Author: Fabrice Zaoui

# Using caRamel on two benchmark tests

#### Fabrice Zaoui

#### January 19 2018

# 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)

    ## Le chargement a nécessité le package : geometry

    ## Le chargement a nécessité le package : parallel

    ## Package 'caRamel' version 1.5

# 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 <-- Tue May 12 11:01:14 2026

    ## Number of variables : 1

    ## Number of functions : 2

    ## Done in 1.71415019035339 secs --> Tue May 12 11:01:16 2026

    ## Size of the Pareto front : 94

    ## 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 <-- Tue May 12 11:01:17 2026

    ## Number of variables : 3

    ## Number of functions : 2

    ## Done in 1.44239926338196 secs --> Tue May 12 11:01:19 2026

    ## Size of the Pareto front : 100

    ## 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