This is a short guide to defining the problem structure for running MOEA/Ds using the MOEADr package. In this document, we cover:

- Structure of the input variable
`problem`

- Structure of objective function routines, given as
`problem$name`

- structure of constraint function routines, given as
`problem$constraint$name`

A general description of the algorithm and the component-based
interpretation behind the MOEADr package is available in our paper^{1}

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 objectives

Fields `$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^{2}. 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:

```
<- list(name = "moeadr_dtlz1", # objective function routine
problem 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 n_{v} ] matrix, where each row represents one
candidate solution. The name of the input argument that receives the
population matrix must be either or .

This routine must return a [ N x n_{f} ] matrix, where each
row contains the n_{f} objective function values for one
solution.

To illustrate these requirements, we provide below the example
function `moeadr_dtlz1`

.^{3} This function is simply a MOEADr-compliant
wrapper for the DTLZ1 implementation available in the `smoof`

package.

```
<- function(X, # population matrix
moeadr_dtlz1 # 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::makeDTLZ1Function(dimensions = 10,
smoof_dtlz1 n.objectives = 2)
# Evaluate points in a vectorized manner:
<- t(apply(X,
Y 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
n_{v} ] matrix, where each row represents one candidate
solution. The name of the input argument that receives the population
matrix must be either or .

This function must return a list object containing the following fields:

`$Cmatrix`

, a [ N x (n_{g}+ n_{h}) ] 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 (n_{g}+ n_{h}) ] 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`

.^{4} Recall that we have a number of different
constraints that were stated in the problem definition:

- 10 inequality constraints regarding the requirement that \(x_i \geq 0\)
- 10 inequality constraints regarding the requirement that \(x_i \leq 1\)
- 1 inequality constraint \(g_1(\mathbf{x}) = x_1^2 + 2x_2^2 - 1.2 \leq 0\)
- 1 equality constraint \(h_1(\mathbf{x}) = x_3x_4 - 0.5 = 0\)

```
<- function(X, # population matrix
my_constraints epsilon = 0, # tolerance for equality constraints
# (defaults to zero if not provided)
...)
{
<- 10 # number of variables of the problem
nv
# Prepare output matrix of constraint function values
<- matrix(numeric(),
Cmatrix 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
<- matrix(0, nrow = nrow(X), ncol = nv)
Xmin <- matrix(1, nrow = nrow(X), ncol = nv)
Xmax
# Calculate "x_i >= 0" and "x_i <= 1" constraints
1:nv] <- Xmin - X
Cmatrix[, + 1):(2 * nv)] <- X - Xmax
Cmatrix[, (nv
# g1 and h1 functions
<- function(X){
g1 return(X[, 1] ^ 2 + 2 * X[, 2] ^ 2 - 1.2)
}<- function(X){
h1 return(X[, 3] * X[, 4] - 0.5)
}
# Calculate g1(x) and h1(x)
2 * nv + 1] <- g1(X)
Cmatrix[, Cmatrix(, 2 * nv + 2) <- h1(X)
# Assemble matrix of *violations*
<- 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
Vmatrix[,
# Return necessary variables
return(list(Cmatrix = Cmatrix,
Vmatrix = Vmatrix,
v = rowSums(Vmatrix)))
}
```

Some **VERY** important points:

- All constraints are expressed in the standard form, that is, \(g_i(\mathbf{x}) \leq 0\) and \(h_j(\mathbf{x}) = 0\). You may choose to
express them differently, but in that case be extra careful with the
calculations of
`Vmatrix`

and`v`

; - All box constraints must be included as part of the constraint
violations routine, otherwise they will be ignored by the constraint
handling approaches. This can in principle be a valid option, for
instance, if i) the problem is actually unconstrained; or ii) a repair
method such as the
*truncate*variation operator is used.

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!

F. Campelo, L.S. Batista and C. Aranha, “A Component-Wise Perspective on Multiobjective Evolutionary Algorithms based on Decomposition”, in preparation.↩︎

https://deap.readthedocs.io/en/master/api/benchmarks.html#deap.benchmarks.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.↩︎Also not available in the MOEADr package, since it does not make much practical sense.↩︎