# Compute several Pareto fronts for a better global result

#### Fabrice Zaoui

#### March 10 2021

# 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)

The Kursawe test function to optimize has two objectives and 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))
    }

For instance, the following caRamel parameters for all the Kursawe optimizations can be:

    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]
    nobj <- 2  # number of objectives
    minmax <- c(FALSE, FALSE) # minimization for both objectives
    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)  # convergence criteria

# Multi-caRamel optimization

In this part we will run caRamel three times on the Kursawe test function and save all the front results.

    nrepeat <- 3  # number of calls to caRamel
    concat_results_objectives <- NULL  # save results for all the calls
    concat_results_parameters <- NULL

    for (i in seq(nrepeat)) {
       optres <- caRamel(nobj,
                   nvar,
                   minmax,
                   bounds,
                   kursawe,
                   popsize,
                   archsize,
                   maxrun,
                   prec,
                   carallel = 0,
                   graph = FALSE,
                   verbose = FALSE)

       concat_results_objectives <- rbind(concat_results_objectives,
                                          optres$objectives)
       concat_results_parameters <- rbind(concat_results_parameters,
                                          optres$parameters)
    }

Then all the results are reduced using the pareto function in order to get a new global front:

    results_objectives <- concat_results_objectives
    results_objectives[, !minmax] <- -results_objectives[, !minmax]  # important !

    is_pareto <- pareto(results_objectives)  # mask

    global_results_objectives <- concat_results_objectives[is_pareto, ]  # front from the three previous fronts
    global_results_parameters <- concat_results_parameters[is_pareto, ]

# Results

All the results can now be plotted:

    plot(concat_results_objectives[, 1], concat_results_objectives[, 2],
         main = "Kursawe Pareto fronts",
         xlab = "Objective #1", ylab = "Objective #2")
    points(global_results_objectives[, 1], global_results_objectives[, 2],
           col = "red", pch = "*")