Title: Evolutionary Multiobjective Optimization Algorithms
Description: Collection of building blocks for the design and analysis of evolutionary multiobjective optimization algorithms.
License: GPL-2
URL: https://github.com/olafmersmann/emoa/
LazyData: yes
Version: 0.5-3
Suggests: testthat
Encoding: UTF-8
RoxygenNote: 7.2.3
NeedsCompilation: yes
Packaged: 2024-09-05 21:21:02 UTC; olafm
Author: Olaf Mersmann ORCID iD [aut, cre]
Maintainer: Olaf Mersmann <olafm@p-value.net>
Repository: CRAN
Date/Publication: 2024-09-06 15:30:02 UTC

The EMOA package

Description

This package provides functions to construct evolutionary multiobjective optimization algorithms (EMOA). The long term goal is to also provide standard implementations of the most common EMOA in use today.

Details

Without the hard work of many researchers who have published their source code under a liberal license, this package would not have been possible. In alphabetical order they are

Author(s)

Olaf Mersmann olafm@statistik.tu-dortmund.de

Functions from the CEC 2009 EMOA competition.

Description

Functions from the CEC 2009 EMOA competition.

Usage

UF1(x)

UF2(x)

UF3(x)

UF4(x)

UF5(x)

UF6(x)

UF7(x)

UF8(x)

UF9(x)

UF10(x)

Arguments

x

Parmater vector.

Value

Function value.

Author(s)

Olaf Mersmann olafm@statistik.tu-dortmund.de

CEC 2007 multiobjective optimization competition results

Description

This data set contains the hypervolume and R2 indicator results of the 8 different algorithms that took part in the CEC 2007 multiobjective optimization benchmark.

Usage

data(cec2007)

Format

A data frame with 456 observations of the following 9 variables.

algo

Abbreviated name of algorithm

fun

Name of benchmark function

d

Dimension of objective space

n

Number of function evaluations

metric

Name of quality metric

pdef

Unique id for each combination of fun, d, n and metric

best

Largest value of metric

median

Median value of metric

worst

Smallest value of metric

mean

Average value of metric

std

Standard deviation of metric

Source

Formerly available at http://web.mysites.ntu.edu.sg/epnsugan/PublicSite/Shared%20Documents/CEC2007-final-pdfs.zip

Examples

  ## Not run: 
  data(cec2007)
  require(lattice)
  print(dotplot(algo ~ median | fun + metric, cec2007, groups=cec2007$n))
  
## End(Not run)

Return first non null argument.

Description

This function is useful when processing complex arguments with multiple possible defaults based on other arguments that may or may not have been provided.

Usage

coalesce(...)

Arguments

...

List of values.

Value

First non null element in ....

Author(s)

Olaf Mersmann olafm@statistik.tu-dortmund.de

Crowding Distance

Description

Calculate crowding distances.

Usage

crowding_distance(front)

Arguments

front

matrix of function values.

Value

crowding distance for each function value.

Author(s)

Olaf Mersmann olafm@statistik.tu-dortmund.de

Calculate the dominance matrix of a set of points

Description

Calculate the dominance matrix of a set of points

Usage

dominance_matrix(points)

Arguments

points

Matrix containing points one per column.

Value

Dominance matrix

Dominated Hypervolume calculation

Description

dominated_hypervolume calculates the dominated hypervolume of the points in points.

Usage

dominated_hypervolume(points, ref)

hypervolume_contribution(points, ref)

Arguments

points

Matrix containing the points one per column.

ref

Optional reference point. If not provided the maximum in each dimension is used.

Details

hypervolume_contribution calculates the hypervolume contribution of each point.

If no reference point ref is given, one is automatically calculated by determening the maximum in each coordinate.

Currently only one general algorithm is implemented due to Fonseca et.al. but work is underway to include others such as the Beume & Rudolph approach as well as the approach by Bradstreet et.al.

The 1D and 2D cases are handle seperately by efficient algorithms. Calculates the exact dominated hypervolume of the points given in x subject to the reference point ref.

Value

For dominated_hypervolume the dominated hypervolume by the points in points with respect to the reference point ref. For hypervolume_contribution a vector giving the hypervolume soley dominated by that point.

Author(s)

Olaf Mersmann olafm@statistik.tu-dortmund.de

References

This code uses version 1.3 of the hypervolume code available from https://lopez-ibanez.eu/hypervolume. For a description of the algorithm see

Carlos M. Fonseca, Luis Paquete, and Manuel Lopez-Ibanez. An improved dimension-sweep algorithm for the hypervolume indicator. In IEEE Congress on Evolutionary Computation, pages 1157-1163, Vancouver, Canada, July 2006.

See Also

nondominated_points to extract the pareto front approximation from a given set of points and nds_hv_selection for a selection strategy based on the hypervolume contribution of each point.

console logger

Description

Logger object that outputs log messages to the console

Usage

emoa_console_logger(...)

Arguments

...

passed to emoa_logger.

Details

This is a wrapper that calls emoa_logger(output=output, ...) internally and returns that logger.

Value

An emoa_logger object.

Basic EMOA control parameters.

Description

The following control parameters are recognized by emoa_control:

logger

emoa_logger object used to log events.

n

Number of parameters, defaults to the length of the longer of upper or lower.

d

Number of dimensions.

Usage

emoa_control(f, upper, lower, ..., control, default)

Arguments

f

Multiobjectve optimization function.

upper

Upper bounds of parameter space.

lower

Lower bounds of parameter space.

...

Further arguments passed to f.

control

List of control parameters.

default

List of default control parameters.

Value

The control list with suitably adjusted arguments. Missing control parameters are taken from default or, if not present there, from an internal default.

Author(s)

Olaf Mersmann olafm@statistik.tu-dortmund.de

generic logger factory

Description

Basic logger object with a flexible output routine.

Usage

emoa_logger(output, every = 10L, ...)

Arguments

output

function used to display logging messages.

every

number of steps of the emoa between evaluations.

...

passed to the parent logger factory.

Value

An emoa_logger object.

See Also

emoa_console_logger and emoa_null_logger for convinience wrappers around emoa_logger providing useful defaults.

null logger

Description

Logger object that discards all log events.

Usage

emoa_null_logger(...)

Arguments

...

ignored.

Value

An emoa_logger object.

Binary quality indicators

Description

Calculates the quality indicator value of the set of points given in x with respect to the set given in o. As with all functions in emoa that deal with sets of objective values these are stored by column.

Usage

hypervolume_indicator(points, o, ref)

epsilon_indicator(points, o)

r1_indicator(points, o, ideal, nadir, lambda, utility = "Tchebycheff")

r2_indicator(points, o, ideal, nadir, lambda, utility = "Tchebycheff")

r3_indicator(points, o, ideal, nadir, lambda, utility = "Tchebycheff")

Arguments

points

Matrix of points for which to calculate the indicator value stored one per column.

o

Matrix of points of the reference set.

ref

Reference point, if omitted, the nadir of the point sets is used.

ideal

Ideal point of true Pareto front. If omited the ideal of both point sets is used.

nadir

Nadir of the true Pareto front. If ommited the nadir of both point sets is used.

lambda

Number of weight vectors to use in estimating the utility.

utility

Name of utility function.

Value

Value of the quality indicator.

Author(s)

Olaf Mersmann olafm@statistik.tu-dortmund.de

References

Zitzler, E., Thiele, L., Laumanns, M., Fonseca, C., and Grunert da Fonseca, V (2003): Performance Assessment of Multiobjective Optimizers: An Analysis and Review. IEEE Transactions on Evolutionary Computation, 7(2), 117-132.

Clip value to a given range

Description

Clip x to the interval [l, u]. This is useful to enforce box constraints.

Usage

inbounds(x, l, u)

Arguments

x

Value to clip.

l

Lower limit.

u

Upper limit.

Value

l if x < l, u if x > u else x.

Author(s)

Olaf Mersmann olafm@statistik.tu-dortmund.de

Pareto dominance checks.

Description

is_dominated returns which points from a set are dominated by another point in the set. %dominates% returns true if x Pareto dominates y and is_maximally_dominated returns TRUE for those points which do not dominate any other points.

Usage

is_dominated(points)

is_maximally_dominated(points)

Arguments

points

Matrix containing points one per column.

Value

For is_dominated and is_maximally_dominated a boolean vector and for %dominates% a single boolean.

Author(s)

Olaf Mersmann olafm@statistik.tu-dortmund.de

Selection strategies

Description

Selection strategies for EMOA.

Usage

nds_hv_selection(values, n = 1, ...)

nds_cd_selection(values, n = 1, ...)

Arguments

values

Matrix of function values.

n

Number of individuals to select for replacement.

...

Optional parameters passed to hypervolume_contribution.

Details

The currently implemented strategies are nondominated sorting followed by either hypervolume contribution or crowding distance based ranking. Both of these implementations are currently limited to selecting a single individual for replacement.

Author(s)

Olaf Mersmann olafm@statistik.tu-dortmund.de

Nondominated sorting ranks

Description

Perform (partial) nondominated sort of the points in points and return the rank of each point.

Usage

nds_rank(points, partial)

nondominated_ordering(points, partial)

Arguments

points

Matrix containing points one per column.

partial

Optional integer specifying the number of points for which the rank should be calculated. Defaults to all points.

Value

Vector containing the ranks of the first partial individuals or all individuals.

Author(s)

Olaf Mersmann olafm@statistik.tu-dortmund.de

Nondominated points

Description

Return those points which are not dominated by another point in points. This is the Pareto front approximation of the point set.

Usage

nondominated_points(points)

Arguments

points

Matrix of points, one per column.

Value

Those points in points which are not dominated by another point.

Author(s)

Olaf Mersmann olafm@statistik.tu-dortmund.de

Scale point cloud

Description

Rescale all points to lie in the box bounded by minval and maxval.

Usage

normalize_points(points, minval, maxval)

Arguments

points

Matrix containing points, one per column.

minval

Optional lower limits for the new bounding box.

maxval

Optional upper limits for the new bounding box.

Value

Scaled points.

Author(s)

Olaf Mersmann olafm@statistik.tu-dortmund.de

Polynomial muation (PM) control parameters

Description

Control parameters:

pm.n

Nu parameter of PM.

pm.p

p parameter of PM.

Usage

pm_control(f, upper, lower, ..., control, default = list())

Arguments

f

Multiobjectve optimization function.

upper

Upper bounds of parameter space.

lower

Lower bounds of parameter space.

...

Further arguments passed to f.

control

List of control parameters.

default

List of default control parameters.

Value

The control list with suitably adjusted arguments. Missing control parameters are taken from default or, if not present there, from an internal default.

Author(s)

Olaf Mersmann olafm@statistik.tu-dortmund.de

Polynomial mutation operator

Description

Returns a polynomial mutation operator with the given parameters.

Usage

pm_operator(n, p, lower, upper)

Arguments

n

Distance parameter mutation distribution (\eta).

p

Probability of one point mutation.

lower

Lower bounds of parameter space.

upper

Upper bounds of parameter space.

Value

Function which implements the specified mutation operator.

Author(s)

Olaf Mersmann olafm@statistik.tu-dortmund.de

Simulated binary crossover (SBX) control parameters

Description

sbx_control interprets the following parameters used to control the behaviour of the simulated binary crossover operator (see sbx_operator):

sbx.n

Nu parameter of SBX.

sbx.p

$p$ parameter of SBX.

Usage

sbx_control(f, upper, lower, ..., control, default = list())

Arguments

f

Multiobjectve optimization function.

upper

Upper bounds of parameter space.

lower

Lower bounds of parameter space.

...

Further arguments passed to f.

control

List of control parameters.

default

List of default control parameters.

Value

The control list with suitably adjusted arguments. Missing control parameters are taken from default or, if not present there, from an internal default.

Author(s)

Olaf Mersmann olafm@statistik.tu-dortmund.de

Simulated binary crossover operator

Description

Returns a simulated binary crossover operator with the given parameters.

Usage

sbx_operator(n, p, lower, upper)

Arguments

n

Distance parameter of crossover distribution (\eta).

p

Probability of one point crossover.

lower

Lower bounds of parameter space.

upper

Upper bounds of parameter space.

Value

Function with one parameter x which takes a matrix containing two sets of parameters and returns a matrix of two sets of parameters which resulted from the crossover operation. As with all emoa functions, the parameter sets are stored in the columns of x. x should therefore always have two columns and a warning will be given if it has more than two columns.

Author(s)

Olaf Mersmann olafm@statistik.tu-dortmund.de

See Also

pm_operator

Steady state EMOA parameters

Description

steady_state_emoa_control interprets the following control parameters:

mu

Population size.

maxeval

Maximum number of function evaluations to use.

Usage

steady_state_emoa_control(f, upper, lower, ..., control, default = list())

Arguments

f

Multiobjectve optimization function.

upper

Upper bounds of parameter space.

lower

Lower bounds of parameter space.

...

Further arguments passed to f.

control

List of control parameters.

default

List of default control parameters.

Value

The control list with suitably adjusted arguments. Missing control parameters are taken from default or, if not present there, from an internal default.

Author(s)

Olaf Mersmann olafm@statistik.tu-dortmund.de

Functions from the CEC 2007 EMOA competition.

Description

Functions from the CEC 2007 EMOA competition.

Usage

sympart(x)

Arguments

x

Parmater vector.

Value

Function value.

Author(s)

Olaf Mersmann olafm@statistik.tu-dortmund.de

Unary R2 indicator

Description

Unary R2 indicator

Usage

unary_r2_indicator(points, weights, ideal)

Arguments

points

Matrix of points for which to calculate the indicator value stored one per column.

weights

Matrix of weight vectors stored one per column.

ideal

Ideal point of true Pareto front. If omited the ideal of points is used.

Value

Value of unary R2 indicator.

Author(s)

Olaf Mersmann olafm@p-value.net

Determine which points are on the edge of a Pareto-front approximation.

Description

Determine which points are on the edge of a Pareto-front approximation.

Usage

which_points_on_edge(front)

Arguments

front

Pareto-front approximation.

Value

An integer vector containing the indicies of the points (columns) of front which are on the edge of the Pareto-front approximation.

Author(s)

Olaf Mersmann olafm@statistik.tu-dortmund.de