="l",col=c("blue","red"), main="Convergence", xlab="Number of calls", ylab="Objectives values")

References

Efstratiadis, A. and Koutsoyiannis, D., The multiobjective evolutionary annealing-simplex method and its application in calibrating hydrological models, EGU General Assembly 2005, Geophysical Research Abstracts, vol.7, Vienna, European Geophysical Union
Reed, P. and Devireddy, D., Groundwater monitoring design: a case study combining epsilon-dominance archiving and automatic parameterization for the NGSA-II, Coello-Coello C editor, Applications of multiobjective evolutionary algorithms, Advances in natural computation series, vol. 1, pp. 79-100, Word Scientific, New-York, 2004

Using caRamel on two benchmark tests

Short Description

Test functions

Schaffer

Kursawe

References