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Title:

Evolutionary Computation in R

Description:

Framework for building evolutionary algorithms for both single- and multi-objective continuous or discrete optimization problems. A set of predefined evolutionary building blocks and operators is included. Moreover, the user can easily set up custom objective functions, operators, building blocks and representations sticking to few conventions. The package allows both a black-box approach for standard tasks (plug-and-play style) and a much more flexible white-box approach where the evolutionary cycle is written by hand.

Version:

2.1.1

Encoding:

UTF-8

Date:

2023-03-08

Maintainer:

Jakob Bossek <j.bossek@gmail.com>

License:

GPL-3

URL:

https://github.com/jakobbossek/ecr2

BugReports:

https://github.com/jakobbossek/ecr2/issues

Depends:

R (≥ 2.10), BBmisc (≥ 1.6), smoof (≥ 1.4), ParamHelpers (≥1.1)

Imports:

checkmate (≥ 1.1), Rcpp (≥ 0.12.16), parallelMap (≥ 1.3),reshape2 (≥ 1.4.1), ggplot2 (≥ 1.0.0), viridis, dplyr,plot3D, plot3Drgl, scatterplot3d, plotly, knitr, kableExtra,lazyeval

Suggests:

testthat (≥ 0.9.1), rmarkdown, mlr, mlbench, randomForest,covr

ByteCompile:

yes

LinkingTo:

Rcpp

VignetteBuilder:

knitr

LazyData:

true

RoxygenNote:

7.2.3

NeedsCompilation:

yes

Packaged:

2023-03-08 16:02:04 UTC; bossek

Author:

Jakob Bossek [aut, cre, cph], Michael H. Buselli [ctb, cph], Wessel Dankers [ctb, cph], Carlos M. Fonseca [ctb, cph], Manuel Lopez-Ibanez [ctb, cph], Luis Paquete [ctb, cph], Joshua Knowles [ctb, cph], Eckart Zitzler [ctb, cph], Olaf Mersmann [ctb]

Repository:

CRAN

Date/Publication:

2023-03-08 22:10:06 UTC

Grouping helpers

Description

Consider a data frame with results of multi-objective stochastic optimizers ona set of problems from different categories/groups (say indicated by column “group”).Occasionally, it is useful to unite the results of several groups into a meta-group.The functionaddUnionGroup aids in generation of such a meta-group whilefunctionaddAllGroup is a wrapper around the former which generates aunion of all groups.

Usage

addUnionGroup(df, col, group, values)addAllGroup(df, col, group = "all")

Arguments

df

[data.frame]
Data frame.

col

[character(1)]
Column name of group-column.

group

[character(1)]
Name for new group.

values

[character(1)]
Subset of values within the value range of columncol.

Value

[data.frame] Modified data frame.

Examples

df = data.frame(  group = c("A1", "A1", "A2", "A2", "B"),  perf = runif(5),  stringsAsFactors = FALSE)df2 = addUnionGroup(df, col = "group", group = "A", values = c("A1", "A2"))df3 = addAllGroup(df, col = "group", group = "ALL")

Reference point approximations.

Description

Helper functions to compute nadir or ideal point from sets ofpoints, e.g., multiple approximation sets.

Usage

approximateNadirPoint(..., sets = NULL)approximateIdealPoint(..., sets = NULL)

Arguments

...

[matrix]
Arbirary number of matrizes.

sets

[list]
List of matrizes. This is an alternative way of passing the sets. Can be usedexclusively or combined with....

Value

[numeric] Reference point.

Helper function to estimate reference points.

Description

E.g., for calculation of dominated hypervolume.

Usage

approximateRefPoints(df, obj.cols = c("f1", "f2"), offset = 0, as.df = FALSE)

Arguments

df

[data.frame]
Data frame with the required structure, i.e. the data frame must contain a problem column "prob" as well as objective column(s).

obj.cols

[character(>= 2)]
Column names of the objective functions.Default isc("f1", "f2"), i.e., the bi-objective case is assumed.

offset

[numeric(1)]
Offset added to reference points.Default is0.

as.df

[logical(1)]
Should a data.frame be returned?Default isFALSE. In this case a named list is returned.

Value

[list |data.frame]

Helper function to estimate reference set(s).

Description

The function takes an data frame with columns at leastspecified byobj.cols and “prob”. The reference set foreach unique problem in column “prob” is then obtained bycombining all approximation sets generated by all considered algorithmsfor the corresponding problem and filtering the non-dominated solutions.

Usage

approximateRefSets(df, obj.cols, as.df = FALSE)

Arguments

df

[data.frame]
Data frame with the required structure.

obj.cols

[character(>= 2)]
Column names of the objective functions.

as.df

[logical(1)]
Should a data.frame be returned?Default isFALSE. In this case a named list is returned.

Value

[list |data.frame] Named list of matrizes(names are the problems) or data frame with columnsobj.cols and “prob”.

Implementation of the NSGA-II EMOA algorithm by Deb.

Description

The AS-EMOA, short for aspiration set evolutionary multi-objectivealgorithm aims to incorporate expert knowledge into multi-objective optimization [1].The algorithm expects an aspiration set, i.e., a set of reference points. Itthen creates an approximation of the pareto front close to the aspiration setutilizing the average Hausdorff distance.

Usage

asemoa(  fitness.fun,  n.objectives = NULL,  minimize = NULL,  n.dim = NULL,  lower = NULL,  upper = NULL,  mu = 10L,  aspiration.set = NULL,  normalize.fun = NULL,  dist.fun = computeEuclideanDistance,  p = 1,  parent.selector = setup(selSimple),  mutator = setup(mutPolynomial, eta = 25, p = 0.2, lower = lower, upper = upper),  recombinator = setup(recSBX, eta = 15, p = 0.7, lower = lower, upper = upper),  terminators = list(stopOnIters(100L)))

Arguments

fitness.fun

[function]
The fitness function.

n.objectives

[integer(1)]
Number of objectives ofobj.fun.Optional ifobj.fun is a benchmark function from packagesmoof.

minimize

[logical(n.objectives)]
Logical vector with ith entryTRUE if the ith objective offitness.funshall be minimized. If a single logical is passed, it is assumed to be validfor each objective.

n.dim

[integer(1)]
Dimension of the decision space.

lower

[numeric]
Vector of minimal values for each parameter of the decision space in caseof float or permutation encoding.Optional ifobj.fun is a benchmark function from packagesmoof.

upper

[numeric]
Vector of maximal values for each parameter of the decision space in caseof float or permutation encoding.Optional ifobj.fun is a benchmark function from packagesmoof.

mu

[integer(1)]
Population size. Default is 10.

aspiration.set

[matrix]
The aspiration set. Each column contains one point of the set.

normalize.fun

[function]
Function used to normalize fitness values of the individualsbefore computation of the average Hausdorff distance.The function must have the formal arguments “set” and “aspiration.set”.Default isNULL, i.e., no normalization at all.

dist.fun

[function]
Distance function used internally by Hausdorff metric to compute distancebetween two points. Expects a single vector of coordinate-wise differencesbetween points.Default iscomputeEuclideanDistance.

p

[numeric(1)]
Parameterp for the average Hausdorff metric. Default is 1.

parent.selector

[ecr_selector]
Selection operator which implements a procedure to copy individuals from agiven population to the mating pool, i. e., allow them to become parents.

mutator

[ecr_mutator]
Mutation operator of typeecr_mutator.

recombinator

[ecr_recombinator]
Recombination operator of typeecr_recombinator.

terminators

[list]
List of stopping conditions of type “ecr_terminator”.Default is to stop after 100 iterations.

Value

[ecr_multi_objective_result]

Note

This is a pure R implementation of the AS-EMOA algorithm. It hides the regularecr interface and offers a more R like interface while still being quiteadaptable.

References

[1] Rudolph, G., Schuetze, S., Grimme, C., Trautmann, H: An Aspiration SetEMOA Based on Averaged Hausdorff Distances. LION 2014: 153-156.[2] G. Rudolph, O. Schuetze, C. Grimme, and H. Trautmann: A MultiobjectiveEvolutionary Algorithm Guided by Averaged Hausdorff Distance to AspirationSets, pp. 261-273 in A.-A. Tantar et al. (eds.): Proceedings of EVOLVE - Abridge between Probability, Set Oriented Numerics and Evolutionary ComputationV, Springer: Berlin Heidelberg 2014.

Assign group membership based on another group membership.

Description

Given a data frame and a grouping column of type factor or character this functiongenerates a new grouping column which groups the groups.

Usage

categorize(df, col, categories, cat.col, keep = TRUE, overwrite = FALSE)

Arguments

df

[data.frame]
Data frame.

col

[character(1)]
Column name of group variable.

categories

[list]
Named list. Names indicate the name of the category while the values are character vectorsof values within the range of thecol column.

cat.col

[character(1)]
Column name for categorization.

keep

[logical(1)]
Keep the source columncol?Default isTRUE.

overwrite

[logical(1)]
IfTRUE,cat.col is set tocol.

Value

[data.frame]df = data.frame(group = c("A1", "A1", "A2", "A2", "B1", "B2"),perf = runif(6),stringsAsFactors = FALSE)df2 = categorize(df, col = "group", categories = list(A = c("A1", "A2"), B = c("B1", "B2")), cat.col = "group2")

Average Hausdorff Distance computation.

Description

Computes the average Hausdroff distance measure between two point sets.

Usage

computeAverageHausdorffDistance(  A,  B,  p = 1,  normalize = FALSE,  dist.fun = computeEuclideanDistance)

Arguments

A

[matrix]
First point set (each column corresponds to a point).

B

[matrix]
Second point set (each column corresponds to a point).

p

[numeric(1)]
Parameter p of the average Hausdoff metric.Default is 1.

normalize

[logical(1)]
Should the front be normalized on basis ofB?Default isFALSE.

dist.fun

[matrix]
Distance function to compute distance between points x and y. Expects a singlenumeric vector d with the coordinate-wise differences di = (xi - yi).Default iscomputeEuclideanDist.

Value

[numeric(1)] Average Hausdorff distance of setsA andB.

Compute the crowding distance of a set of points.

Description

The crowding distance is a measure of spread of solutions in theapproximation of the Pareto front. It is used, e.g., in the NSGA-II algorithmas a second selection criterion.

Usage

computeCrowdingDistance(x)

Arguments

x

[matrix]
Numeric matrix with each column representing a point.

Value

[numeric] Vector of crowding distance values.

References

K. Deb, A. Pratap, S. Agarwal, T. Meyarivan, A fast and elitistmultiobjective genetic algorithm: NSGA-II, IEEE Transactions on EvolutionaryComputation In Evolutionary Computation, IEEE Transactions on, Vol. 6, No. 2.(07 April 2002), pp. 182-197, doi:10.1109/4235.996017

Computes distance between a single point and set of points.

Description

Helper to compute distance between a single point and a point set.

Usage

computeDistanceFromPointToSetOfPoints(  a,  B,  dist.fun = computeEuclideanDistance)

Arguments

a

[numeric(1)]
Point given as a numeric vector.

B

[matrix]
Point set (each column corresponds to a point).

dist.fun

[matrix]
Distance function to compute distance between points x and y. Expects a singlenumeric vector d with the coordinate-wise differences di = (xi - yi).Default iscomputeEuclideanDist.

Value

[numeric(1)]

Ranking of approximation sets.

Description

Ranking is performed by merging all approximation sets over allalgorithms and runs per instance. Next, each approximation setC is assigned arank which is 1 plus the number of approximation sets that are better thanC. A setD is better thanC, if for each pointx \in C thereexists a point iny \in D which weakly dominatesx.Thus, each approximation set is reduced to a number – its rank. This rank distributionmay act for first comparrison of multi-objecitve stochastic optimizers.See [1] for more details.This function makes use ofparallelMap toparallelize the computation of dominance ranks.

Usage

computeDominanceRanking(df, obj.cols)

Arguments

df

[data.frame]
Data frame with columns at least “prob”, “algorithm”, “repl” andcolumn names specified via parameterobj.cols.

obj.cols

[character(>= 2)]
Column names indf which store the objective function values.

Value

[data.frame] Reduceddf with columns “prob”, “algorithm”, “repl”and “rank”.

Note

Since pairwise non-domination checks are performed over all algorithms andalgorithm runs this function may take some time if the number of problems, algorithmsand/or replications is high.

References

[1] Knowles, J., Thiele, L., & Zitzler, E. (2006). A Tutorial on the Performance Assessmentof Stochastic Multiobjective Optimizers. Retrieved from https://sop.tik.ee.ethz.ch/KTZ2005a.pdf

Computes Generational Distance.

Description

Helper to compute the Generational Distance (GD) between two sets of points.

Usage

computeGenerationalDistance(  A,  B,  p = 1,  normalize = FALSE,  dist.fun = computeEuclideanDistance)

Arguments

A

[matrix]
First point set (each column corresponds to a point).

B

[matrix]
Second point set (each column corresponds to a point).

p

[numeric(1)]
Parameter p of the average Hausdoff metric.Default is 1.

normalize

[logical(1)]
Should the front be normalized on basis ofB?Default isFALSE.

dist.fun

[matrix]
Distance function to compute distance between points x and y. Expects a singlenumeric vector d with the coordinate-wise differences di = (xi - yi).Default iscomputeEuclideanDist.

Value

[numeric(1)]

Functions for the calculation of the dominated hypervolume (contribution).

Description

The functioncomputeHV computes the dominatedhypervolume of a set of points given a reference set wherebycomputeHVContr computes the hypervolume contributionof each point.

If no reference point is given the nadir point of the setx isdetermined and a positive offset with default 1 is added. This is to ensurethat the reference point dominates all of the points in the reference set.

Usage

computeHV(x, ref.point = NULL, ...)computeHVContr(x, ref.point = NULL, offset = 1)

Arguments

x

[matrix]
Matrix of points (column-wise).

ref.point

[numeric |NULL]
Reference point.Set to the maximum in each dimension by default if not provided.

...

[any]
Not used at the moment.

offset

[numeric(1)]
Offset to be added to each component of the reference point only in the casewhere no reference is provided and one is calculated automatically.

Value

[numeric(1)] Dominated hypervolume in the case ofcomputeHV and the dominated hypervolume contributionsfor each point in the case ofcomputeHVContr.

Note

: Keep in mind that this function assumes all objectives to be minimized.In case at least one objective is to be maximized the matrixx needsto be transformed accordingly in advance.

Computation of EMOA performance indicators.

Description

Given a data.frame of Pareto-front approximations for differentsets of problems, algorithms and replications, the function computes setsof unary and binary EMOA performance indicators.This function makes use ofparallelMap toparallelize the computation of indicators.

Usage

computeIndicators(  df,  obj.cols = c("f1", "f2"),  unary.inds = NULL,  binary.inds = NULL,  normalize = FALSE,  offset = 0,  ref.points = NULL,  ref.sets = NULL)

Arguments

df

[data.frame]
Data frame with columnsobj.cols, “prob”, “algorithm”and “repl”.

obj.cols

[character(>= 2)]
Column names of the objective functions.Default isc("f1", "f2"), i.e., the bi-objective case is assumed.

unary.inds

[list]
Named list of unary indicators which shall be calculated.Each component must be another list with mandatory argumentfun (thefunction which calculates the indicator) and optional argumentpars (a namedlist of parameters forfun). Functionfun must have thesigniture “function(points, arg1, ..., argk, ...)”.The arguments “points” and “...” are mandatory, the remaining areoptional.The names of the components on the first level are used for the column namesof the output data.frame.Default islist(HV = list(fun = computeHV)), i.e., the dominatedHypervolume indicator.

binary.inds

[list]
Named list of binary indicators which shall be applied for each algorithmcombination. Parameterbinary.inds needs the same structure asunary.inds.However, the function signature offun is slighly different:“function(points1, points2, arg1, ..., argk, ...)”.See functionemoaIndEps for an example.Default islist(EPS = list(fun = emoaIndEps)).

normalize

[logical(1)]
Normalize approximation sets to[0, 1]^p wherep is the number ofobjectives? Normalization is done on the union of all approximation sets for eachproblem.Default isFALSE.

offset

[numeric(1)]
Offset added to reference point estimations.Default is 0.

ref.points

[list]
Named list of numeric vectors (the reference points). The names must be theunique problem names indf$prob or a subset of these.IfNULL (the default), reference points are estimated from theapproximation sets for each problem.

ref.sets

[list]
Named list matrizes (the reference sets). The names must be theunique problem names indf$prob or a subset of these.IfNULL (the default), reference points are estimated from theapproximation sets for each problem.

Value

[list] List with components “unary” (data frame ofunary indicators), “binary” (list of matrizes of binary indicators),“ref.points” (list of reference points used) and “ref.sets”(reference sets used).

References

[1] Knowles, J., Thiele, L., & Zitzler, E. (2006). A Tutorial on the Performance Assessmentof Stochastic Multiobjective Optimizers. Retrieved from https://sop.tik.ee.ethz.ch/KTZ2005a.pdf[2] Knowles, J., & Corne, D. (2002). On Metrics for Comparing Non-Dominated Sets.In Proceedings of the 2002 Congress on Evolutionary Computation Conference (CEC02)(pp. 711–716). Honolulu, HI, USA: Institute of Electrical and Electronics Engineers.[3] Okabe, T., Yaochu, Y., & Sendhoff, B. (2003). A Critical Survey of PerformanceIndices for Multi-Objective Optimisation. In Proceedings of the 2003 Congress on EvolutionaryComputation Conference (CEC03) (pp. 878–885). Canberra, ACT, Australia: IEEE.

Computes Inverted Generational Distance.

Description

Helper to compute the Inverted Generational Distance (IGD) between two setsof points.

Usage

computeInvertedGenerationalDistance(  A,  B,  p = 1,  normalize = FALSE,  dist.fun = computeEuclideanDistance)

Arguments

A

[matrix]
First point set (each column corresponds to a point).

B

[matrix]
Second point set (each column corresponds to a point).

p

[numeric(1)]
Parameter p of the average Hausdoff metric.Default is 1.

normalize

[logical(1)]
Should the front be normalized on basis ofB?Default isFALSE.

dist.fun

[matrix]
Distance function to compute distance between points x and y. Expects a singlenumeric vector d with the coordinate-wise differences di = (xi - yi).Default iscomputeEuclideanDist.

Value

[numeric(1)]

Fast non-dominated sorting algorithm.

Description

Fast non-dominated sorting algorithm proposed by Deb. Non-dominated sortingexpects a set of points and returns aset of non-dominated fronts. In short words this is done as follows: thenon-dominated points of the entire set are determined and assigned rank 1.Afterwards all points with the current rank are removed, the rank is increasedby one and the procedure starts again. This is done until the set is empty, i.~e.,each point is assigned a rank.

Usage

doNondominatedSorting(x)

Arguments

x

[matrix]
Numeric matrix of points. Each column contains one point.

Value

[list]List with the following components

ranks: Integer vector of ranks of lengthncol(x). The higherthe rank, the higher the domination front the corresponding point islocated on.
dom.counter: Integer vector of lengthncol(x). The i-th elementis the domination number of the i-th point.

Note

This procedure is the key survival selection of the famous NSGA-II multi-objectiveevolutionary algorithm (seensga2).

References

[1] Deb, K., Pratap, A., and Agarwal, S. A Fast and Elitist Multiobjective GeneticAlgorithm: NSGA-II. IEEE Transactions on Evolutionary Computation, 6 (8) (2002),182-197.

Check for pareto dominance.

Description

These functions take a numeric matrixx where each column corresponds toa point and return a logical vector. The i-th position of the latter isTRUE if the i-th point is dominated by at least one other point fordominated andFALSE fornonDominated.

Usage

dominated(x)nondominated(x)

Arguments

x

[matrix]
Numeric (d x n) matrix where d is the number of objectives and n is thenumber of points.

Value

[logical]

Dominance relation check.

Description

Check if a vector dominates another (dominates) or isdominated by another (isDominated). There are corresponding infixoperatorsdominates andisDominatedBy.

Usage

dominates(x, y)isDominated(x, y)x %dominates% yx %isDominatedBy% y

Arguments

x

[numeric]
First vector.

y

[numeric]
Second vector.

Value

[logical(1)]

Interface to ecr similar to the optim function.

Description

The most flexible way to setup evolutionary algorithms with ecr is byexplicitely writing the evolutionary loop utilizing various ecr utlity functions.However, in everyday life R users frequently need to optimize a single-objective R function.Theecr function thus provides a more R like interface for singleobjective optimization similar to the interface of theoptimfunction.

Usage

ecr(  fitness.fun,  minimize = NULL,  n.objectives = NULL,  n.dim = NULL,  lower = NULL,  upper = NULL,  n.bits,  representation,  mu,  lambda,  perm = NULL,  p.recomb = 0.7,  p.mut = 0.3,  survival.strategy = "plus",  n.elite = 0L,  log.stats = list(fitness = list("min", "mean", "max")),  log.pop = FALSE,  monitor = NULL,  initial.solutions = NULL,  parent.selector = NULL,  survival.selector = NULL,  mutator = NULL,  recombinator = NULL,  terminators = list(stopOnIters(100L)),  ...)

Arguments

fitness.fun

[function]
The fitness function.

minimize

[logical(n.objectives)]
Logical vector with ith entryTRUE if the ith objective offitness.funshall be minimized. If a single logical is passed, it is assumed to be validfor each objective.

n.objectives

[integer(1)]
Number of objectives ofobj.fun.Optional ifobj.fun is a benchmark function from packagesmoof.

n.dim

[integer(1)]
Dimension of the decision space.

lower

[numeric]
Vector of minimal values for each parameter of the decision space in caseof float or permutation encoding.Optional ifobj.fun is a benchmark function from packagesmoof.

upper

[numeric]
Vector of maximal values for each parameter of the decision space in caseof float or permutation encoding.Optional ifobj.fun is a benchmark function from packagesmoof.

n.bits

[integer(1)]
Number of bits to use for binary representation.

representation

[character(1)]
Genotype representation of the parameters. Available are “binary”,“float”, “permutation” and “custom”.

mu

[integer(1)]
Number of individuals in the population.

lambda

[integer(1)]
Number of individuals generated in each generation.

perm

[integer(1) |vector]
Either a single integer number. In this case the set is assumed to be1:perm.Alternatively, a set, i.e., a vector of elements can be passed which should form eachindividual.

p.recomb

[numeric(1)]
Probability of two parents to perform crossover.Default is 0.7.

p.mut

[numeric(1)]
The probability that the mutation operator will be applied to a child. Refers only to the application of the mutation operator, not to the probability of mutating individual genes of the respective child.Default is 0.1.

survival.strategy

[character(1)]
Determines the survival strategy used by the EA. Possible are “plus” fora classical (mu + lambda) strategy and “comma” for (mu, lambda).Default is “plus”.

n.elite

[integer(1)]
Number of fittest individuals of the current generation that shall be copied to thenext generation without changing. Keep in mind, that the algorithmdoes not care about this option if thesurvival.strategy is set to 'plus'.Default is 0.

log.stats

[list]
(Named) list of scalar functions to compute statistics on the fitness valuesin each generation. SeeinitLogger for more information.Default is to log fitness minimum, mean and maximum values.

log.pop

[logical(1)]
Shall the entire population be saved in each generation?Default isFALSE.

monitor

[function]
Monitoring function.Default isNULL, i.e. no monitoring.

initial.solutions

[list]
List of individuals which should be placed in the initial population.If the number of passed individualsis lower thanmu, the population will be filled upby individuals generated by the corresponding generator.Default isNULL, i.e., the entire population is generated by thepopulation generator.

parent.selector

[ecr_selector]
Selection operator which implements a procedure to copy individuals from agiven population to the mating pool, i. e., allow them to become parents.

survival.selector

[ecr_selector]
Selection operator which implements a procedurce to extract individuals froma given set, which should survive and set up the next generation.

mutator

[ecr_mutator]
Mutation operator of typeecr_mutator.

recombinator

[ecr_recombinator]
Recombination operator of typeecr_recombinator.

terminators

[list]
List of stopping conditions of type “ecr_terminator”.Default is to stop after 100 iterations.

...

[any]
Further arguments passed down tofitness.fun.

Value

[ecr_result]

Examples

fn = function(x) {   sum(x^2)}lower = c(-5, -5); upper = c(5, 5)res = ecr(fn, n.dim = 2L, n.objectives = 1L, lower = lower, upper = lower, representation = "float", mu = 20L, lambda = 10L,  mutator = setup(mutGauss, lower = lower, upper = upper))

Parallelization in ecr

Description

In ecr it is possible to parallelize the fitness function evaluationto make use, e.g., of multiple CP cores or nodes in a HPC cluster.For maximal flexibility this is realized by means of theparallelMap package(see theofficialGitHub page for instructions on how to set up parallelization).The different levels of parallelization can be specified in theparallelStart* function. At them moment only the level“ecr.evaluateFitness” is supported.

Keep in mind that parallelization comes along with some overhead. Thus activatingparallelization, e.g., for evaluation a fitness function which is evaluatedlightning-fast, may result in higher computation time. However, if the functionevaluations are computationally more expensive, parallelization leads tosignificant running time benefits.

Result object.

Description

S3 object returned byecr containing the best foundparameter setting and value in the single-objective case and the Pareto-front/-setin case of a multi-objective optimization problem. Moreover a set of furtherinformation, e.g., reason of termination, the control object etc. are returned.

The single objective result object contains the following fields:

task: Theecr_optimization_task.
best.x: Overall best parameter setting.
best.y: Overall best objective value.
log: Logger object.
last.population: Last population.
last.fitness: Numeric vector of fitness values of the last population.
message: Character string describing the reason of termination.

In case of a solved multi-objective function the result object contains thefollowing fields:

task: Theecr_optimization_task.
log: Logger object.
pareto.idx: Indizes of the non-dominated solutions in the last population.
pareto.front: (n x d) matrix of the approximated non-dominated front where nis the number of non-dominated points and d is the number of objectives.
pareto.set: Matrix of decision space values resulting with objective valuesgiven in pareto.front.
last.population: Last population.
message: Character string describing the reason of termination.

EMOA performance indicators

Description

Functions for the computation of unary and binary measures whichare useful for the evaluation of the performace of EMOAs. See the referencessection for literature on these indicators.

Given a set of pointspoints,emoaIndEps computes theunary epsilon-indicator provided a set of reference pointsref.points.

TheemoaIndHV function computes the hypervolume indicatorHyp(X, R, r). Given a set of points X (points), another set of referencepoints R (ref.points) (which maybe the true Pareto front) and a referencepoint r (ref.point) it is defined as Hyp(X, R, r) = HV(R, r) - HV(X, r).

FunctionemoaIndR1,emoaIndR2 andemoaIndR3 calculate theR1, R2 and R3 indicator respectively.

FunctionemoaIndMD computes the minimum distance indicator, i.e., the minimumEuclidean distance between two points of the setpoints while functionemoaIndM1 determines the mean Euclidean distance betweenpointsand points from a reference setref.points.

FunctionemoaIndC calculates the coverage of the setspoints (A) andref.points (B). This is the ratio of points in B which are dominated byat least one solution in A.

emoaIndONVG calculates the “Overall Non-dominated Vector Generation”indicator. Despite its complicated name it is just the number of non-dominated pointsinpoints.

FunctionsemoaIndSP andemoaIndDelta calculate spacing indicators.The former was proposed by Schott: first calculate the sum of squared distancesbetween minimal distancesof points to all other points and the mean of these minimaldistance. Next, normalize by the number of points minus 1 and finally calculate thesquare root. In contrast, Delta-indicator

Usage

emoaIndEps(points, ref.points, ...)emoaIndHV(points, ref.points, ref.point = NULL, ...)emoaIndR1(  points,  ref.points,  ideal.point = NULL,  nadir.point = NULL,  lambda = NULL,  utility = "tschebycheff",  ...)emoaIndR2(  points,  ref.points,  ideal.point = NULL,  nadir.point = NULL,  lambda = NULL,  utility = "tschebycheff",  ...)emoaIndR3(  points,  ref.points,  ideal.point = NULL,  nadir.point = NULL,  lambda = NULL,  utility = "tschebycheff",  ...)emoaIndMD(points, ...)emoaIndC(points, ref.points, ...)emoaIndM1(points, ref.points, ...)emoaIndONVG(points, ...)emoaIndGD(  points,  ref.points,  p = 1,  normalize = FALSE,  dist.fun = computeEuclideanDistance,  ...)emoaIndIGD(  points,  ref.points,  p = 1,  normalize = FALSE,  dist.fun = computeEuclideanDistance,  ...)emoaIndDeltap(  points,  ref.points,  p = 1,  normalize = FALSE,  dist.fun = computeEuclideanDistance,  ...)emoaIndSP(points, ...)emoaIndDelta(points, ...)

Arguments

points

[matrix]
Matrix of points.

ref.points

[matrix]
Set of reference points.

...

[any]
Not used at the moment.

ref.point

[numeric]
A single reference point used, e.g., for the computation of the hypervolumeindicator viaemoaIndHV. IfNULL thenadir point of the union of thepoints andref.points is used.

ideal.point

[numeric]
The utopia point of the true Pareto front, i.e., each component of the pointcontains the best value if the other objectives are neglected.

nadir.point

[numeric]
Nadir point of the true Pareto front.

lambda

[integer(1)]
Number of weight vectors to use in estimating the utility function.

utility

[character(1)]
Name of the utility function to use. Must be one of “weightedsum”,“tschebycheff” or “augmented tschbycheff”.

p

[numeric(1)]
Parameter p of the average Hausdoff metric.Default is 1.

normalize

[logical(1)]
Should the front be normalized on basis ofB?Default isFALSE.

dist.fun

[matrix]
Distance function to compute distance between points x and y. Expects a singlenumeric vector d with the coordinate-wise differences di = (xi - yi).Default iscomputeEuclideanDist.

Value

[numeric(1)] Epsilon indicator.

Computes the fitness value(s) for each individual of a given set.

Description

This function expects a list of individuals, computes the fitness and alwaysreturns a matrix of fitness values; even in single-objective optimization a(1 x n) matrix is returned for consistency, where n is the number of individuals.This function makes use ofparallelMap toparallelize the fitness evaluation.

Usage

evaluateFitness(control, inds, ...)

Arguments

control

[ecr_control]
Control object.

inds

[list]
List of individuals.

...

[any]
Optional parameters passed down to fitness function.

Value

[matrix].

Explode/implode data frame column(s).

Description

Given a data frame and a column name, functionexplode splits the content of a column by a specifieddelimiter (thus exploded) into multiple columns. Functionimplodedoes vice versa, i.e., given a non-empty set of column names ornumbers, the function glues together the columns. Hence, functionsexplode andimplode are kind of inverse to each other.

Usage

explode(df, col, by = ".", keep = FALSE, col.names = NULL)implode(df, cols, by = ".", keep = FALSE, col.name)

Arguments

df

[data.frame]
Data frame.

col

[character(1)]
Name of column which should be exploded.

by

[character(1)]
Delimeter used to split cell entries (forexplode) orglue them together (forimplode).

keep

[logical(1)]
Should exploded or imploded source column be kept?Default isFALSE.

col.names

[character]
Names of new columns.Default is “col.1”, ..., “col.k”, wherek is the number of elements each cell in columncol is split into.

cols

[character(1)]
Names of columns (or column number) which should be imploded.

col.name

[character(1)]
Name of new column.

Value

[data.frame] Modified data frame.

Examples

df = data.frame(x = 1:3, y = c("a.c", "a.b", "a.c"))df.ex = explode(df, col = "y", col.names = c("y1", "y2"))df.im = implode(df.ex, cols = c("y1", "y2"), by = "---", col.name = "y", keep = TRUE)

Filter approximation sets by duplicate objective vectors.

Description

Filter approximation sets by duplicate objective vectors.

Usage

filterDuplicated(x, ...)## S3 method for class 'data.frame'filterDuplicated(x, ...)## S3 method for class 'matrix'filterDuplicated(x, ...)## S3 method for class 'ecr_multi_objective_result'filterDuplicated(x, ...)## S3 method for class 'list'filterDuplicated(x, ...)

Arguments

x

[object]
Object of type data frame (objectives column-wise), matrix (objectives row-wise),ecr_multi_objective_result orlist (with components “pareto.front”)and “pareto.set”.

...

[any]
Not used at the moment

Value

[object] Modified inputx.

Note

Note that this may be misleading if there can be solutions with identicalobjective function values but different values in decision space.

Helper functions for offspring generation

Description

Functionmutate expects a control object, a list of individuals, and a mutationprobability. The mutation operator registered in the control object is then appliedwith the given probability to each individual.Functionrecombinate expects a control object, a list of individuals as well astheir fitness matrix and createslambda offspring individuals by recombining parentsfrominds. Which parents take place in the parent selection depends ontheparent.selector registered in the control object.Finally, functiongenerateOffspring is a wrapper for bothrecombinateandmutate. Both functions are applied subsequently to generate new individualsby variation and mutation.

Usage

generateOffspring(control, inds, fitness, lambda, p.recomb = 0.7, p.mut = 0.1)mutate(control, inds, p.mut = 0.1, slot = "mutate", ...)recombinate(  control,  inds,  fitness,  lambda = length(inds),  p.recomb = 0.7,  slot = "recombine",  ...)

Arguments

control

[ecr_control]
Control object.

inds

[list]
List of individuals.

fitness

[matrix]
Matrix of fitness values (each column contains the fitness value(s) of oneindividual).

lambda

[integer(1)]
Number of individuals generated in each generation.

p.recomb

[numeric(1)]
Probability of two parents to perform crossover.Default is 0.7.

p.mut

slot

[character(1)]
The slot of the control object which contains the registered operator to use.Default is “mutate” formutate and “recombine” forrecombinate.In most cases there is no need to change this. However, it might be useful if you make useof different mutation operators registerted, e.g., in the slots “mutate1” and “mutate2”.

...

[any]
Furhter arguments passed down to recombinator/mutator.There parameters will overwrite parameters inpar.list.

Value

[list] List of individuals.

Does the recombinator generate multiple children?

Description

Returns as to whether the recombinator generates multiple Children.

Usage

generatesMultipleChildren(recombinator)

Arguments

recombinator

[function]
Actual mutation operator.

Value

[logical]Boolean

Population generators

Description

Utility functions to build a set of individuals. The functiongen expects an R expression and a number n in order to create a listof n individuals based on the given expression. FunctionsgenBin,genPerm andgenReal are shortcuts for initializing populationsof binary strings, permutations or real-valued vectors respectively.

Usage

gen(expr, n)genBin(n, n.dim)genPerm(n, n.dim)genReal(n, n.dim, lower, upper)

Arguments

expr

[R expression]
Expression to generate a single individual.

n

[integer(1)]
Number of individuals to create.

n.dim

[integer(1)]
Dimension of the decision space.

lower

[numeric]
Vector of minimal values for each parameter of the decision space in caseof float encoding.

upper

[numeric]
Vector of maximal values for each parameter of the decision space in caseof float encoding.

Value

[list]

Extract fitness values from Pareto archive.

Description

Get all non-dominated points in objective space, i.e., an (m x n)matrix of fitness with m being the number of objectives and n being the numberof non-dominated points in the Pareto archive.

Usage

getFront(x)

Arguments

x

[ecr_pareto_archive]
Pareto archive.

Value

[matrix]

Extract individuals from Pareto archive.

Description

Get the non-dominated individuals logged in the Pareto archive.

Usage

getIndividuals(x)

Arguments

x

[ecr_pareto_archive]
Pareto archive.

Value

[list]

Number of children

Description

Returns the number children generated by the recombinator

Usage

getNumberOfChildren(recombinator)

Arguments

recombinator

[function]
Actual mutation operator.

Value

[numeric]Number of children generated

Number of parents needed for mating

Description

Returns the number of parents needed for mating.

Usage

getNumberOfParentsNeededForMating(recombinator)

Arguments

recombinator

[function]
Actual mutation operator.

Value

[numeric]Number of Parents need for mating

Access to logged population fitness.

Description

Returns the fitness values of all individuals as a data.framewith columns f1, ..., fo, where o is the number of objectives and column“gen” for generation.

Usage

getPopulationFitness(log, trim = TRUE)

Arguments

log

[ecr_logger]
The log generated byinitLogger.

trim

[logical(1)]
Should unused lines in the logged be cut off?Usually one wants this behaviour.Thus the default isTRUE.

Value

[list] List of populations.

Access to logged populations.

Description

Simple getter for the logged populations.

Usage

getPopulations(log, trim = TRUE)

Arguments

log

[ecr_logger]
The log generated byinitLogger.

trim

[logical(1)]
Should unused lines in the logged be cut off?Usually one wants this behaviour.Thus the default isTRUE.

Details

This functions throws an error if the logger was initialized withlog.pop = FALSE (seeinitLogger).

Value

[list] List of populations.

Get size of Pareto-archive.

Description

Returns the number of stored individuals in Pareto archive.

Usage

getSize(x)

Arguments

x

[ecr_pareto_archive]
Pareto archive.

Value

[integer(1)]

Access the logged statistics.

Description

Simple getter for the logged fitness statistics.

Usage

getStatistics(log, trim = TRUE)

Arguments

log

[ecr_logger]
The log generated byinitLogger.

trim

[logical(1)]
Should unused lines in the logged be cut off?Usually one wants this behaviour.Thus the default isTRUE.

Value

[data.frame] Logged statistics.

Get supported representations.

Description

Returns the character vector of representation which the operator supports.

Usage

getSupportedRepresentations(operator)

Arguments

operator

[ecr_operator]
Operator object.

Value

[character]Vector of representation types.

Control object generator.

Description

The control object keeps information on the objective function and a set ofevolutionary components, i.e., operators.

Usage

initECRControl(fitness.fun, n.objectives = NULL, minimize = NULL)

Arguments

fitness.fun

[function]
The fitness function.

n.objectives

[integer(1)]
Number of objectives ofobj.fun.Optional ifobj.fun is a benchmark function from packagesmoof.

minimize

[logical(n.objectives)]
Logical vector with ith entryTRUE if the ith objective offitness.funshall be minimized. If a single logical is passed, it is assumed to be validfor each objective.

Value

[ecr_control]

Initialize a log object.

Description

Logging is a central aspect of each EA. Besides the final solution(s)especially in research often we need to keep track of different aspects of theevolutionary process, e.g., fitness statistics. The logger of ecr keepstrack of different user-defined statistics and the population.It may also be used to check stopping conditions (seemakeECRTerminator). Mostimportant this logger is used internally by theecr black-box interface.

Usage

initLogger(  control,  log.stats = list(fitness = list("min", "mean", "max")),  log.extras = NULL,  log.pop = FALSE,  init.size = 1000L)

Arguments

control

[ecr_control]
Control object.

log.stats

[list]
List of lists for statistic computation on attributes of the individualsof the population. Each entry should be named by the attribute it should bebased on, e.g., fitness, and should contain a list of R functions as acharacter string or a a list with elementsfun for the function, andpars for additionalparameters which shall be passed to the corresponding function.Each function is required to return a scalar numeric value.By default the minimum, mean and maximum of the fitness values is computed.Since fitness statistics are the most important ones these do not have tobe stored as attributes, but can be passed as a matrix to the update function.

log.extras

[character]
Possibility to instruct the logger to store additionalscalar values in each generation. Named character vector where the namesindicate the value to store and the value indicates the corresponding data types.Currently we support all atomic modes ofvector expect “factor”and “raw”.

log.pop

[logical(1)]
Shall the entire population be saved in each generation?Default isFALSE.

init.size

[integer(1)]
Initial number of rows of the slot of the logger, where the fitnessstatistics are stored. The size of the statistics log is doubled each time anoverflow occurs.Default is 1000.

Value

[ecr_logger]An S3 object of classecr_logger with the following components:

log.stats: Thelog.stats list.
log.pop: Thelog.pop parameter.
init.size: Initial size of the log.
env: The actual log. This is an R environment which ensures, thatin-place modification is possible.

Note

Statistics are logged in adata.frame.

Examples

control = initECRControl(function(x) sum(x), minimize = TRUE,  n.objectives = 1L)control = registerECROperator(control, "mutate", mutBitflip, p = 0.1)control = registerECROperator(control, "selectForMating", selTournament, k = 2)control = registerECROperator(control, "selectForSurvival", selGreedy)log = initLogger(control,  log.stats = list(    fitness = list("mean", "myRange" = function(x) max(x) - min(x)),    age = list("min", "max")  ), log.pop = TRUE, init.size = 1000L) # simply pass stuff down to control object constructorpopulation = initPopulation(mu = 10L, genBin, n.dim = 10L)fitness = evaluateFitness(control, population)# append fitness to individuals and init agefor (i in seq_along(population)) {  attr(population[[i]], "fitness") = fitness[, i]  attr(population[[i]], "age") = 1L}for (iter in seq_len(10)) {  # generate offspring  offspring = generateOffspring(control, population, fitness, lambda = 5)  fitness.offspring = evaluateFitness(control, offspring)  # update age of population  for (i in seq_along(population)) {    attr(population[[i]], "age") = attr(population[[i]], "age") + 1L  }  # set offspring attributes  for (i in seq_along(offspring)) {    attr(offspring[[i]], "fitness") = fitness.offspring[, i]    # update age    attr(offspring[[i]], "age") = 1L  }  sel = replaceMuPlusLambda(control, population, offspring)  population = sel$population  fitness = sel$fitness  # do some logging  updateLogger(log, population, n.evals = 5)}head(getStatistics(log))

Initialize Pareto Archive.

Description

A Pareto archive is usually used to store all / a part of thenon-dominated points stored during a run of an multi-objective evolutionaryalgorithm.

Usage

initParetoArchive(control, max.size = Inf, trunc.fun = NULL)

Arguments

control

[ecr_control]
Control object.

max.size

[integer(1)]
Maximum capacity of the Pareto archive, i.e., the maximal number of non-dominatedpoints which can be stored in the archive. Default isInf, i.e., (theoretically)unbounded capacity.

trunc.fun

[function(archive, inds, fitness, ...)]
In case the archive is limited in capacity, i.e.,max.size is not infinite,this function is called internally if an archive overflow occurs. This functionexpects thearchive, a list of individualsinds, a matrix of fitnessvalues (each column contains the fitness value(s) of one individual)fitnessand further optional arguments... which may be used by the internalsoftrunc.fun. The function must return a list with components “fitness”and “inds” which shall be the subsets offitness andindsrespectively, which should be kept by the archive.

Value

[ecr_pareto_archive]

Helper function to build initial population.

Description

Generates the initial population. Optionally a set of initial solutionscan be passed.

Usage

initPopulation(mu, gen.fun, initial.solutions = NULL, ...)

Arguments

mu

[integer(1)]
Number of individuals in the population.

gen.fun

[function]
Function used to generate initial solutions, e.g.,genBin.

initial.solutions

...

[any]
Further parameters passed togen.fun.

Value

[ecr_population]

Check if ecr operator supports given representation.

Description

Check if the given operator supportds a certain representation, e.g., “float”.

Usage

is.supported(operator, representation)

Arguments

operator

[ecr_operator]
Object of typeecr_operator.

representation

[character(1)]
Representation as a string.

Value

[logical(1)]TRUE, if operator supports the representation type.

Check if given function is an ecr operator.

Description

Checks if the passed object is of typeecr_operator.

Usage

isEcrOperator(obj)

Arguments

obj

[any]
Arbitrary R object to check.

Value

[logical(1)]

Factory method for monitor objects.

Description

Monitor objects serve for monitoring the optimization process, e.g., printingsome status messages to the console. Each monitor includes the functionsbefore,step andafter, each being a function and expectinga loglog of typeecr_logger and... as the only parameters.This way the logger has access to the entire log.

Usage

makeECRMonitor(before = NULL, step = NULL, after = NULL, ...)

Arguments

before

[function]
Function called one time after initialization of the EA.

step

[function]
Function applied after each iteration of the algorithm.

after

[function]
Function applied after the EA terminated.

...

[any]
Not used.

Value

[ecr_monitor]Monitor object.

Constructor for EMOA indicators.

Description

Simple wrapper for function which computeperformance indicators for multi-objective stochastic algorithm.Basically this function appends some meta information to thepassed functionfun.

Usage

makeEMOAIndicator(fun, minimize, name, latex.name)

Arguments

fun

[function(points, ...)]
Function which expects a numeric matrix “points” as first argument.Optional named arguments (often “ref.point” for a referencepoint or “ref.points” for a reference set, e.g., the true Pareto-front)are allowed. See implementations of existing indicators for examples.

minimize

[logical(1)]
Lower values indicate better performance?

name

[character(1)]
Short name of the indicator. Used, e.g., as column name for the indicatorin the data.frame returned bycomputeIndicators.

latex.name

[character(1)]
LaTeX representation of the indicator. Used in LaTeX-table output statisticaltests (seetoLatex).

Value

[function(points, ...)] Argumentfun with all otherarguments appended.

Construct a mutation operator.

Description

Helper function which constructs a mutator, i. e., a mutation operator.

Usage

makeMutator(mutator, supported = getAvailableRepresentations())

Arguments

mutator

[function]
Actual mutation operator.

supported

[character]
Vector of strings/names of supported parameter representations. Possible choices:“permutation”, “float”, “binary” or “custom”.

Value

[ecr_mutator]Mutator object.

Construct evolutionary operator.

Description

Helper function which constructs an evolutionary operator.

Usage

makeOperator(operator, supported = getAvailableRepresentations())

Arguments

operator

[function]
Actual operator.

supported

[character]
Vector of names of supported parameter representations. Possible choices:“permutation”, “float”, “binary” or “custom”.

Value

[ecr_operator] Operator object.

Note

In general you will not need this function, but rather one of itsderiviatives likemakeMutator ormakeSelector.

Creates an optimization task.

Description

An optimization task consists of the fitness/objective function, thenumber of objectives, the “direction” of optimization, i.e.,which objectives should be minimized/maximized and the names of theobjectives.

Usage

makeOptimizationTask(  fun,  n.objectives = NULL,  minimize = NULL,  objective.names = NULL)

Arguments

fun

[function |smoof_function]
Fitness/objective function.

n.objectives

[integer(1)]
Number of objectives. This must be a positive integer value unlessfunis of typesmoof_function.

minimize

[logical]
A logical vector indicating which objectives to minimize/maximize. By defaultall objectives are assumed to be minimized.

objective.names

[character]
Names for the objectuves.Default isNULL. In this case the names are set to y1, ..., yn withn equal ton.objectives and simply y in the single-objective case.

Value

[ecr_optimization_task]

Construct a recombination operator.

Description

Helper function which constructs a recombinator, i. e., a recombination operator.

Usage

makeRecombinator(  recombinator,  supported = getAvailableRepresentations(),  n.parents = 2L,  n.children = NULL)

Arguments

recombinator

[function]
Actual mutation operator.

supported

[character]
Vector of strings/names of supported parameter representations. Possible choices:“permutation”, “float”, “binary” or “custom”.

n.parents

[integer(1)]
Number of parents supported.

n.children

[integer(1)]
How many children does the recombinator produce?Default is1.

Value

[ecr_recombinator]Recombinator object.

Note

If a recombinator returns more than one child, themultiple.childrenparameter needs to beTRUE, which is the default. In case of multiplechildren produced these have to be placed within a list.

Construct a selection operator.

Description

Helper function which defines a selector method, i. e., an operator whichtakes the population and returns a part of it for mating or survival.

Usage

makeSelector(  selector,  supported = getAvailableRepresentations(),  supported.objectives,  supported.opt.direction = "minimize")

Arguments

selector

[function]
Actual selection operator.

supported

[character]
Vector of strings/names of supported parameter representations. Possible choices:“permutation”, “float”, “binary” or “custom”.

supported.objectives

[character]
At least one of “single-objective” or “multi-objective”.

supported.opt.direction

[character(1-2)]
Does the selector work for maximization tasks xor minimization tasks or both?Default is “minimize”, which means that the selector selectsin favour of low fitness values.

Value

[ecr_selector]Selector object.

Generate stopping condition.

Description

Wrap a function within a stopping condition object.

Usage

makeTerminator(condition.fun, name, message)

Arguments

condition.fun

[function]
Function which takes a logger objectlog (seeinitLogger)and returns a single logical.

name

[character(1)]
Identifier for the stopping condition.

message

[character(1)]
Message which should be stored in the termination object, if the stoppingcondition is met.

Value

[ecr_terminator]

mcMST

Description

Pareto-front approximations for some graph problems obtained by severalalgorithms for the multi-criteria minimum spanning tree (mcMST) problem.

Usage

mcMST

Format

A data frame with four variables:

f1: First objective (to be minimized).
f2: Second objective (to be minimized).
algorithm: Short name of algorithm used.
prob: Short name of problem instance.
repl: Algorithm run.

The data is based on themcMST package.

Bitplip mutator.

Description

This operator works only on binary representation and flips each bitwith a given probabilityp \in (0, 1). Usually it is recommended tosetp = \frac{1}{n} wheren is the number of bits in therepresentation.

Usage

mutBitflip(ind, p = 0.1)

Arguments

ind

[binary]
Binary vector, i.e., vector with elements 0 and 1 only.

p

[numeric(1)]
Probability to flip a single bit.Default is0.1.

Value

[binary]

References

[1] Eiben, A. E. & Smith, James E. (2015). Introduction to EvolutionaryComputing (2nd ed.). Springer Publishing Company, Incorporated. 52.

Gaussian mutator.

Description

Default Gaussian mutation operator known from Evolutionary Algorithms.This mutator is applicable only forrepresentation="float". Givenan individual\mathbf{x} \in R^l this mutator adds a Gaussiandistributed random value to each component of\mathbf{x}, i.~e.,\tilde{\mathbf{x}}_i = \mathbf{x}_i + \sigma \mathcal{N}(0, 1).

Usage

mutGauss(ind, p = 1L, sdev = 0.05, lower, upper)

Arguments

ind

[numeric]
Numeric vector / individual to mutate.

p

[numeric(1)]
Probability of mutation for the gauss mutation operator.

sdev

[numeric(1)
Standard deviance of the Gauss mutation, i. e., the mutation strength.

lower

[numeric]
Vector of minimal values for each parameter of the decision space.

upper

[numeric]
Vector of maximal values for each parameter of the decision space.

Value

[numeric]

References

[1] Beyer, Hans-Georg & Schwefel, Hans-Paul (2002). Evolution strategies. Kluwer Academic Publishers.

[2] Mateo, P. M. & Alberto, I. (2011). A mutation operator basedon a Pareto ranking for multi-objective evolutionary algorithms. Springer Science+Business Meda. 57.

Insertion mutator.

Description

The Insertion mutation operator selects a position random and inserts it ata random position.

Usage

mutInsertion(ind)

Arguments

ind

[integer]
Permutation of integers, i.e., vector of integer values.

Value

[integer]

Inversion mutator.

Description

The Inversion mutation operator selects two positions within the chromosome atrandom and inverts the elements inbetween.

Usage

mutInversion(ind)

Arguments

ind

[integer]
Permutation of integers, i.e., vector of integer values.

Value

[integer]

Jump mutator.

Description

The jump mutation operator selects two positions within the chromosome atrandom, saya andb witha < b. Next, all elements atpositionsb-1, b-2, ..., a are shifted to the right by one positionand finally the element at positionb is assigned at positiona.

Usage

mutJump(ind)

Arguments

ind

[integer]
Permutation of integers, i.e., vector of integer values.

Value

[integer]

Polynomial mutation.

Description

Performs an polynomial mutation as used in the SMS-EMOA algorithm.Polynomial mutation tries to simulate the distribution of the offspring of binary-encoded bit flip mutations based on real-valued decision variables. Polynomial mutation favors offspring nearer to the parent.

Usage

mutPolynomial(ind, p = 0.2, eta = 10, lower, upper)

Arguments

ind

[numeric]
Numeric vector / individual to mutate.

p

[numeric(1)]
Probability of mutation for each gene of an offspring. In other words, the probability that the value (allele) of a given gene will change.Default is 0.2

eta

[numeric(1)
Distance parameter to control the shape of the mutation distribution. Larger values generate offspring closer to the parents.Default is 10.

lower

[numeric]
Vector of minimal values for each parameter of the decision space.Must have the same length asind.

upper

[numeric]
Vector of maximal values for each parameter of the decision space.Must have the same length asind.

Value

[numeric]

References

[1] Deb, Kalyanmoy & Goyal, Mayank. (1999). A Combined Genetic Adaptive Search (GeneAS) for Engineering Design. Computer Science and Informatics. 26.Retrieved from http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.27.767&rep=rep1&type=pdf

Scramble mutator.

Description

The Scramble mutation operator selects two positions within the chromosome atrandom and randomly intermixes the subsequence between these positions.

Usage

mutScramble(ind)

Arguments

ind

[integer]
Permutation of integers, i.e., vector of integer values.

Value

[integer]

Swap mutator.

Description

Chooses two positions at random and swaps the genes.

Usage

mutSwap(ind)

Arguments

ind

[integer]
Permutation of integers, i.e., vector of integer values.

Value

[integer]

Uniform mutator.

Description

This mutation operator works on real-valued genotypes only. It selects a positionin the solution vector at random and replaced it with a uniformally chosen valuewithin the box constraints of the corresponding parameter.This mutator may proof beneficial in early stages of the optimization process,since it distributes points widely within the box constraints and thus mayhinder premature convergence. However, in later stages - when fine tuning isnecessary, this feature is disadvantegous.

Usage

mutUniform(ind, lower, upper)

Arguments

ind

[numeric]
Numeric vector / individual to mutate.

lower

[numeric]
Vector of minimal values for each parameter of the decision space.

upper

[numeric]
Vector of maximal values for each parameter of the decision space.

Value

[numeric]

Formatter for table cells of LaTeX tables.

Description

This formatter function should be applied totables where each table cell contains ap-value ofa statistical significance test. SeetoLatexfor an application.

Usage

niceCellFormater(cell, alpha = 0.05)

Arguments

cell

[any]
Cell value. In the majority of cases this will be a numeric value.

alpha

[numeric(1)]
Significance level of underlying statistical test.Default is 0.05.

Value

Formatted output.

Normalize approximations set(s).

Description

Normalization is done by subtracting themin.value for each dimensionand dividing by the differencemax.value - min.value for each dimensionCertain EMOA indicators require all elements to be strictly positive. Hence, an optionaloffset is added to each element which defaults to zero.

Usage

normalize(x, obj.cols, min.value = NULL, max.value = NULL, offset = NULL)

Arguments

x

[matrix |data.frame]
Either a numericmatrix (each column corresponds to a point) or adata.frame with columns at leastobj.cols.

obj.cols

[character(>= 2)]
Column names of the objective functions.

min.value

[numeric]
Vector of minimal values of lengthnrow(x).Only relevant ifx is a matrix.Default is the row-wise minimum ofx.

max.value

[numeric]
Vector of maximal values of lengthnrow(x).Only relevant ifx is a matrix.Default is the row-wise maximum ofx.

offset

[numeric]
Numeric constant added to each normalized element.Useful to make all objectives strictly positive, e.g., located in[1,2].

Value

[matrix |data.frame]

Note

In case a data.frame is passed and a “prob” column exists, normalizationis performed for each unique element of the “prob” column independently (ifexistent).

Implementation of the NSGA-II EMOA algorithm by Deb.

Description

The NSGA-II merges the current population and the generated offspring andreduces it by means of the following procedure: It first applies the nondominated sorting algorithm to obtain the nondominated fronts. Starting withthe first front, it fills the new population until the i-th front does not fit.It then applies the secondary crowding distance criterion to select the missingindividuals from the i-th front.

Usage

nsga2(  fitness.fun,  n.objectives = NULL,  n.dim = NULL,  minimize = NULL,  lower = NULL,  upper = NULL,  mu = 100L,  lambda = mu,  mutator = setup(mutPolynomial, eta = 25, p = 0.2, lower = lower, upper = upper),  recombinator = setup(recSBX, eta = 15, p = 0.7, lower = lower, upper = upper),  terminators = list(stopOnIters(100L)),  ...)

Arguments

fitness.fun

[function]
The fitness function.

n.objectives

[integer(1)]
Number of objectives ofobj.fun.Optional ifobj.fun is a benchmark function from packagesmoof.

n.dim

[integer(1)]
Dimension of the decision space.

minimize

[logical(n.objectives)]
Logical vector with ith entryTRUE if the ith objective offitness.funshall be minimized. If a single logical is passed, it is assumed to be validfor each objective.

lower

[numeric]
Vector of minimal values for each parameter of the decision space in caseof float or permutation encoding.Optional ifobj.fun is a benchmark function from packagesmoof.

upper

[numeric]
Vector of maximal values for each parameter of the decision space in caseof float or permutation encoding.Optional ifobj.fun is a benchmark function from packagesmoof.

mu

[integer(1)]
Number of individuals in the population.Default is 100.

lambda

[integer(1)]
Offspring size, i.e., number of individuals generated by variation operatorsin each iteration.Default is 100.

mutator

[ecr_mutator]
Mutation operator of typeecr_mutator.

recombinator

[ecr_recombinator]
Recombination operator of typeecr_recombinator.

terminators

[list]
List of stopping conditions of type “ecr_terminator”.Default is to stop after 100 iterations.

...

[any]
Further arguments passed down to fitness function.

Value

[ecr_multi_objective_result]

Note

This is a pure R implementation of the NSGA-II algorithm. It hides the regularecr interface and offers a more R like interface while still being quiteadaptable.

References

Deb, K., Pratap, A., and Agarwal, S. A Fast and Elitist Multiobjective GeneticAlgorithm: NSGA-II. IEEE Transactions on Evolutionary Computation, 6 (8) (2002),182-197.

Plot distribution of EMOA indicators.

Description

Visualizes of empirical distributions of unary EMOA indicatorbased on the results ofcomputeIndicators.

Usage

plotDistribution(  inds,  plot.type = "boxplot",  fill = "algorithm",  facet.type = "grid",  facet.args = list(),  logscale = character())

Arguments

inds

[data.frame]
Data frame with columns “algorithm”, “prob”, “repl” andone additional column per EMOA indicator.

plot.type

[character(1)]
Either “boxplot” (the default) for boxplots or “violin” forviolin plots.

fill

[character(1)]
Variable used to fill boxplots.Default is “algorithm”.

facet.type

[character(1)]
Which faceting method to use? Pass “wrap” forfacet_wrapor “grid” forfacet_grid.Default is “wrap”.

facet.args

[list]
Named list of arguments passed down tofacet_wrap orfacet_grid respectively (depends onfacet.type).E.g.,nrow to change layout.Default is the empty list. In this case data is grouped by problem and indicator.

logscale

[character]
Vector of indicator names which should be log-transformed prior tovisualization.Default is the empty character vector.

Value

[ggplot]

Draw scatterplot of Pareto-front approximation

Description

The function expects a data.frame or a matrix. By default the first2 or 3 columns/rows are assumed to contain the elements of the approximation sets.Depending on the number of numeric columns (in case of a data.frame) or thenumber of rows (in case of a matrix) the function internally callsplotScatter2d orplotScatter3d.

Usage

plotFront(x, obj.names = NULL, minimize = TRUE, ...)

Arguments

x

[matrix |data.frame]
Object which contains the approximations sets.

obj.names

[character]
Optional objectives names.Default isc("f1", "f2").

minimize

[logical]
Logical vector with ith entryTRUE if the ith objective shall be minimized.If a single logical is passed, it is assumed to be valid for each objective.If the matrix is of typeecr_fitness_matrix (this is the case if it isproduced by one of ecr2's utility functions, e.g.evaluateFitness),the appendedminimize attribute is the default.

...

[any]
Not used at the moment.

Value

[ggplot]ggplot object.

Plot heatmap.

Description

Given a matrix or list of matrizesx this functionvisualizes each matrix with a heatmap.

Usage

plotHeatmap(x, value.name = "Value", show.values = FALSE)

Arguments

x

[matrix |list[matrix]]
Either a matrix or a list of matrizes.

value.name

[character(1)]
Name for the values represented by the matrix.Internally, the matrix is transformed into adata.frameviamelt in order to obtain a formatwhich may be processed byggplot easily.Default is “Value”.

show.values

[logical(1L)]
Should the values be printed within the heatmap cells?Default isFALSE.

Value

[ggplot] ggplot object.

Examples

# simulate two (correlation) matrizesx = matrix(runif(100), ncol = 10)y = matrix(runif(100), ncol = 10)## Not run: pl = plotHeatmap(x)pl = plotHeatmap(list(x, y), value.name = "Correlation")pl = plotHeatmap(list(MatrixX = x, MatrixY = y), value.name = "Correlation")## End(Not run)

Visualize bi-objective Pareto-front approximations.

Description

Given a data frame with the results of (multiple) runs of (multiple)different multi-objective optimization algorithms on (multiple) problem instancesthe function generatesggplot plots of the obtainedPareto-front approximations.

Usage

plotScatter2d(  df,  obj.cols = c("f1", "f2"),  shape = "algorithm",  colour = NULL,  highlight.algos = NULL,  offset.highlighted = 0,  title = NULL,  subtitle = NULL,  facet.type = "wrap",  facet.args = list())

Arguments

df

[data.frame]
Data.frame with columns at leastobj.cols, “prob” and “algorithm”.

obj.cols

[character(>= 2)]
Column names of the objective functions.Default isc("f1", "f2").

shape

[character(1)]
Name of column which shall be used to define shape of points.Default is “algorithm”.

colour

[character(1)]
Name of column which shall be used to define colour of points.Default isNULL, i.e., coloring is deactivated.

highlight.algos

[character(1)]
Name of algorithm to highlight exclusively. Useful to highlight, e.g., thetrue Pareto-optimal front (if known) or some reference set.Default isNULL, i.e., unknown.

offset.highlighted

[numeric(1)]
Numeric offset used to shift set (seehighlight.algos)which should be highlighted.Even though this produces objective vectors itmay be used to make visible reference sets which otherwise wouldbe hidden by overlap of multiple other approximation sets.

title

[character(1)]
Plot title.

subtitle

[character(1)]
Plot subtitle.

facet.type

[character(1)]
Which faceting method to use? Pass “wrap” forfacet_wrapor “grid” forfacet_grid.Default is “wrap”.

facet.args

Value

[ggplot] A ggplot object.

Note

At the moment only approximations of bi-objective functions are supported.

Examples

## Not run: # load examplary datadata(mcMST)print(head(mcMST))# no customization; use the defaultspl = plotFronts(mcMST)# algo PRIM is obtained by weighted sum scalarization# Since the front is (mainly) convex we highlight these solutionspl = plotFronts(mcMST, highlight.algos = "PRIM")# customize layoutpl = plotFronts(mcMST, title = "Pareto-approximations",  subtitle = "based on different mcMST algorithms.", facet.args = list(nrow = 2))## End(Not run)

Visualize three-objective Pareto-front approximations.

Description

Given a data frame with the results of (multiple) runs of (multiple)different three-objective optimization algorithms on (multiple) problem instancesthe function generates 3D scatterplots of the obtained Pareto-front approximations.

Usage

plotScatter3d(  df,  obj.cols = c("f1", "f2", "f3"),  max.in.row = 4L,  package = "scatterplot3d",  ...)

Arguments

df

[data.frame]
Data.frame with columns at leastobj.cols, “prob” and “algorithm”.

obj.cols

[character(>= 3)]
Column names of the objective functions.Default isc("f1", "f2", "f3").

max.in.row

[integer(1)]
Maximum number of plots to be displayed side by side in a row.Default is 4.

package

[character(1L)]
Which package to use for 3d scatterplot generation?Possible choices are “scatterplot3d”, “plot3D”, “plot3Drgl”or “plotly”.Default is “scatterplot3d”.

...

[any]
Further arguments passed down to scatterplot function.

Value

Nothing

Generate line plot of logged statistics.

Description

Expects a data.frame of logged statistics, e.g., extracted froma logger object by callinggetStatistics, and generates a basicline plot. The plot is generated with theggplot2 package and the ggplotobject is returned.

Usage

plotStatistics(x, drop.stats = character(0L))

Arguments

x

[ecr_statistics |ecr_logger]
Logger object or statistics data frame from logger object.

drop.stats

[character]
Names of logged statistics to be dropped.Default is the empty character, i.e., not to drop any stats.

One-point crossover recombinator.

Description

The one-point crossover recombinator is defined for float and binaryrepresentations. Given two real-valued/binary vectors of length n, theselector samples a random position i between 1 and n-1. In the next stepit creates two children. The first part of the first child contains of thesubvector from position 1 to position i of the first parent, the second partfrom position i+1 to n is taken from the second parent. The second childis build analogously.If the parents are list of real-valued/binary vectors, the procedure describedabove is applied to each element of the list.

Usage

recCrossover(inds)

Arguments

inds

[list]
Parents, i.e., list of exactly two numeric or binary vectors of equal length.

Value

[list]

Indermediate recombinator.

Description

Intermediate recombination computes the component-wise mean value of thek given parents. It is applicable only for float representation.

Usage

recIntermediate(inds)

Arguments

inds

[list]
Parents, i.e., list of exactly two numeric vectors of equal length.

Value

[numeric] Single offspring.

Ordered-Crossover (OX) recombinator.

Description

This recombination operator is specifically designed for permutations.The operators chooses two cut-points at random and generates two offspringas follows: a) copy the subsequence of one parent and b) remove the copiednode indizes from the entire sequence of the second parent from the sescondcut point and b) fill the remaining gaps with this trimmed sequence.

Usage

recOX(inds)

Arguments

inds

[list]
Parents, i.e., list of exactly two permutations (vectors of integer values) of equal length.

Value

[list]

Partially-Mapped-Crossover (PMX) recombinator.

Description

This recombination operator is specifically designed for permutations.The operators chooses two cut-points at random and generates two offspringas follows: a) copy the subsequence of one parent and b) fill the remainingpositions while preserving the order and position of as many genes as possible.

Usage

recPMX(inds)

Arguments

inds

[numeric]
Parents, i.e., list of exactly two permutations of equal length.

Value

[ecr_recombinator]

Simulated Binary Crossover (SBX) recombinator.

Description

The Simulated Binary Crossover was first proposed by [1]. It i suited forfloat representation only and creates two offspring. Given parentsp_1, p_2the offspring are generated asc_{1/2} = \bar{x} \pm \frac{1}{2}\beta(p_2 - p_1)where\bar{x} = \frac{1}{2}(p_1 + p_2), p_2 > p_1. This way\bar{c} = \bar{x}is assured.

Usage

recSBX(inds, eta = 5, p = 1, lower, upper)

Arguments

inds

[numeric]
Parents, i.e., list of exactly two numeric vectors of equal length.

eta

[numeric(1)]
Parameter eta, i.e., the distance parameters of the crossover distribution.

p

[numeric(1)]
Crossover probability for each gene. Default is1.0.

lower

[numeric]
Vector of minimal values for each parameter of the decision space.

upper

[numeric]
Vector of maximal values for each parameter of the decision space.

Value

[ecr_recombinator]

Note

This is the default recombination operator used in the NSGA-II EMOA (seensga2).

References

[1] Deb, K. and Agrawal, R. B. (1995). Simulated binary crossover for continuoussearch space. Complex Systems 9(2), 115-148.

Uniform crossover recombinator.

Description

Produces two child individuals. The i-th gene is from parent1 with probabilityp and from parent2 with probability1-p.

Usage

recUnifCrossover(inds, p = 0.5)

Arguments

inds

[list]
Parents, i.e., list of exactly two numeric or binary vectors of equal length.

p

[numeric(1)]
Probability to select gene from parent1.

Value

[list]

Combine multiple data frames into a single data.frame.

Description

Combine multiple data frames into a single data.frame.

Usage

reduceToSingleDataFrame(res = list(), what = NULL, group.col.name)

Arguments

res

[list]
List of data frames or other lists which contain a data frame as one of thecomponents which is selected bywhat. Ifres is a named listthose names are used to fill the group column. Otherwise the names are 1 tolength(res) by default.

what

[character(1)]
Which component of each list element inres to choose. Set this toNULL, ifres is not complex, i.e., is not a list of lists.

group.col.name

[character(1)]
Name for the grouping column.

Register operators to control object.

Description

In ecr the control object stores information on the fitnessfunction and serves as a storage for evolutionary components used by your evolutionaryalgorithm. This function handles the registration process.

Usage

registerECROperator(control, slot, fun, ...)

Arguments

control

[ecr_control]
Control object.

slot

[character(1)]
Name of the field in the control object where to store the operator.

fun

[function]
Actual operator. In order to use the various helper functions of ecr one needsto stick to a simple convention: The first argument offunction shouldbe the individual to mutate, a list of individuals for recombination or a matrixof fitness values for recombination. If one does not want to use the correspondinghelpers, e.g.,mutate, the signature of the function does not matter. However,in this case you are responsible to pass arguments correctly.

...

[any]
Further arguments forfun. These arguments are stored in the control objectand passed on tofun.

Value

[ecr_control]

(mu + lambda) selection

Description

Takes a population of mu individuals and another set of lambdaoffspring individuals and selects mu individuals out of the union set accordingto the survival selection strategy stored in the control object.

Usage

replaceMuPlusLambda(  control,  population,  offspring,  fitness = NULL,  fitness.offspring = NULL)replaceMuCommaLambda(  control,  population,  offspring,  fitness = NULL,  fitness.offspring = NULL,  n.elite = base::max(ceiling(length(population) * 0.1), 1L))

Arguments

control

[ecr_control]
Control object.

population

[list]
Current set of individuals.

offspring

[list]
Another set of individuals.

fitness

[matrix]
Matrix of fitness values for the individuals frompopulation.This is only optional in the case that each individual inpopulation hasan attribute “fitness”.

fitness.offspring

[matrix]
Matrix of fitness values for the individuals fromoffspring.This is only optional in the case that each individual inoffspring hasan attribute “fitness”.

n.elite

Value

[list] List with selected population and corresponding fitness matrix.

Dominated Hypervolume selector.

Description

Performs non-dominated sorting and drops the individual from the last frontwith minimal hypervolume contribution. This selector is the basis of theS-Metric Selection Evolutionary Multi-Objective Algorithm, termed SMS-EMOA(seesmsemoa).

Usage

selDomHV(fitness, n.select, ref.point)

Arguments

fitness

[matrix]
Matrix of fitness values (each column contains the fitness value(s) of oneindividual).

n.select

[integer(1)]
Number of elements to select.

ref.point

[numeric]
Reference point for hypervolume computation.

Value

[integer] Vector of survivor indizes.

Note

Note that the current implementation expectsn.select = ncol(fitness) - 1and the selection process quits with an error message ifn.select is greaterthan 1.

Simple selector.

Description

Sorts the individuals according to their fitness value in increasing orderand selects the best ones.

Usage

selGreedy(fitness, n.select)

Arguments

fitness

[matrix]
Matrix of fitness values (each column contains the fitness value(s) of oneindividual).

n.select

[integer(1)]
Number of elements to select.

Value

[integer] Vector of survivor indizes.

Non-dominated sorting selector.

Description

Applies non-dominated sorting of the objective vectors and subsequent crowdingdistance computation to select a subset of individuals. This is the selector usedby the famous NSGA-II EMOA (seensga2).

Usage

selNondom(fitness, n.select)

Arguments

fitness

[matrix]
Matrix of fitness values (each column contains the fitness value(s) of oneindividual).

n.select

[integer(1)]
Number of elements to select.

Value

[setOfIndividuals]

Rank Selection Operator

Description

Rank-based selection preserves a constant selection pressure by sorting thepopulation on the basis of fitness, and then allocating selectionprobabilities to individuals according to their rank, rather than accordingto their actual fitness values.

Usage

selRanking(fitness, n.select, s = 1.5, scheme = "linear")

Arguments

fitness

[matrix]
Matrix of fitness values (each column contains the fitness value(s) of oneindividual).

n.select

[integer(1)]
Number of elements to select.

s

[numeric(1)]
Selection pressure for linear ranking scheme with value range[0,1].Ignored ifscheme is set to “exponential”.Default is 1.5.

scheme

[character(1)]
Mapping from rank number to selection probability, either“linear” or “exponential”.

Value

[setOfIndividuals]

References

Eiben, A. E., & Smith, J. E. (2007). Introduction to evolutionary computing.Berlin: Springer.

Roulette-wheel / fitness-proportional selector.

Description

The chance of an individual to get selected is proportional to its fitness,i.e., better individuals get a higher chance to take part in the reproductionprocess. Low-fitness individuals however, have a positive fitness as well.

Usage

selRoulette(fitness, n.select, offset = 0.1)

Arguments

fitness

[matrix]
Matrix of fitness values (each column contains the fitness value(s) of oneindividual).

n.select

[integer(1)]
Number of elements to select.

offset

[numeric(1)]
In case of negative fitness values all values are shifted towards positivevalues by adding the negative of the minimal fitness value. However, in thiscase the minimal fitness value after the shifting process is zero. Theoffset is a positive numeric value which is added additionally to eachshifted fitness value. This way even the individual with the smallest fitnessvalue has a positive porbability to be selected.Default is0.1.

Details

Fitness proportional selection can be naturally applied to single objectivemaximization problems. However, negative fitness values can are problematic.The Roulette-Wheel selector thus works with the following heuristic: ifnegative values occur, the negative of the smallest fitness value is addedto each fitness value. In this case to avoid the smallest shifted fitnessvalue to be zero and thus have a zero probability of being selected an additionalpositive constantoffset is added (see parameters).

Value

[setOfIndividuals]

Simple (naive) selector.

Description

Just for testing. Actually does not really select, but instead returns a randomsample ofncol(fitness) indizes.

Usage

selSimple(fitness, n.select)

Arguments

fitness

[matrix]
Matrix of fitness values (each column contains the fitness value(s) of oneindividual).

n.select

[integer(1)]
Number of elements to select.

Value

[setOfIndividuals]

k-Tournament selector.

Description

k individuals from the population are chosen randomly and the best one isselected to be included into the mating pool. This process is repeated untilthe desired number of individuals for the mating pool is reached.

Usage

selTournament(fitness, n.select, k = 3L)

Arguments

fitness

[matrix]
Matrix of fitness values (each column contains the fitness value(s) of oneindividual).

n.select

[integer(1)]
Number of elements to select.

k

[integer(1)]
Number of individuals to participate in each tournament. Default is2L.

Value

[integer] Vector of survivor indizes.

Select individuals.

Description

This utility functions expect a control object, a matrix offitness values - each column containing the fitness value(s) of one individual -and the number of individuals to select.The corresponding selector, i.e., mating selector forselectForMatingor survival selector forselectForSurvival is than called internallyand a vector of indizes of selected individuals is returned.

Usage

selectForMating(control, fitness, n.select)selectForSurvival(control, fitness, n.select)

Arguments

control

[ecr_control]
Control object.

fitness

[matrix]
Matrix of fitness values (each column contains the fitness value(s) of oneindividual).

n.select

[integer(1)]
Number of individuals to select.

Details

Both functions check the optimization directions stored in the taskinside the control object, i.e., whether to minimize or maximize each objective,and transparently prepare/transform thefitness matrix for the selector.

Value

[integer] Integer vector with the indizes of selected individuals.

Check if one set is better than another.

Description

The function checks, whether each points of the second set of pointsis dominated by at least one point from the first set.

Usage

setDominates(x, y)

Arguments

x

[matrix]
First set of points.

y

[matrix]
Second set of points.

Value

[logical(1)]

Set up parameters for evolutionary operator.

Description

This function builds a simple wrapper around an evolutionary operator, i.e.,mutator, recombinator or selector and defines its parameters. The result is afunction that does not longer depend on the parameters. E.g.,fun = setup(mutBitflip, p = 0.3)initializes a bitflip mutator with mutation probability 0.3. Thus,the following calls have the same behaviour:fun(c(1, 0, 0)) andmutBitflip(fun(c(1, 0, 0), p = 0.3).Basically, this type of preinitialization is only neccessary if operatorswith additional parameters shall be initialized in order to use the black-boxecr.

Usage

setup(operator, ...)

Arguments

operator

[ecr_operator]
Evolutionary operator.

...

[any]
Furhter parameters foroperator.

Value

[function] Wrapper evolutionary operator with parametersx and....

Examples

# initialize bitflip mutator with p = 0.3bf = setup(mutBitflip, p = 0.3)# sample binary stringx = sample(c(0, 1), 100, replace = TRUE)set.seed(1)# apply preinitialized functionprint(bf(x))set.seed(1)# apply raw functionprint(mutBitflip(x, p = 0.3))# overwrite preinitialized values with mutatectrl = initECRControl(fitness.fun = function(x) sum(x), n.objectives = 1L)# here we define a mutation probability of 0.3ctrl = registerECROperator(ctrl, "mutate", setup(mutBitflip, p = 0.3))# here we overwrite with 1, i.e., each bit is flippedprint(x)print(mutate(ctrl, list(x), p.mut = 1, p = 1)[[1]])

Default monitor.

Description

Default monitor object that outputs messages to the consolebased on a default logger (seeinitLogger).

Usage

setupECRDefaultMonitor(step = 10L)

Arguments

step

[integer(1)]
Number of steps of the EA between monitoring.Default is 10.

Value

[ecr_monitor]

Implementation of the SMS-EMOA by Emmerich et al.

Description

Pure R implementation of the SMS-EMOA. This algorithm belongs to the groupof indicator based multi-objective evolutionary algorithms. In each generation,the SMS-EMOA selects two parents uniformly at, applies recombination and mutationand finally selects the best subset of individuals among all subsets by maximizingthe Hypervolume indicator.

Usage

smsemoa(  fitness.fun,  n.objectives = NULL,  n.dim = NULL,  minimize = NULL,  lower = NULL,  upper = NULL,  mu = 100L,  ref.point = NULL,  mutator = setup(mutPolynomial, eta = 25, p = 0.2, lower = lower, upper = upper),  recombinator = setup(recSBX, eta = 15, p = 0.7, lower = lower, upper = upper),  terminators = list(stopOnIters(100L)),  ...)

Arguments

fitness.fun

[function]
The fitness function.

n.objectives

[integer(1)]
Number of objectives ofobj.fun.Optional ifobj.fun is a benchmark function from packagesmoof.

n.dim

[integer(1)]
Dimension of the decision space.

minimize

[logical(n.objectives)]
Logical vector with ith entryTRUE if the ith objective offitness.funshall be minimized. If a single logical is passed, it is assumed to be validfor each objective.

lower

[numeric]
Vector of minimal values for each parameter of the decision space in caseof float or permutation encoding.Optional ifobj.fun is a benchmark function from packagesmoof.

upper

[numeric]
Vector of maximal values for each parameter of the decision space in caseof float or permutation encoding.Optional ifobj.fun is a benchmark function from packagesmoof.

mu

[integer(1)]
Number of individuals in the population.Default is 100.

ref.point

[numeric]
Reference point for the hypervolume computation. Default is (11, ..., 11)'with the corresponding dimension.

mutator

[ecr_mutator]
Mutation operator of typeecr_mutator.

recombinator

[ecr_recombinator]
Recombination operator of typeecr_recombinator.

terminators

[list]
List of stopping conditions of type “ecr_terminator”.Default is to stop after 100 iterations.

...

[any]
Further arguments passed down to fitness function.

Value

[ecr_multi_objective_result]

Note

This helper function hides the regular ecr interface and offers a moreR like interface of this state of the art EMOA.

References

Beume, N., Naujoks, B., Emmerich, M., SMS-EMOA: Multiobjective selection basedon dominated hypervolume, European Journal of Operational Research, Volume 181,Issue 3, 16 September 2007, Pages 1653-1669.

Sort Pareto-front approximation by objective.

Description

Sort Pareto-front approximation by objective.

Usage

sortByObjective(x, obj = 1L, ...)## S3 method for class 'data.frame'sortByObjective(x, obj = 1L, ...)## S3 method for class 'matrix'sortByObjective(x, obj = 1L, ...)## S3 method for class 'ecr_multi_objective_result'sortByObjective(x, obj = 1L, ...)## S3 method for class 'list'sortByObjective(x, obj = 1L, ...)

Arguments

x

[object]
Object of type data frame (objectives column-wise), matrix (objectives row-wise),ecr_multi_objective_result orlist (with components “pareto.front”)and “pareto.set”.

obj

[integer(1) | character(1)]
Either the row/column number to sort by or the column name, e.g., for data frames.

...

[any]
Further arguments passed down toorder.

Value

Modified object.

Stopping conditions

Description

Stop the EA after a fixed number of fitness function evaluations, aftera predefined number of generations/iterations, a given cutoff time orif the known optimal function value is approximated (only for single-objective optimization).

Usage

stopOnEvals(max.evals = NULL)stopOnIters(max.iter = NULL)stopOnOptY(opt.y, eps)stopOnMaxTime(max.time = NULL)

Arguments

max.evals

[integer(1)]
Maximal number of function evaluations.Default isInf.

max.iter

[integer(1)]
Maximal number of iterations/generations.Default isInf.

opt.y

[numeric(1)]
Optimal scalar fitness function value.

eps

[numeric(1)]
Stop if absolute deviation fromopt.y is lower thaneps.

max.time

[integer(1)]
Time limit in seconds.Default isInf.

Value

[ecr_terminator]

Transform to long format.

Description

Transform the data.frame of logged statistics from wide toggplot2-friendly long format.

Usage

toGG(x, drop.stats = character(0L))

Arguments

x

[ecr_statistics |ecr_logger]
Logger object or statistics data frame from logger object.

drop.stats

[character]
Names of logged statistics to be dropped.Default is the empty character, i.e., not to drop any stats.

Value

[data.frame]

Export results of statistical tests to LaTeX table(s).

Description

Returns high-quality LaTeX-tables of the test results ofstatistical tests performed with functionteston per-instance basis. I.e., a table is returned for each instances combiningthe results of different indicators.

Usage

toLatex(  stats,  stat.cols = NULL,  probs = NULL,  type = "by.instance",  cell.formatter = NULL)## S3 method for class 'list'toLatex(  stats,  stat.cols = NULL,  probs = NULL,  type = "by.instance",  cell.formatter = NULL)## S3 method for class 'data.frame'toLatex(  stats,  stat.cols = NULL,  probs = NULL,  type = "by.instance",  cell.formatter = NULL)

Arguments

stats

[list]
Data frame (return value ofcomputeIndicators) or named list of list as returned bytest.

stat.cols

[character]
Names of the indicators to consider.Defaults to all indicators available instats.

probs

[character]
Filtering: vector of problem instances. This way one can restrict thesize of the table(s).Defaults to all problems available instats.Ignored ifstats is a data frame.

type

[character(1)]
Type of tables. At the moment only option “by.instance” is available.I.e., a separate LaTeX-table is generated for each instance specified viaprobs.Ignored ifstats is a data frame.

cell.formatter

[function(cell, ...)]
Function which is used to format table cells. This function is applied to eachtable cell and may be used to customize the output. Default isniceCellFormater.Ignored ifstats is a data frame.

Value

[list] Named list of strings (LaTeX tables). Names correspond to theselected problem instances inprobs.

Convert matrix to Pareto front data frame.

Description

Inside ecr EMOA algorithms the fitness is maintained in an(o, n) matrixwhereo is the number of objectives andn is the number of individuals.This function basically transposes such a matrix and converts it into a data frame.

Usage

toParetoDf(x, filter.dups = FALSE)

Arguments

x

[matrix]
Matrix.

filter.dups

[logical(1)]
Shall duplicates be removed?Default isFALSE.

Value

[data.frame]

Fitness transformation / scaling.

Description

Some selectors support maximization only, e.g., roulette wheel selector, orminimization (most others). This function computes a factor from -1, 1 foreach objective to match supported selector optimization directions andthe actual objectives of the task.

Usage

transformFitness(fitness, task, selector)

Arguments

fitness

[matrix]Matrix of fitness values with the fitness vector of individual i in the i-thcolumn.

task

[ecr_optimization_task]Optimization task.

selector

[ecr_selector] Selector object.

Value

[matrix] Transformed / scaled fitness matrix.

Update the log.

Description

This function modifies the log in-place, i.e., without makingcopies.

Usage

updateLogger(log, population, fitness = NULL, n.evals, extras = NULL, ...)

Arguments

log

[ecr_logger]
The log generated byinitLogger.

population

[list]
List of individuals.

fitness

[matrix]
Optional matrix of fitness values (each column contains the fitness value(s) forone individual) ofpopulation. If no matrix is passed and the log shallstore information of the fitness, each individual needs to have an attribute fitness.

n.evals

[integer(1)]
Number of fitness function evaluations performed in the last generation.

extras

[list]
Optional named list of additional scalar values to log.Seelog.extras argument ofinitLogger for details.

...

[any]
Furhter arguments. Not used at the moment.

Update Pareto Archive.

Description

This function updates a Pareto archive, i.e., an archive of non-dominatedpoints. It expects the archive, a set of individuals, a matrix of fitness values(each column corresponds to the fitness vector of one individual) and updatesthe archive “in-place”. If the archive has unlimited capacity all non-dominated points ofthe union of archive and passed individuals are stored. Otherwise, i.e., in casethe archive is limited in capacity (argumentmax.size ofinitParetoArchive was set to an integer value greater zero), thetrunc.fun function passed toinitParetoArchive is applied toall non-dominated points to determine which points should be dropped.

Usage

updateParetoArchive(archive, inds, fitness, ...)

Arguments

archive

[ecr_pareto_archive]
The archive generated byinitParetoArchive.

inds

[list]
List of individuals.

fitness

[matrix]
Matrix of fitness values (each column contains the fitness value(s) forone individual) ofinds.

...

[any]
Furhter arguments passed down totrunc.fun (set viainitParetoArchive).

Determine which points of a set are (non)dominated.

Description

Given a matrix with one point per columnwhich.dominated returns thecolumn numbers of the dominated points andwhich.nondominated the columnnumbers of the nondominated points. FunctionisMaximallyDominated returnsa logical vector withTRUE for each point which is located on the lastnon-domination level.

Usage

which.dominated(x)which.nondominated(x)isMaximallyDominated(x)

Arguments

x

[matrix]
Numeric (n x d) matrix where n is the number of points and d is the numberof objectives.

Value

[integer]

Examples

  data(mtcars)  # assume we want to maximize horsepower and minimize gas consumption  cars = mtcars[, c("mpg", "hp")]  cars$hp = -cars$hp  idxs = which.nondominated(as.matrix(cars))  print(mtcars[idxs, ])

Wrap the individuals constructed by a recombination operator.

Description

Should be used if the recombinator returns multiple children.

Usage

wrapChildren(...)

Arguments

...

[any]
Individuals.

Value

[list] List of individuals.

Movatterモバイル変換

Grouping helpers

Description

Usage

Arguments

Value

Examples

Reference point approximations.

Description

Usage

Arguments

Value

See Also

Helper function to estimate reference points.

Description

Usage

Arguments

Value

See Also

Helper function to estimate reference set(s).

Description

Usage

Arguments

Value

See Also

Implementation of the NSGA-II EMOA algorithm by Deb.

Description

Usage

Arguments

Value

Note

References

Assign group membership based on another group membership.

Description

Usage

Arguments

Value

Average Hausdorff Distance computation.

Description

Usage

Arguments

Value

Compute the crowding distance of a set of points.

Description

Usage

Arguments

Value

References

Computes distance between a single point and set of points.

Description

Usage

Arguments

Value

Ranking of approximation sets.

Description

Usage

Arguments

Value

Note

References

See Also

Computes Generational Distance.

Description

Usage

Arguments

Value

Functions for the calculation of the dominated hypervolume (contribution).

Description

Usage

Arguments

Value

Note

Computation of EMOA performance indicators.

Description

Usage

Arguments

Value

References

Computes Inverted Generational Distance.

Description