Dealing with constraints

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.

Dealing with constraints is possible with caRamel by returning a NaN value for an infeasible solution. See the example below.

library(caRamel)

Test functions

Constr-Ex problem

Constr-Ex test function has two objectives with two variables and two inequality constraints.

constr_ex <- function(i) {
  # functions f1 and f2
  s1 <- x[i,1]
  s2 <- (1. + x[i,2]) / x[i,1]
  # now test for the feasibility
  # constraint g1
  if((x[i,2] + 9. * x[i,1] - 6.) < 0. | (-x[i,2] + 9. * x[i,1] -1.) < 0.) {
    s1 <- NaN
    s2 <- NaN
  }
  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 [0.1, 1] and [0, 5]:

nvar <- 2 # number of variables
bounds <- matrix(data = 0., nrow = nvar, ncol = 2) # upper and lower bounds
bounds[1, 1] <- 0.1
bounds[1, 2] <- 1.
bounds[2, 1] <- 0.
bounds[2, 2] <- 5.

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 Constr_Ex 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,
          constr_ex,
          popsize,
          archsize,
          maxrun,
          prec,
          carallel=FALSE) # no parallelism

## Beginning of caRamel optimization <-- Mon Jul 29 10:00:28 2024

## Number of variables : 2

## Number of functions : 2

## Done in 2.61916065216064 secs --> Mon Jul 29 10:00:30 2024

## Size of the Pareto front : 93

## Number of calls : 1000

Test if the convergence is successful:

print(results$success==TRUE)

## [1] TRUE

Plot the Pareto front:

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

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