moead: MOEA/D

Description

MOEA/D implementation in R

Usage

moead(
  preset = NULL,
  problem = NULL,
  decomp = NULL,
  aggfun = NULL,
  neighbors = NULL,
  variation = NULL,
  update = NULL,
  constraint = NULL,
  scaling = NULL,
  stopcrit = NULL,
  showpars = NULL,
  seed = NULL,
  ...
)

Value

List object of class moead containing:

information on the final population (X), its objective values (Y) and constraint information list (V) (see evaluate_population() for details);
Archive population list containing its corresponding X, Y and V fields (only if update$UseArchive = TRUE).
Estimates of the ideal and nadir points, calculated for the final population;
Number of function evaluations, iterations, and total execution time;
Random seed employed in the run, for reproducibility

Arguments

preset: List object containing preset values for one or more of the other parameters of the moead function. Values provided in the preset list will override any other value provided. Presets should be generated by the preset_moead() function.
problem: List containing the problem parameters. See Problem Description for details.
decomp: List containing the decomposition method parameters See Decomposition methods for details.
aggfun: List containing the aggregation function parameters See Scalarization methods for details.
neighbors: List containing the decomposition method parameters See Neighborhood strategies for details.
variation: List containing the variation operator parameters See Variation operators for details.
update: List containing the population update parameters See Update strategies for details.
constraint: List containing the constraint handing parameters See Constraint operators for details.
scaling: List containing the objective scaling parameters See Objective scaling for details.
stopcrit: list containing the stop criteria parameters. See Stop criteria for details.
showpars: list containing the echoing behavior parameters. See print_progress() for details.
seed: seed for the pseudorandom number generator. Defaults to NULL, in which case as.integer(Sys.time()) is used for the definition.
...: Other parameters (useful for development and debugging, not necessary in regular use)

Problem Description

The problem parameter consists of a list with all necessary definitions for the multiobjective optimization problem to be solved. problem must contain at least the following fields:

problem$name: name of the problem instance function, that is, a routine that calculates Y = f(X);
problem$xmin: vector of lower bounds of each variable
problem$xmax: vector of upper bounds of each variable
problem$m: integer indicating the number of objectives

Besides these fields, problem should contain any other relevant inputs for the routine listed in $name. problem may also contain the (optional) field problem$constraints, which is a list object containing information about the problem constraints. If present, this list must have the following fields:

problem$constraints$name - (required) name of the function that calculates the constraint values (see below for details)
problem$constraints$epsilon - (optional) a small non-negative value indicating the tolerance to be considered for equality constraints. Defaults to zero.

Besides these fields, problem$constraint should contain any other relevant inputs for the routine listed in problem$constraint$name.

Detailed instructions for defining the routines for calculating the objective and constraint functions are provided in the vignette Defining Problems in the MOEADr Package. Check that documentation for details.

Decomposition Methods

The decomp parameter is a list that defines the method to be used for the generation of the weight vectors. decomp must have at least the $name parameter. Currently available methods can be verified using get_decomposition_methods(). Check generate_weights() and the information provided by get_decomposition_methods() for more details.

Neighborhood Strategies

The neighbors parameter is a list that defines the method for defining the neighborhood relations among subproblems. neighbors must have at least three parameters:

neighbors$name, name of the strategy used to define the neighborhoods. Currently available methods are: - $name = "lambda": uses the distances between weight vectors. The calculation is performed only once for the entire run, since the weight vectors are assumed static. - $name = "x": uses the distances between the incumbent solutions associated with each subproblem. In this case the calculation is performed at each iteration, since incumbent solutions may change.
neighbors$T: defines the neighborhood size. This parameter must receive a value smaller than the number of subproblems defined for the MOEA/D.
neighbors$delta.p: parameter that defines the probability of sampling from the neighborhood when performing variation.

Check define_neighborhood() for more details.

Variation Operators

The variation parameter consists of a list vector, in which each sublist defines a variation operator to be used as part of the variation block. Each sublist must have at least a field $name, containing the name of the i-th variation operator to be applied. Use get_variation_operators() to generate a list of available operators, and consult the vignette Variation Stack in the MOEADr Package for more details.

Scalar Aggregation Functions

The aggfun parameter is a list that defines the scalar aggregation function to be used. aggfun must have at least the $name parameter. Currently available methods can be verified using get_scalarization_methods(). Check scalarize_values() and the information provided by get_scalarization_methods() for more details.

Update Methods

The update parameter is a list that defines the population update strategy to be used. update must have at least the $name parameter. Currently available methods can be verified using get_update_methods(). Check update_population() and the information provided by get_update_methods() for more details.

Another (optional) field of the update parameter is update$UseArchive, which is a binary flag defining whether the algorithm should keep an external solution archive (TRUE) or not (FALSE). Since it adds to the computational burden and memory requirements of the algorithm, the use of an archive population is recommended only in the case of constrained problems with constraint handling method that can occasionally accept unfeasible solutions, leading to the potential loss of feasible efficient solutions for certain subproblems (e.g., constraint_vbr() with type = "sr" or "vt").

Constraint Handling Methods

The constraint parameter is a list that defines the constraint-handling technique to be used. constraint must have at least the $name parameter. Currently available methods can be verified using get_constraint_methods(). Check update_population() and the information provided by get_constraint_methods() for more details.

Objective Scaling

Objective scaling refers to the re-scaling of the objective values at each iteration, which is generally considered to prevent problems arising from differently-scaled objective functions. scaling is a list that must have at least the $name parameter. Currently available options are $name = "none", which does not perform any scaling, and $name = "simple", which performs a simple linear scaling of the objectives to the interval [0, 1].

Stop Criteria

The stopcrit parameter consists of a list vector, in which each sublist defines a termination criterion to be used for the MOEA/D. Each sublist must have at least a field $name, containing the name of the i-th criterion to be verified. The iterative cycle of the MOEA/D is terminated whenever any criterion is met. Use get_stop_criteria() to generate a list of available criteria, and check the information provided by that function for more details.

Echoing Options

The showpars parameter is a list that defines the echoing options of the MOEA/D. showpars must contain two fields:

showpars$show.iters, defining the type of echoing output. $show.iters can be set as "none", "numbers", or "dots".
showpars$showevery, defining the period of echoing (in iterations). $showevery must be a positive integer.

Details

Component-wise implementation of the Multiobjective Evolutionary Algorithm based on decomposition - MOEA/D.

References

F. Campelo, L.S. Batista, C. Aranha (2020): The MOEADr Package: A Component-Based Framework for Multiobjective Evolutionary Algorithms Based on Decomposition. Journal of Statistical Software tools:::Rd_expr_doi("10.18637/jss.v092.i06")

Examples

Run this code

## Prepare a test problem composed of minimization of the (shifted)
## sphere and Rastrigin functions
sphere     <- function(x){sum((x + seq_along(x) * 0.1) ^ 2)}
rastringin <- function(x){
                x.shift <- x - seq_along(x) * 0.1
                sum((x.shift) ^ 2 - 10 * cos(2 * pi * x.shift) + 10)}
problem.sr <- function(X){
                t(apply(X, MARGIN = 1,
                FUN = function(X){c(sphere(X), rastringin(X))}))}


## Set the input parameters for the moead() routine
## This reproduces the Original MOEA/D of Zhang and Li (2007)
## (with a few changes in the computational budget, to make it run faster)
problem   <- list(name       = "problem.sr",
                  xmin       = rep(-1, 30),
                  xmax       = rep(1, 30),
                  m          = 2)
decomp    <- list(name       = "SLD", H = 49) # <-- H = 99 in the original
neighbors <- list(name       = "lambda",
                  T          = 20,
                  delta.p    = 1)
aggfun    <- list(name       = "wt")
variation <- list(list(name  = "sbx",
                       etax  = 20, pc = 1),
                  list(name  = "polymut",
                       etam  = 20, pm = 0.1),
                  list(name  = "truncate"))
update    <- list(name       = "standard", UseArchive = FALSE)
scaling   <- list(name       = "none")
constraint<- list(name       = "none")
stopcrit  <- list(list(name  = "maxiter",
                    maxiter  = 50))      # <-- maxiter = 200 in the original
showpars  <- list(show.iters = "dots",
                  showevery  = 10)
seed      <- 42

## Run MOEA/D
out1 <- moead(preset = NULL,
              problem, decomp, aggfun, neighbors, variation, update,
              constraint, scaling, stopcrit, showpars, seed)

## Examine the output:
summary(out1)

## Alternatively, the standard MOEA/D could also be set up using the
## preset_moead() function. The code below runs the original MOEA/D with
## exactly the same configurations as in Zhang and Li (2007).
if (FALSE) {
  out2 <- moead(preset   = preset_moead("original"),
                problem  = problem,
                showpars = showpars,
                seed     = 42)

  ## Examine the output:
  summary(out2)
  plot(out2, suppress.pause = TRUE)
}

# Rerun with MOEA/D-DE configuration and AWT scalarization
out3 <- moead(preset   = preset_moead("moead.de"),
              problem  = problem,
              aggfun   = list(name = "awt"),
              stopcrit = list(list(name    = "maxiter",
                                   maxiter = 50)),
              seed    = seed)
plot(out3, suppress.pause = TRUE)

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