This is a short guide to defining the problem structure for running MOEA/Ds using the MOEADr package. In this document, we cover:
problem
problem$name
problem$constraint$name
A general description of the algorithm and the component-based interpretation behind the MOEADr package is available in our paper^[F. Campelo, L.S. Batista and C. Aranha, "A Component-Wise Perspective on Multiobjective Evolutionary Algorithms based on Decomposition", in preparation.]
In the MOEADr package, all information regarding the problem is input as a list
variable problem
. This parameter must contain enough information for the
algorithm to correctly evaluate all the objective and constraint functions, as
documented in function moead()
and replicated below for convenience.
The problem
list must contain the following required fields:
problem$name
, with the name of the problem instance function, that is, a routine that calculates Y = f(X). The details of this routine are documented in Section Objective Functions Routine, see below. problem$xmin
, containing a vector of lower bounds of each variable of the problem. problem$xmax
, containing a vector of upper bounds of each variable of the problem. problem$m
, containing a positive integer m>1
with the number of objectivesFields $xmin
and $xmax
provide information for the internal variable standardization performed by the MOEADr package, as well as information about the number of problem variables. This information is also used for the definition of the initial population, but it does not guarantee that the points will remain within this interval as the iterations progress - for that, variation operators such as truncate
or explicit constraints (see below, Section Constraint Functions Routine) must be employed.
Besides these fields, problem
should contain any other relevant inputs for the routine listed in problem$name
.
problem
may also contain the optional field problem$constraints
, which is itself a list object containing information about the problem constraints. If present, problem$constraints
must contain the following fields:
problem$constraints$name
- (required) name of the function that
calculates the constraint values. The details of this routine are documented in Section Constraint Functions Routine, see below. 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$constraints
should contain any other relevant inputs for the routine listed in problem$constraints$name
.
To guide us through the steps required to define a problem structure for the MOEADr package, assume that we want to use the MOEADr framework to solve a 10-variable, 2-objective DTLZ1 benchmark function^[https://deap.readthedocs.io/en/master/api/benchmarks.html#deap.benchmarks.dtlz1]. Assume that the feasible space is defined by the interval [0, 1] for all variables, and by $x_1^2 + 2x_2^2 \leq 1.2$ and $x_3x_4 = 0.5$. In this case, the problem
variable would be defined as:
problem <- list(name = "moeadr_dtlz1", # objective function routine xmin = rep(0, 10), # lower limits xmax = rep(1, 10), # upper limits m = 2, # number of objectives constraints = list( name = "my_constraints",# constraint function routine epsilon = 0.05)) # tolerance for equality constraints
The specific requirements regarding functions problem$name
and problem$constraints$name
are provided in the following sections.
The routine indicated in problem$name
must be able to receive a [ N x nv ] matrix, where each row represents one candidate solution. The name of the input argument that receives the population matrix must be either \code{X} or \code{x}.
This routine must return a [ N x nf ] matrix, where each row contains the nf objective function values for one solution.
To illustrate these requirements, we provide below the example function moeadr_dtlz1
.^[This function is not available in the MOEADr package - instead we provide the more general function make_vectorized_smoof()
. See the documentation for details.] This function is simply a MOEADr-compliant wrapper for the DTLZ1 implementation available in the smoof
package.
moeadr_dtlz1 <- function(X, # population matrix ... # allow function to receive extra parameters. # These are unused in most cases, but it is useful # for preventng errors due to unwanted parameters # being passed ){ # "smoof" is listed in the Suggests field MOEADr's DESCRIPTION, but we need to # be sure that it is available, so: if(!("smoof" %in% rownames(utils::installed.packages()))){ stop("Please install package 'smoof' to continue") } # make 10-variable, 2-objective DTLZ1 smoof_dtlz1 <- smoof::makeDTLZ1Function(dimensions = 10, n.objectives = 2) # Evaluate points in a vectorized manner: Y <- t(apply(X, MARGIN = 1, FUN = smoof_dtlz1)) # Return [N x n_f] matrix return(Y) }
Notice that the objective functions routine does not use the information from
xmin
, xmax
, m
, or constraints
- these fields are used elsewhere in the
MOEADr structure to define the initial population, weight matrices, truncation operators etc.
As in the objective functions case, the routine indicated in problem$constraints$name
must be able to receive a [ N x nv ] matrix, where each row represents one candidate solution. The name of the input argument that receives the population matrix must be either \code{X} or \code{x}.
This function must return a list object containing the following fields:
$Cmatrix
, a [ N x (ng + nh) ] matrix, where each row contains the individual constraint function values for one solution. The names of each column should ideally be informative regardind to which constraint the function refers (this is not mandatory, but it is a good practice that can save the user a great deal of time). $Vmatrix
, a [ N x (ng + nh) ] matrix, where each row contains the individual constraint violations for one solution.$v
, a vector [N x 1], where each component contains the total violation of one solution, that is, the value of:$$v[k] = v(x_k) = \sum_i max(~g_i(x_k),~~0) + \sum_j max(~|h_j(x_k)| - \epsilon,~~0)$$
v
is calculated simply as rowsums(Vmatrix)
, but returning it prevents having to re-compute v
in different places of the MOEA/D structure.
To illustrate these requirements, we provide below the example function my_constraints
.^[Also not available in the MOEADr package, since it does not make much practical sense.] Recall that we have a number of different constraints that were stated in the problem definition:
my_constraints <- function(X, # population matrix epsilon = 0, # tolerance for equality constraints # (defaults to zero if not provided) ...) { nv <- 10 # number of variables of the problem # Prepare output matrix of constraint function values Cmatrix <- matrix(numeric(), nrow = nrow(X), ncol = 2 * nv + 2) # 20 inequality box constraints, plus g1 and h1 # Set informative column names (be nice to your users!) colnames(Cmatrix) <- c(paste0("x", rep(1:nv, times = 2), rep(c("min","max"), each = nv)), "g1", "h1") # Box limits of the feasible space Xmin <- matrix(0, nrow = nrow(X), ncol = nv) Xmax <- matrix(1, nrow = nrow(X), ncol = nv) # Calculate "x_i >= 0" and "x_i <= 1" constraints Cmatrix[, 1:nv] <- Xmin - X Cmatrix[, (nv + 1):(2 * nv)] <- X - Xmax # g1 and h1 functions g1 <- function(X){ return(X[, 1] ^ 2 + 2 * X[, 2] ^ 2 - 1.2) } h1 <- function(X){ return(X[, 3] * X[, 4] - 0.5) } # Calculate g1(x) and h1(x) Cmatrix[, 2 * nv + 1] <- g1(X) Cmatrix(, 2 * nv + 2) <- h1(X) # Assemble matrix of *violations* Vmatrix <- Cmatrix Vmatrix[, 1:(2 * nv + 1)] <- pmax(Vmatrix[, 1:(2 * nv + 1)], 0) # inequality constraints Vmatrix[, 2 * nv + 2] <- pmax(abs(Vmatrix[, 2 * nv + 2]) - epsilon, 0) # equality constraint h1 # Return necessary variables return(list(Cmatrix = Cmatrix, Vmatrix = Vmatrix, v = rowSums(Vmatrix))) }
Some VERY important points:
Vmatrix
and v
;The MOEADr package already provides a convenient implementation for a "box constraints" (function box_constraints()
) and "unitary constraints" (function unitary_constraints()
). See the specific documentation for details.
To use these functions, simple make constraints = list(name = "box_constraints")
(or "unitary_constraints"
, if that is the case) in your definition of the problem
input. And don't forget the epsilon
in the case of unitary constraints!
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