| Title: | Multi-Objective Optimization in R |
|---|---|
| Description: | The 'rmoo' package is a framework for multi- and many-objective optimization, which allows researchers and users versatility in parameter configuration, as well as tools for analysis, replication and visualization of results. The 'rmoo' package was built as a fork of the 'GA' package by Luca Scrucca(2017) <DOI:10.32614/RJ-2017-008> and implementing the Non-Dominated Sorting Genetic Algorithms proposed by K. Deb's. |
| Authors: | Francisco Benitez [aut, cre],
Diego P. Pinto-Roa [aut] |
| Maintainer: | Francisco Benitez <[email protected]> |
| License: | GPL (>= 2) |
| Version: | 0.3.0 |
| Built: | 2025-01-26 06:13:21 UTC |
| Source: | https://github.com/evolutionary-optimization-laboratory/rmoo |
It will use when other algorithms are implemented. Equivalent to a Abstract class in other languages.
Function that associates each member of the population with a reference point. The function calculates the perpendicular distance of each individual from each of the reference lines. This code section corresponds to Algorithm 3 of the referenced paper.
associate_to_niches(object, utopian_epsilon = 0) compute_perpendicular_distance(x, y) compute_niche_count(n_niches, niche_of_individuals)associate_to_niches(object, utopian_epsilon = 0) compute_perpendicular_distance(x, y) compute_niche_count(n_niches, niche_of_individuals)
object |
An object of class |
utopian_epsilon |
The epsilon used for decrease the ideal point to get the utopian point. |
x |
Individuals to calculate their niche. |
y |
Reference points. |
n_niches |
Number of reference points. |
niche_of_individuals |
The niche count of individuals, except the last front. |
Returns a list with the niche count of individuals and the distances between them.
Francisco Benitez
J. Blank and K. Deb, "Pymoo: Multi-Objective Optimization in Python," in IEEE Access, vol. 8, pp. 89497-89509, 2020, doi: 10.1109/ACCESS.2020.2990567.
K. Deb and H. Jain, "An Evolutionary Many-Objective Optimization Algorithm Using Reference-Point-Based Nondominated Sorting Approach, Part I: Solving Problems With Box Constraints," in IEEE Transactions on Evolutionary Computation, vol. 18, no. 4, pp. 577-601, Aug. 2014, doi: 10.1109/TEVC.2013.2281535.
A Crowded-comparison approach.
crowding_distance(object, nObj)crowding_distance(object, nObj)
object, nObj
|
An object of class 'nsga2', usually resulting from a call to function nsga2. Fitness Function Objective Numbers |
The crowded-comparison operator guides the selection process at the various stages of the algorithm toward a uniformly spread-out Pareto-optimal front
A vector with the crowding-distance between individuals of a population.
Francisco Benitez [email protected]
K. Deb, A. Pratap, S. Agarwal and T. Meyarivan, 'A fast and elitist multiobjective genetic algorithm: NSGA-II,' in IEEE Transactions on Evolutionary Computation, vol. 6, no. 2, pp. 182-197, April 2002, doi: 10.1109/4235.996017.
A implementation of Das and Dennis's Reference Points Generation.
generate_reference_points(m, h, scaling = NULL)generate_reference_points(m, h, scaling = NULL)
m, h, scaling
|
Number of reference points 'h' in M-objective problems, and scaling that is the scale on which the points are distributed. |
The implemented Reference Point Generation is based on the Das and Dennis's systematic approach that places points on a normalized hyper-plane which is equally inclined to all objective axes and has an intercept of one on each axis.
A matrix with the reference points uniformly distributed.
Francisco Benitez [email protected]
K. Deb and H. Jain, 'An Evolutionary Many-Objective Optimization Algorithm Using Reference-Point-Based Nondominated Sorting Approach, Part I: Solving Problems With Box Constraints,' in IEEE Transactions on Evolutionary Computation, vol. 18, no. 4, pp. 577-601, Aug. 2014, doi: 10.1109/TEVC.2013.2281535.
Das, Indraneel & Dennis, J. (2000). Normal-Boundary Intersection: A New Method for Generating the Pareto Surface in Nonlinear Multicriteria Optimization Problems. SIAM Journal on Optimization. 8. 10.1137/S1052623496307510.
non_dominated_fronts() and get_fixed_rowsum_integer_matrix()
Implementation of the recursive function in Generation of Reference points of Das and Dennis..
get_fixed_rowsum_integer_matrix(m, h)get_fixed_rowsum_integer_matrix(m, h)
m, h
|
Number of reference points 'h' in M-objective problems |
The implemented Reference Point Generation is based on the Das and Dennis's systematic approach that places points on a normalized hyper-plane which is equally inclined to all objective axes and has an intercept of one on each axis.
A matrix with the reference points uniformly distributed.
Francisco Benitez [email protected]
K. Deb and H. Jain, 'An Evolutionary Many-Objective Optimization Algorithm Using Reference-Point-Based Nondominated Sorting Approach, Part I: Solving Problems With Box Constraints,' in IEEE Transactions on Evolutionary Computation, vol. 18, no. 4, pp. 577-601, Aug. 2014, doi: 10.1109/TEVC.2013.2281535.
Das, Indraneel & Dennis, J.. (2000). Normal-Boundary Intersection: A New Method for Generating the Pareto Surface in Nonlinear Multicriteria Optimization Problems. SIAM Journal on Optimization. 8. 10.1137/S1052623496307510.
non_dominated_fronts() and generate_reference_points()
Accessor methods to the crowding distance for NSGA-II results
getCrowdingDistance(obj) ## S4 method for signature 'nsga2' getCrowdingDistance(obj)getCrowdingDistance(obj) ## S4 method for signature 'nsga2' getCrowdingDistance(obj)
obj |
an object resulting from the execution of NSGA-II algorithm |
Returns a vector with the crowding distances of class nsga2. See nsga2 for a description of available slots information.
Francisco Benitez [email protected]
# Where 'out' is an object resulting from the execution of the NSGA-II algorithm. # # getCrowdingDistance(out) ## Where 'out' is an object resulting from the execution of the NSGA-II algorithm. # # getCrowdingDistance(out) #
Accessor methods to the dummy fitness for NSGA-I results
getDummyFitness(obj) ## S4 method for signature 'nsga1' getDummyFitness(obj)getDummyFitness(obj) ## S4 method for signature 'nsga1' getDummyFitness(obj)
obj |
an object resulting from the execution of NSGA-I algorithm |
Returns a matrix with the dummy fitness of class nsga1. See nsga1 for a description of available slots information.
Francisco Benitez [email protected]
# Where 'out' is an object resulting from the execution of the NSGA-I algorithm. # # getDummyFitness(out) ## Where 'out' is an object resulting from the execution of the NSGA-I algorithm. # # getDummyFitness(out) #
Accessor methods to the fitness for rmoo results
getFitness(obj)getFitness(obj)
obj |
an object resulting from the execution of NSGA-I, NSGA-II or NSGA-III algorithm |
Prints the resulting fitness and when the result of the method-call is assigned to a variable, the fitness is stored as a data frame. See nsga1 nsga2, nsga3 for a description of available slots information.
Francisco Benitez [email protected]
# Where 'out' is an object resulting from the execution of the rmoo. # # fitness_result <- getFitness(out) # # fitness_result# Where 'out' is an object resulting from the execution of the rmoo. # # fitness_result <- getFitness(out) # # fitness_result
Accessor methods to the metrics evaluated during execution
getMetrics(obj) ## S4 method for signature 'nsga' getMetrics(obj)getMetrics(obj) ## S4 method for signature 'nsga' getMetrics(obj)
obj |
an object resulting from the execution of NSGA-I, NSGA-II or NSGA-III algorithm. During the execution of the performance metrics must be evaluated. |
A dataframe with performance metrics evaluated iteration by iteration.
Francisco Benitez [email protected]
# Where 'out' is an object resulting from the execution of the rmoo. # # metrics_result <- getMetrics(out) # # metrics_result# Where 'out' is an object resulting from the execution of the rmoo. # # metrics_result <- getMetrics(out) # # metrics_result
Accessor methods to the population for rmoo results
getPopulation(obj) ## S4 method for signature 'nsga' getPopulation(obj) ## S4 method for signature 'nsga' getFitness(obj)getPopulation(obj) ## S4 method for signature 'nsga' getPopulation(obj) ## S4 method for signature 'nsga' getFitness(obj)
obj |
an object resulting from the execution of NSGA-I, NSGA-II or NSGA-III algorithm |
Prints the resulting population and when the result of the method-call is assigned to a variable, the population is stored as a data frame. See nsga1 nsga2, nsga3 for a description of available slots information.
Francisco Benitez [email protected]
# Where 'out' is an object resulting from the execution of rmoo. # # population_result <- getPopulation(out) # # population_result# Where 'out' is an object resulting from the execution of rmoo. # # population_result <- getPopulation(out) # # population_result
A dataset containing the coord and section of 100 cities
kroA100kroA100
A data frame with 100 rows and 2 variables:
City Coordinates
City Section
Reinelt, G. (1991). TSPLIB—A traveling salesman problem library. ORSA journal on computing, 3(4), 376-384
A dataset containing the coord and section of 100 cities
kroB100kroB100
A data frame with 100 rows and 2 variables:
City Coordinates
City Section
Reinelt, G. (1991). TSPLIB—A traveling salesman problem library. ORSA journal on computing, 3(4), 376-384
A dataset containing the coord and section of 100 cities
kroC100kroC100
A data frame with 100 rows and 2 variables:
City Coordinates
City Section
Reinelt, G. (1991). TSPLIB—A traveling salesman problem library. ORSA journal on computing, 3(4), 376-384
A Crowded-comparison approach.
modifiedCrowdingDistance( object, epsilon, weights = NULL, normalization = "front", extreme_points_as_ref_dirs = FALSE )modifiedCrowdingDistance( object, epsilon, weights = NULL, normalization = "front", extreme_points_as_ref_dirs = FALSE )
object |
An object of class 'rnsga2', usually resulting from a call to function r-nsga2. Fitness Function Objective Numbers |
extreme_points_as_ref_dirs |
The crowded-comparison operator guides the selection process at the various stages of the algorithm toward a uniformly spread-out Pareto-optimal front
A vector with the crowding-distance between individuals of a population.
Francisco Benitez [email protected]
Kalyanmoy Deb and J. Sundar. 2006. Reference point based multi-objective optimization using evolutionary algorithms. In Proceedings of the 8th annual conference on Genetic and evolutionary computation (GECCO '06). Association for Computing Machinery, New York, NY, USA, 635–642. doi: 10.1145/1143997.1144112
Generation of niche, by associating reference points to population members
niching(pop, n_remaining, niche_count, niche_of_individuals, dist_to_niche)niching(pop, n_remaining, niche_count, niche_of_individuals, dist_to_niche)
pop |
Last Front Population |
n_remaining |
Number of points to choose |
niche_count |
Niche count of individuals with the reference point |
niche_of_individuals |
Count of the closest reference point to the last front objective values |
dist_to_niche |
Distance between closest reference point to last front objective values |
Niching procesure is a algorithms proposed by K. Deb and H. Jain in 2013.
Returns the association of reference points to each individual in the population.
Francisco Benitez [email protected]
K. Deb and H. Jain, 'An Evolutionary Many-Objective Optimization Algorithm Using Reference-Point-Based Nondominated Sorting Approach, Part I: Solving Problems With Box Constraints,' in IEEE Transactions on Evolutionary Computation, vol. 18, no. 4, pp. 577-601, Aug. 2014, doi: 10.1109/TEVC.2013.2281535.
Scrucca, L. (2017) On some extensions to 'GA' package: hybrid optimisation, parallelisation and islands evolution. The R Journal, 9/1, 187-206. doi: 10.32614/RJ-2017-008
Felix-Antoine Fortin, Francois-Michel De Rainville, Marc-André Gardner Gardner, Marc Parizeau, and Christian Gagne. 2012. DEAP: evolutionary algorithms made easy. J. Mach. Learn. Res. 13, 1 (January 2012), 2171–2175.
associate_to_niches(), PerformScalarizing()
A fast approach for calculate Non-Dominated Fronts.
non_dominated_fronts(object)non_dominated_fronts(object)
object |
An object of class 'nsga', usually resulting from a call to function nsga, nsga2 and nsga3. |
Function to determine the non-dominated fronts of a population and the aptitude value.
A list with 'non-dominated fronts' and 'occupied positions' on the fronts.
Francisco Benitez [email protected]
K. Deb, A. Pratap, S. Agarwal and T. Meyarivan, 'A fast and elitist multiobjective genetic algorithm: NSGA-II,' in IEEE Transactions on Evolutionary Computation, vol. 6, no. 2, pp. 182-197, April 2002, doi: 10.1109/4235.996017.
Minimization of a fitness function using Non-Dominated Genetic algorithms (NSGA). Local search using general-purpose optimisation algorithms can be applied stochastically to exploit interesting regions.
nsga( type = c("binary", "real-valued", "permutation"), fitness, ..., lower, upper, nBits, population = rmooControl(type)$population, selection = rmooControl(type)$selection, crossover = rmooControl(type)$crossover, mutation = rmooControl(type)$mutation, popSize = 50, nObj = NULL, dshare, pcrossover = 0.8, pmutation = 0.1, maxiter = 100, run = maxiter, maxFitness = Inf, names = NULL, suggestions = NULL, monitor = if (interactive()) rmooMonitor else FALSE, summary = FALSE, seed = NULL )nsga( type = c("binary", "real-valued", "permutation"), fitness, ..., lower, upper, nBits, population = rmooControl(type)$population, selection = rmooControl(type)$selection, crossover = rmooControl(type)$crossover, mutation = rmooControl(type)$mutation, popSize = 50, nObj = NULL, dshare, pcrossover = 0.8, pmutation = 0.1, maxiter = 100, run = maxiter, maxFitness = Inf, names = NULL, suggestions = NULL, monitor = if (interactive()) rmooMonitor else FALSE, summary = FALSE, seed = NULL )
type |
the type of genetic algorithm to be run depending on the nature of decision variables. Possible values are:
|
fitness |
the fitness function, any allowable R function which takes as input an individual string representing a potential solution, and returns a numerical value describing its “fitness”. |
... |
additional arguments to be passed to the fitness function. This allows to write fitness functions that keep some variables fixed during the search. |
lower |
a vector of length equal to the decision variables providing the lower bounds of the search space in case of real-valued or permutation encoded optimizations. |
upper |
a vector of length equal to the decision variables providing the upper bounds of the search space in case of real-valued or permutation encoded optimizations. |
nBits |
a value specifying the number of bits to be used in binary encoded optimizations. |
population |
an R function for randomly generating an initial population.
See |
selection |
an R function performing selection, i.e. a function which
generates a new population of individuals from the current population
probabilistically according to individual fitness.
See |
crossover |
an R function performing crossover, i.e. a function which
forms offsprings by combining part of the genetic information from
their parents. See |
mutation |
an R function performing mutation, i.e. a function which
randomly alters the values of some genes in a parent chromosome.
See |
popSize |
the population size. |
nObj |
number of objective in the fitness function. |
dshare |
the maximun phenotypic distance allowed between any two individuals to become members of a niche. |
pcrossover |
the probability of crossover between pairs of chromosomes. Typically this is a large value and by default is set to 0.8. |
pmutation |
the probability of mutation in a parent chromosome. Usually mutation occurs with a small probability, and by default is set to 0.1. |
maxiter |
the maximum number of iterations to run before the NSGA search is halted. |
run |
the number of consecutive generations without any improvement in the best fitness value before the NSGA is stopped. |
maxFitness |
the upper bound on the fitness function after that the NSGA search is interrupted. |
names |
a vector of character strings providing the names of decision variables. |
suggestions |
a matrix of solutions strings to be included in the initial population. If provided the number of columns must match the number of decision variables. |
monitor |
a logical or an R function which takes as input the current state of the nsga-class object and show the evolution of the search. By default, for interactive sessions the function rmooMonitor prints the average and best fitness values at each iteration. If set to plot these information are plotted on a graphical device. Other functions can be written by the user and supplied as argument. In non interactive sessions, by default monitor = FALSE so any output is suppressed. |
summary |
If there will be a summary generation after generation. |
seed |
an integer value containing the random number generator state. This argument can be used to replicate the results of a NSGA search. Note that if parallel computing is required, the doRNG package must be installed. |
The Non-dominated genetic algorithms is a meta-heuristic proposed by N. Srinivas and K. Deb in 1994. The purpose of the algorithms is to find an efficient way to optimize multi-objectives functions (two or more).
Returns an object of class nsga1-class. See nsga1 for a description of available slots information.
Francisco Benitez [email protected]
N. Srinivas and K. Deb, "Multiobjective Optimization Using Nondominated Sorting in Genetic Algorithms, in Evolutionary Computation, vol. 2, no. 3, pp. 221-248, Sept. 1994, doi: 10.1162/evco.1994.2.3.221.
Scrucca, L. (2017) On some extensions to 'GA' package: hybrid optimisation, parallelisation and islands evolution. The R Journal, 9/1, 187-206. doi: 10.32614/RJ-2017-008
#Example #Two Objectives - Real Valued zdt1 <- function (x) { if (is.null(dim(x))) { x <- matrix(x, nrow = 1) } n <- ncol(x) g <- 1 + rowSums(x[, 2:n, drop = FALSE]) * 9/(n - 1) return(cbind(x[, 1], g * (1 - sqrt(x[, 1]/g)))) } #Not run: ## Not run: result <- nsga(type = "real-valued", fitness = zdt1, lower = c(0,0), upper = c(1,1), popSize = 100, nObj = 2, dshare = 1, monitor = FALSE, maxiter = 500) ## End(Not run)#Example #Two Objectives - Real Valued zdt1 <- function (x) { if (is.null(dim(x))) { x <- matrix(x, nrow = 1) } n <- ncol(x) g <- 1 + rowSums(x[, 2:n, drop = FALSE]) * 9/(n - 1) return(cbind(x[, 1], g * (1 - sqrt(x[, 1]/g)))) } #Not run: ## Not run: result <- nsga(type = "real-valued", fitness = zdt1, lower = c(0,0), upper = c(1,1), popSize = 100, nObj = 2, dshare = 1, monitor = FALSE, maxiter = 500) ## End(Not run)
The 'nsga' class is the parent superclass of the nsga1, nsga2, and nsga3 classes
callan object of class 'call' representing the matched call.
typea character string specifying the type of genetic algorithm used.
lowera vector providing for each decision variable the lower bounds of the search space in case of real-valued or permutation encoded optimisations.
uppera vector providing for each decision variable the upper bounds of the search space in case of real-valued or permutation encoded optimizations.
nBitsa value specifying the number of bits to be used in binary encoded optimizations.
namesa vector of character strings providing the names of decision variables (optional).
nvarsa
popSizethe population size.
frontRank of individuals on the non-dominated front.
fFront of individuals on the non-dominated front.
iterthe actual (or final) iteration of NSGA search.
runthe number of consecutive generations without any improvement in the best fitness value before the NSGA is stopped.
maxiterthe maximum number of iterations to run before the NSGA search is halted.
suggestionsa matrix of user provided solutions and included in the initial population.
populationthe current (or final) population.
pcrossoverthe crossover probability.
pmutationthe mutation probability.
fitnessthe values of fitness function for the current (or final) population.
summarya matrix of summary statistics for fitness values at each iteration (along the rows).
fitnessValuethe best fitness value at the final iteration.
solutionthe value(s) of the decision variables giving the best fitness at the final iteration.
execution_timea
Since it is a virtual Class, no objects may be created from it.
showClass('nsga')showClass('nsga')
The class 'nsga1' is instantiated within the execution of rmoo and will be returned as a result of it. All data generated during execution will be stored in it.
dumFitnessa large dummy fitness value assigned to individuals from the nondominated front.
dSharethe maximun phenotypic distance allowed between any two individuals to become members of a niche.
deltaDummyvalue to decrease the dummy fitness of individuals by non-dominated fronts.
showClass('nsga1')showClass('nsga1')
Minimization of a fitness function using non-dominated sorting genetic algorithms - II (NSGA-IIs). Multiobjective evolutionary algorithms
nsga2( type = c("binary", "real-valued", "permutation"), fitness, ..., lower, upper, nBits, population = rmooControl(type)$population, selection = rmooControl(type)$selection, crossover = rmooControl(type)$crossover, mutation = rmooControl(type)$mutation, popSize = 50, nObj = NULL, pcrossover = 0.8, pmutation = 0.1, maxiter = 100, run = maxiter, maxFitness = Inf, names = NULL, suggestions = NULL, parallel = FALSE, monitor = if (interactive()) rmooMonitor else FALSE, summary = FALSE, seed = NULL )nsga2( type = c("binary", "real-valued", "permutation"), fitness, ..., lower, upper, nBits, population = rmooControl(type)$population, selection = rmooControl(type)$selection, crossover = rmooControl(type)$crossover, mutation = rmooControl(type)$mutation, popSize = 50, nObj = NULL, pcrossover = 0.8, pmutation = 0.1, maxiter = 100, run = maxiter, maxFitness = Inf, names = NULL, suggestions = NULL, parallel = FALSE, monitor = if (interactive()) rmooMonitor else FALSE, summary = FALSE, seed = NULL )
type |
the type of genetic algorithm to be run depending on the nature of decision variables. Possible values are:
|
fitness |
the fitness function, any allowable R function which takes as input an individual string representing a potential solution, and returns a numerical value describing its 'fitness'. |
... |
additional arguments to be passed to the fitness function. This allows to write fitness functions that keep some variables fixed during the search |
lower |
a vector of length equal to the decision variables providing the lower bounds of the search space in case of real-valued or permutation encoded optimizations. |
upper |
a vector of length equal to the decision variables providing the upper bounds of the search space in case of real-valued or permutation encoded optimizations. |
nBits |
a value specifying the number of bits to be used in binary encoded optimizations |
population |
an R function for randomly generating an initial population.
See |
selection |
an R function performing selection, i.e. a function which
generates a new population of individuals from the current population
probabilistically according to individual fitness. See |
crossover |
an R function performing crossover, i.e. a function which
forms offsprings by combining part of the genetic information from their
parents. See |
mutation |
an R function performing mutation, i.e. a function which
randomly alters the values of some genes in a parent chromosome.
See |
popSize |
the population size. |
nObj |
number of objective in the fitness function. |
pcrossover |
the probability of crossover between pairs of chromosomes. Typically this is a large value and by default is set to 0.8. |
pmutation |
the probability of mutation in a parent chromosome. Usually mutation occurs with a small probability, and by default is set to 0.1. |
maxiter |
the maximum number of iterations to run before the NSGA search is halted. |
run |
the number of consecutive generations without any improvement in the best fitness value before the NSGA is stopped |
maxFitness |
the upper bound on the fitness function after that the NSGA search is interrupted. |
names |
a vector of character strings providing the names of decision variables. |
suggestions |
a matrix of solutions strings to be included in the initial population. If provided the number of columns must match the number of decision variables. |
parallel |
An optional argument which allows to specify if the NSGA-II should be run sequentially or in parallel. |
monitor |
a logical or an R function which takes as input the current state of the nsga-class object and show the evolution of the search. By default, for interactive sessions the function rmooMonitor prints the average and best fitness values at each iteration. If set to plot these information are plotted on a graphical device. Other functions can be written by the user and supplied as argument. In non interactive sessions, by default monitor = FALSE so any output is suppressed. |
summary |
If there will be a summary generation after generation. |
seed |
an integer value containing the random number generator state. This argument can be used to replicate the results of a NSGA search. Note that if parallel computing is required, the doRNG package must be installed. |
The Non-dominated genetic algorithms II is a meta-heuristic proposed by K. Deb, A. Pratap, S. Agarwal and T. Meyarivan in 2002. The purpose of the algorithms is to find an efficient way to optimize multi-objectives functions (two or more).
Returns an object of class nsga2-class. See nsga2 for a description of available slots information.
Francisco Benitez [email protected]
K. Deb, A. Pratap, S. Agarwal and T. Meyarivan, 'A fast and elitist multiobjective genetic algorithm: NSGA-II,' in IEEE Transactions on Evolutionary Computation, vol. 6, no. 2, pp. 182-197, April 2002, doi: 10.1109/4235.996017.
Scrucca, L. (2017) On some extensions to 'GA' package: hybrid optimisation, parallelisation and islands evolution. The R Journal, 9/1, 187-206. doi: 10.32614/RJ-2017-008
#Example #Two Objectives - Real Valued zdt1 <- function (x) { if (is.null(dim(x))) { x <- matrix(x, nrow = 1) } n <- ncol(x) g <- 1 + rowSums(x[, 2:n, drop = FALSE]) * 9/(n - 1) return(cbind(x[, 1], g * (1 - sqrt(x[, 1]/g)))) } #Not run: ## Not run: result <- nsga2(type = "real-valued", fitness = zdt1, lower = c(0,0), upper = c(1,1), popSize = 100, nObj = 2, monitor = FALSE, maxiter = 500) ## End(Not run) #Example 2 #Three Objectives - Real Valued dtlz1 <- function (x, nobj = 3){ if (is.null(dim(x))) { x <- matrix(x, 1) } n <- ncol(x) y <- matrix(x[, 1:(nobj - 1)], nrow(x)) z <- matrix(x[, nobj:n], nrow(x)) g <- 100 * (n - nobj + 1 + rowSums((z - 0.5)^2 - cos(20 * pi * (z - 0.5)))) tmp <- t(apply(y, 1, cumprod)) tmp <- cbind(t(apply(tmp, 1, rev)), 1) tmp2 <- cbind(1, t(apply(1 - y, 1, rev))) f <- tmp * tmp2 * 0.5 * (1 + g) return(f) } #Not run: ## Not run: result <- nsga2(type = "real-valued", fitness = dtlz1, lower = c(0,0,0), upper = c(1,1,1), popSize = 92, nObj = 3, monitor = FALSE, maxiter = 500) ## End(Not run)#Example #Two Objectives - Real Valued zdt1 <- function (x) { if (is.null(dim(x))) { x <- matrix(x, nrow = 1) } n <- ncol(x) g <- 1 + rowSums(x[, 2:n, drop = FALSE]) * 9/(n - 1) return(cbind(x[, 1], g * (1 - sqrt(x[, 1]/g)))) } #Not run: ## Not run: result <- nsga2(type = "real-valued", fitness = zdt1, lower = c(0,0), upper = c(1,1), popSize = 100, nObj = 2, monitor = FALSE, maxiter = 500) ## End(Not run) #Example 2 #Three Objectives - Real Valued dtlz1 <- function (x, nobj = 3){ if (is.null(dim(x))) { x <- matrix(x, 1) } n <- ncol(x) y <- matrix(x[, 1:(nobj - 1)], nrow(x)) z <- matrix(x[, nobj:n], nrow(x)) g <- 100 * (n - nobj + 1 + rowSums((z - 0.5)^2 - cos(20 * pi * (z - 0.5)))) tmp <- t(apply(y, 1, cumprod)) tmp <- cbind(t(apply(tmp, 1, rev)), 1) tmp2 <- cbind(1, t(apply(1 - y, 1, rev))) f <- tmp * tmp2 * 0.5 * (1 + g) return(f) } #Not run: ## Not run: result <- nsga2(type = "real-valued", fitness = dtlz1, lower = c(0,0,0), upper = c(1,1,1), popSize = 92, nObj = 3, monitor = FALSE, maxiter = 500) ## End(Not run)
The class 'nsga2' is instantiated within the execution of rmoo and will be returned as a result of it. All data generated during execution will be stored in it.
crowdingDistanceCrowding-comparison approach to estimate of the perimeter of the cuboid formed by using the nearest neighbors as the vertices.
showClass('nsga2')showClass('nsga2')
Minimization of a fitness function using non-dominated sorting genetic algorithms - III (NSGA-IIIs). Multiobjective evolutionary algorithms
nsga3( type = c("binary", "real-valued", "permutation"), fitness, ..., lower, upper, nBits, population = rmooControl(type)$population, selection = rmooControl(type)$selection, crossover = rmooControl(type)$crossover, mutation = rmooControl(type)$mutation, popSize = 50, nObj = NULL, n_partitions = NULL, pcrossover = 0.8, pmutation = 0.1, reference_dirs = generate_reference_points, maxiter = 100, run = maxiter, maxFitness = Inf, names = NULL, suggestions = NULL, parallel = FALSE, monitor = if (interactive()) rmooMonitor else FALSE, summary = FALSE, seed = NULL )nsga3( type = c("binary", "real-valued", "permutation"), fitness, ..., lower, upper, nBits, population = rmooControl(type)$population, selection = rmooControl(type)$selection, crossover = rmooControl(type)$crossover, mutation = rmooControl(type)$mutation, popSize = 50, nObj = NULL, n_partitions = NULL, pcrossover = 0.8, pmutation = 0.1, reference_dirs = generate_reference_points, maxiter = 100, run = maxiter, maxFitness = Inf, names = NULL, suggestions = NULL, parallel = FALSE, monitor = if (interactive()) rmooMonitor else FALSE, summary = FALSE, seed = NULL )
type |
the type of genetic algorithm to be run depending on the nature of decision variables. Possible values are:
|
fitness |
the fitness function, any allowable R function which takes as input an individual string representing a potential solution, and returns a numerical value describing its “fitness”. |
... |
additional arguments to be passed to the fitness function. This allows to write fitness functions that keep some variables fixed during the search |
lower |
a vector of length equal to the decision variables providing the lower bounds of the search space in case of real-valued or permutation encoded optimizations. |
upper |
a vector of length equal to the decision variables providing the upper bounds of the search space in case of real-valued or permutation encoded optimizations. |
nBits |
a value specifying the number of bits to be used in binary encoded optimizations. |
population |
an R function for randomly generating an initial population.
See |
selection |
an R function performing selection, i.e. a function which
generates a new population of individuals from the current population
probabilistically according to individual fitness. See |
crossover |
an R function performing crossover, i.e. a function which
forms offsprings by combining part of the
genetic information from their parents. See |
mutation |
an R function performing mutation, i.e. a function which
randomly alters the values of some genes in a parent chromosome.
See |
popSize |
the population size. |
nObj |
number of objective in the fitness function. |
n_partitions |
Partition number of generated reference points |
pcrossover |
the probability of crossover between pairs of chromosomes. Typically this is a large value and by default is set to 0.8. |
pmutation |
the probability of mutation in a parent chromosome. Usually mutation occurs with a small probability, and by default is set to 0.1. |
reference_dirs |
Function to generate reference points using Das and Dennis approach or matrix with supplied reference points. |
maxiter |
the maximum number of iterations to run before the NSGA search is halted. |
run |
the number of consecutive generations without any improvement in the best fitness value before the NSGA is stopped |
maxFitness |
the upper bound on the fitness function after that the NSGA search is interrupted. |
names |
a vector of character strings providing the names of decision variables. |
suggestions |
a matrix of solutions strings to be included in the initial population. If provided the number of columns must match the number of decision variables. |
parallel |
An optional argument which allows to specify if the NSGA-II should be run sequentially or in parallel. |
monitor |
a logical or an R function which takes as input the current state of the nsga-class object and show the evolution of the search. By default, for interactive sessions the function rmooMonitor prints the average and best fitness values at each iteration. If set to plot these information are plotted on a graphical device. Other functions can be written by the user and supplied as argument. In non interactive sessions, by default monitor = FALSE so any output is suppressed. |
summary |
If there will be a summary generation after generation. |
seed |
an integer value containing the random number generator state. This argument can be used to replicate the results of a NSGA search. Note that if parallel computing is required, the doRNG package must be installed. |
The Non-dominated genetic algorithms III is a meta-heuristic proposed by K. Deb and H. Jain in 2013. The purpose of the algorithms is to find an efficient way to optimize multi-objectives functions (more than three).
Returns an object of class nsga3-class. See nsga3 for a description of available slots information.
Francisco Benitez [email protected]
K. Deb and H. Jain, "An Evolutionary Many-Objective Optimization Algorithm Using Reference-Point-Based Nondominated Sorting Approach, Part I: Solving Problems With Box Constraints," in IEEE Transactions on Evolutionary Computation, vol. 18, no. 4, pp. 577-601, Aug. 2014, doi: 10.1109/TEVC.2013.2281535.
Scrucca, L. (2017) On some extensions to 'GA' package: hybrid optimisation, parallelisation and islands evolution. The R Journal, 9/1, 187-206. doi: 10.32614/RJ-2017-008
#Example 1 #Two Objectives - Real Valued zdt1 <- function (x) { if (is.null(dim(x))) { x <- matrix(x, nrow = 1) } n <- ncol(x) g <- 1 + rowSums(x[, 2:n, drop = FALSE]) * 9/(n - 1) return(cbind(x[, 1], g * (1 - sqrt(x[, 1]/g)))) } #Not run ## Not run: result <- nsga3(type = "real-valued", fitness = zdt1, lower = c(0,0), upper = c(1,1), popSize = 100, nObj = 2, n_partitions = 100, monitor = FALSE, maxiter = 500) ## End(Not run) #Example 2 #Three Objectives - Real Valued dtlz1 <- function (x, nobj = 3, ...){ if (is.null(dim(x))) { x <- matrix(x, 1) } n <- ncol(x) y <- matrix(x[, 1:(nobj - 1)], nrow(x)) z <- matrix(x[, nobj:n], nrow(x)) g <- 100 * (n - nobj + 1 + rowSums((z - 0.5)^2 - cos(20 * pi * (z - 0.5)))) tmp <- t(apply(y, 1, cumprod)) tmp <- cbind(t(apply(tmp, 1, rev)), 1) tmp2 <- cbind(1, t(apply(1 - y, 1, rev))) f <- tmp * tmp2 * 0.5 * (1 + g) return(f) } #Not Run ## Not run: result <- nsga3(type = "real-valued", fitness = dtlz1, lower = c(0,0,0), upper = c(1,1,1), popSize = 92, nObj = 3, n_partitions = 12, monitor = FALSE, maxiter = 500) ## End(Not run)#Example 1 #Two Objectives - Real Valued zdt1 <- function (x) { if (is.null(dim(x))) { x <- matrix(x, nrow = 1) } n <- ncol(x) g <- 1 + rowSums(x[, 2:n, drop = FALSE]) * 9/(n - 1) return(cbind(x[, 1], g * (1 - sqrt(x[, 1]/g)))) } #Not run ## Not run: result <- nsga3(type = "real-valued", fitness = zdt1, lower = c(0,0), upper = c(1,1), popSize = 100, nObj = 2, n_partitions = 100, monitor = FALSE, maxiter = 500) ## End(Not run) #Example 2 #Three Objectives - Real Valued dtlz1 <- function (x, nobj = 3, ...){ if (is.null(dim(x))) { x <- matrix(x, 1) } n <- ncol(x) y <- matrix(x[, 1:(nobj - 1)], nrow(x)) z <- matrix(x[, nobj:n], nrow(x)) g <- 100 * (n - nobj + 1 + rowSums((z - 0.5)^2 - cos(20 * pi * (z - 0.5)))) tmp <- t(apply(y, 1, cumprod)) tmp <- cbind(t(apply(tmp, 1, rev)), 1) tmp2 <- cbind(1, t(apply(1 - y, 1, rev))) f <- tmp * tmp2 * 0.5 * (1 + g) return(f) } #Not Run ## Not run: result <- nsga3(type = "real-valued", fitness = dtlz1, lower = c(0,0,0), upper = c(1,1,1), popSize = 92, nObj = 3, n_partitions = 12, monitor = FALSE, maxiter = 500) ## End(Not run)
The class 'nsga3' is instantiated within the execution of rmoo and will be returned as a result of it. All data generated during execution will be stored in it.
ideal_pointNadir point estimate used as lower bound in normalization.
worst_pointWorst point generated over generations.
sminIndex used to obtain the extreme points.
extreme_pointsare selected using the ASF in the (PerformScalarizing()).
Necessary in the nadir point generation.
worst_of_populationThe worst individuals generated by objectives in the current generation.
worst_of_frontThe worst individuals in the first front generated by objectives in the current generation.
nadir_pointNadir point estimate used as upper bound in normalization.
reference_pointsNSGA-III uses a predefined set of reference points to ensure diversity in obtained solutions. The chosen refenrece points can be predefined in structured manner or supplied by the user. We use the Das and Dennis procedure.
showClass('nsga3')showClass('nsga3')
The class 'numberOrNAOrMatrix' is a simple class union (setClassUnion())
of 'numeric', 'logical', 'logical' and 'matrix'.
Since it is a virtual Class, no objects may be created from it.
showClass('numberOrNAOrMatrix')showClass('numberOrNAOrMatrix')
Functions to evaluate the quality of the results obtained by the algorithms, evaluating their diversity and convergence, providing or not some parameters to compare.
generational_distance(front, true_pareto_front, p, inverted, plus)generational_distance(front, true_pareto_front, p, inverted, plus)
front |
a N×M matrix where N is the number of points and M is the number of objectives. |
true_pareto_front |
a N×M matrix where N is the number of points and M is the number of objectives. |
p |
is the power in which the normalized distance is calculated. |
inverted |
if TRUE then computes IGD. |
plus |
if TRUE then computes the GD+. |
A vector with the measurement metric.
Francisco Benitez
Lamont, G., & Veldhuizen, D.V. (1999). Multiobjective evolutionary algorithms: classifications, analyses, and new innovations.
Method used to visualize the fitness of the individuals during the execution of the algorithms.
plot(x, y, ...) ## S4 method for signature 'nsga,missing' plot(x, y = "missing", type = c("scatter", "pcp", "heatmap", "polar"), ...) ## S4 method for signature 'nsga1,missing' plot(x, y = "missing", type = c("scatter", "pcp", "heatmap", "polar"), ...) ## S4 method for signature 'nsga2,missing' plot(x, y = "missing", type = c("scatter", "pcp", "heatmap", "polar"), ...) ## S4 method for signature 'nsga3,missing' plot(x, y = "missing", type = c("scatter", "pcp", "heatmap", "polar"), ...) ## S4 method for signature 'rnsga2,missing' plot(x, y = "missing", type = c("scatter", "pcp", "heatmap", "polar"), ...)plot(x, y, ...) ## S4 method for signature 'nsga,missing' plot(x, y = "missing", type = c("scatter", "pcp", "heatmap", "polar"), ...) ## S4 method for signature 'nsga1,missing' plot(x, y = "missing", type = c("scatter", "pcp", "heatmap", "polar"), ...) ## S4 method for signature 'nsga2,missing' plot(x, y = "missing", type = c("scatter", "pcp", "heatmap", "polar"), ...) ## S4 method for signature 'nsga3,missing' plot(x, y = "missing", type = c("scatter", "pcp", "heatmap", "polar"), ...) ## S4 method for signature 'rnsga2,missing' plot(x, y = "missing", type = c("scatter", "pcp", "heatmap", "polar"), ...)
x, y
|
|
... |
other arguments passed on to methods
|
type |
Type of graph to draw, the graphs can be of the type "scatter", "pcp", "heatmap", or "polar" |
The following plots are available:
"Scatter Plot"
"Parallel Coordinate Plot"
"Heat Map"
"Polar Coordinate"
A graph of the evaluated type.
Francisco Benitez [email protected]
# Where 'out' is an object of class nsga1, nsga2, or nsga3. # The plot method will by default plot a scatter plot. # # plot(out) # # The Parallel Coordinate Plot will be plotted if "pcp" is passed as a parameter to "type". # # plot(out, type="pcp") # # A heat map plot will be plotted if "heatmap" is passed as a parameter to "type" # and a vector with the individuals to plot to "individual" # # plot(out, type = "heatmap", individual = c(1:5)) # # A polar coordinate plot will be plotted if "polar" is passed as a parameter to "type" # and a vector with the individuals to plot to "individual" # # plot(out, type = "polar", individual = c(1:5))# Where 'out' is an object of class nsga1, nsga2, or nsga3. # The plot method will by default plot a scatter plot. # # plot(out) # # The Parallel Coordinate Plot will be plotted if "pcp" is passed as a parameter to "type". # # plot(out, type="pcp") # # A heat map plot will be plotted if "heatmap" is passed as a parameter to "type" # and a vector with the individuals to plot to "individual" # # plot(out, type = "heatmap", individual = c(1:5)) # # A polar coordinate plot will be plotted if "polar" is passed as a parameter to "type" # and a vector with the individuals to plot to "individual" # # plot(out, type = "polar", individual = c(1:5))
Method used to print the slots and relevant values of the object.
print(x, ...) ## S4 method for signature 'nsga' print(x, ...) ## S4 method for signature 'nsga1' print(x, ...) ## S4 method for signature 'nsga3' print(x, ...)print(x, ...) ## S4 method for signature 'nsga' print(x, ...) ## S4 method for signature 'nsga1' print(x, ...) ## S4 method for signature 'nsga3' print(x, ...)
x |
|
... |
other arguments passed on to methods |
Print the slots and relevant values of the object.
Francisco Benitez [email protected]
# Where 'out' is an object of class nsga1, nsga2, or nsga3 # # print(out)# Where 'out' is an object of class nsga1, nsga2, or nsga3 # # print(out)
Method used to save the progress of the evaluation results, similar to the summary method. Passing additional arguments to the progress method evaluates performance metrics per iteration. This method cannot be called outside of rmoo execution.
progress(object, ...) ## S4 method for signature 'nsga' progress(object, ...) ## S4 method for signature 'nsga1' progress(object, ...) ## S4 method for signature 'nsga2' progress(object, ...) ## S4 method for signature 'nsga3' progress(object, ...)progress(object, ...) ## S4 method for signature 'nsga' progress(object, ...) ## S4 method for signature 'nsga1' progress(object, ...) ## S4 method for signature 'nsga2' progress(object, ...) ## S4 method for signature 'nsga3' progress(object, ...)
object |
|
... |
other arguments passed on to methods. Passing |
A list of length equal to the number of iterations, where the progress made during execution is saved.
Francisco Benitez [email protected]
# Where 'out' is an object of class nsga1, nsga2, or nsga3, and callArgs are # the additional arguments passed when calling the rmoo function, for the # evaluation of performance metrics, reference points are expected to be passed # as an argument to reference_dirs. # # progress(object, callArgs) ## Where 'out' is an object of class nsga1, nsga2, or nsga3, and callArgs are # the additional arguments passed when calling the rmoo function, for the # evaluation of performance metrics, reference points are expected to be passed # as an argument to reference_dirs. # # progress(object, callArgs) #
A implementation of Multi-layer Reference Points Generation.
reference_point_multi_layer(...)reference_point_multi_layer(...)
... |
The different layers provided by the user |
The Multi-layer reference point implementation is based on Blank and Deb's pymoo library, the approach generates different layers of references point at different scales, provided by the user.
A matrix with the multi-layer reference points
Francisco Benitez [email protected]
J. Blank and K. Deb, "Pymoo: Multi-Objective Optimization in Python," in IEEE Access, vol. 8, pp. 89497-89509, 2020, doi: 10.1109/ACCESS.2020.2990567.
Das, Indraneel & Dennis, J. (2000). Normal-Boundary Intersection: A New Method for Generating the Pareto Surface in Nonlinear Multicriteria Optimization Problems. SIAM Journal on Optimization. 8. 10.1137/S1052623496307510.
generate_reference_points() and get_fixed_rowsum_integer_matrix()
Functions implementing crossover non-dominated genetic operator.
rmoo_spCrossover(object, parents) rmoobin_spCrossover(object, parents) rmooreal_spCrossover(object, parents) rmooreal_sbxCrossover(object, parents, eta = 20, indpb = 0.5) rmooperm_oxCrossover(object, parents)rmoo_spCrossover(object, parents) rmoobin_spCrossover(object, parents) rmooreal_spCrossover(object, parents) rmooreal_sbxCrossover(object, parents, eta = 20, indpb = 0.5) rmooperm_oxCrossover(object, parents)
object |
An object of class |
parents |
A two-rows matrix of values indexing the parents from the current population. |
eta |
Distribution parameter for crossover operator. |
indpb |
Independent crossover probability. |
Return a list with two elements:
children |
a matrix of dimension 2 times the number of decision variables containing the generated offsprings; |
fitness |
a vector of length 2 containing the fitness values for the offsprings. A value |
Francisco Benitez
Scrucca, L. (2017) On some extensions to 'GA' package: hybrid optimisation, parallelisation and islands evolution. The R Journal, 9/1, 187-206, doi: 10.32614/RJ-2017-008.
Main function of rmoo, based on the parameters it will call the different
algorithms implemented in the package. Optimization algorithms will minimize
a fitness function. For more details of the algorithms
see nsga2(), nsga3(), rnsga2().
rmoo( type = c("binary", "real-valued", "permutation", "discrete"), algorithm = c("NSGA-II", "NSGA-III", "R-NSGA-II"), fitness, ..., lower, upper, nBits, nvars, population = rmooControl(type)$population, selection = rmooControl(type)$selection, crossover = rmooControl(type)$crossover, mutation = rmooControl(type)$mutation, pcrossover = 0.8, pmutation = 0.1, popSize = 50, maxiter = 100, nObj = NULL, names = NULL, suggestions = NULL, monitor = if (interactive()) rmooMonitor else FALSE, parallel = FALSE, summary = FALSE, seed = NULL, reference_dirs = NULL, epsilon = 0.001, normalization = NULL, extreme_points_as_ref_dirs = FALSE, weights = NULL )rmoo( type = c("binary", "real-valued", "permutation", "discrete"), algorithm = c("NSGA-II", "NSGA-III", "R-NSGA-II"), fitness, ..., lower, upper, nBits, nvars, population = rmooControl(type)$population, selection = rmooControl(type)$selection, crossover = rmooControl(type)$crossover, mutation = rmooControl(type)$mutation, pcrossover = 0.8, pmutation = 0.1, popSize = 50, maxiter = 100, nObj = NULL, names = NULL, suggestions = NULL, monitor = if (interactive()) rmooMonitor else FALSE, parallel = FALSE, summary = FALSE, seed = NULL, reference_dirs = NULL, epsilon = 0.001, normalization = NULL, extreme_points_as_ref_dirs = FALSE, weights = NULL )
type |
the type of genetic algorithm to be run depending on the nature of decision variables. Possible values are:
|
algorithm |
the type of genetic algorithm to be run depending on the nature of decision variables. Possible values are:
|
fitness |
the fitness function, any allowable R function which takes as input an individual string representing a potential solution, and returns a numerical value describing its 'fitness'. |
... |
argument in which all the values necessary for the configuration will be passed as parameters. The user is encouraged to see the documentations. |
lower |
a vector of length equal to the decision variables providing the lower bounds of the search space in case of real-valued or permutation encoded optimizations. |
upper |
a vector of length equal to the decision variables providing the upper bounds of the search space in case of real-valued or permutation encoded optimizations. |
nBits |
a value specifying the number of bits to be used in binary encoded optimizations. |
nvars |
a value . |
population |
an R function for randomly generating an initial population.
See |
selection |
an R function performing selection, i.e. a function which
generates a new population of individuals from the current population
probabilistically according to individual fitness. See |
crossover |
an R function performing crossover, i.e. a function which
forms offsprings by combining part of the genetic information from their
parents. See |
mutation |
an R function performing mutation, i.e. a function which
randomly alters the values of some genes in a parent chromosome.
See |
pcrossover |
the probability of crossover between pairs of chromosomes. Typically this is a large value and by default is set to 0.8. |
pmutation |
the probability of mutation in a parent chromosome. Usually mutation occurs with a small probability, and by default is set to 0.1. |
popSize |
the population size. |
maxiter |
the maximum number of iterations to run before the NSGA search is halted. |
nObj |
number of objective in the fitness function. |
names |
a vector of character strings providing the names of decision variables. |
suggestions |
a matrix of solutions strings to be included in the initial population. If provided the number of columns must match the number of decision variables. |
monitor |
a logical or an R function which takes as input the current state of the nsga-class object and show the evolution of the search. By default, for interactive sessions the function rmooMonitor prints the average and best fitness values at each iteration. If set to plot these information are plotted on a graphical device. Other functions can be written by the user and supplied as argument. In non interactive sessions, by default monitor = FALSE so any output is suppressed. |
parallel |
An optional argument which allows to specify if the NSGA-II should be run sequentially or in parallel. |
summary |
If there will be a summary generation after generation. |
seed |
an integer value containing the random number generator state. This argument can be used to replicate the results of a NSGA search. Note that if parallel computing is required, the doRNG package must be installed. |
reference_dirs |
Function to generate reference points using Das and Dennis approach or matrix with supplied reference points. |
epsilon |
controls the extent of obtained solutions by grouping all solutions that have a normalized difference sum in objective values of epsilon or less. |
normalization |
of the ideal points and nadir. They can be:
|
extreme_points_as_ref_dirs |
flag to use extreme points as reference points. |
weights |
vector specifies the importance of one objective function over
the other, by default all objectives have equal weights.
of |
Multi- and Many-Optimization of a fitness function using Non-dominated Sorting Genetic Algorithms. The algorithms currently implemented by rmoo are: NSGA-II, NSGA-III and R-NSGA-II
The Non-dominated genetic algorithms II (NSGA-II) is a meta-heuristic proposed by K. Deb, A. Pratap, S. Agarwal and T. Meyarivan in 2002. The purpose of the algorithms is to find an efficient way to optimize multi-objectives functions (two or more).
The Non-dominated genetic algorithms III (NSGA-III) is a meta-heuristic proposed by K. Deb and H. Jain in 2013. The purpose of the algorithms is to find an efficient way to optimize multi-objectives functions (more than three).
The Reference point-based Non-dominated genetic algorithms II (R-NSGA-II) is a meta-heuristic proposed by K. Deb and J. Sundar in 2006. It is a modification of NSGA-II based on reference points in which the decision-maker supplies one or more preference points and a weight vector that will guide the solutions towards regions desired by the user.
Returns an object of class nsga2-class, rnsga2-class or nsga3-class. See nsga2, rnsga2, nsga3 for a description of available slots information.
Francisco Benitez [email protected]
Scrucca, L. (2017) On some extensions to 'GA' package: hybrid optimisation, parallelisation and islands evolution. The R Journal, 9/1, 187-206. doi: 10.32614/RJ-2017-008
Kalyanmoy Deb and J. Sundar. 2006. Reference point based multi-objective optimization using evolutionary algorithms. In Proceedings of the 8th annual conference on Genetic and evolutionary computation (GECCO '06). Association for Computing Machinery, New York, NY, USA, 635–642. doi: 10.1145/1143997.1144112
K. Deb, A. Pratap, S. Agarwal and T. Meyarivan, 'A fast and elitist multiobjective genetic algorithm: NSGA-II,' in IEEE Transactions on Evolutionary Computation, vol. 6, no. 2, pp. 182-197, April 2002, doi: 10.1109/4235.996017.
K. Deb and H. Jain, "An Evolutionary Many-Objective Optimization Algorithm Using Reference-Point-Based Nondominated Sorting Approach, Part I: Solving Problems With Box Constraints," in IEEE Transactions on Evolutionary Computation, vol. 18, no. 4, pp. 577-601, Aug. 2014, doi: 10.1109/TEVC.2013.2281535.
#Example 1 #Two Objectives - Real Valued zdt1 <- function (x) { if (is.null(dim(x))) { x <- matrix(x, nrow = 1) } n <- ncol(x) g <- 1 + rowSums(x[, 2:n, drop = FALSE]) * 9/(n - 1) return(cbind(x[, 1], g * (1 - sqrt(x[, 1]/g)))) } #Not run: ## Not run: result <- rmoo(type = "real-valued", fitness = zdt1, algorithm = "NSGA-II", lower = c(0,0), upper = c(1,1), popSize = 100, nObj = 2, monitor = FALSE, maxiter = 500) ## End(Not run) #Example 2 #Three Objectives - Real Valued dtlz1 <- function (x, nobj = 3){ if (is.null(dim(x))) { x <- matrix(x, 1) } n <- ncol(x) y <- matrix(x[, 1:(nobj - 1)], nrow(x)) z <- matrix(x[, nobj:n], nrow(x)) g <- 100 * (n - nobj + 1 + rowSums((z - 0.5)^2 - cos(20 * pi * (z - 0.5)))) tmp <- t(apply(y, 1, cumprod)) tmp <- cbind(t(apply(tmp, 1, rev)), 1) tmp2 <- cbind(1, t(apply(1 - y, 1, rev))) f <- tmp * tmp2 * 0.5 * (1 + g) return(f) } #Define uniformly distributed reference points. ref_points <- generate_reference_points(3,12) #Not Run ## Not run: result <- rmoo(type = "real-valued", fitness = dtlz1, algorithm = "NSGA-III", lower = c(0,0,0), upper = c(1,1,1), popSize = 92, nObj = 3, reference_dirs = ref_points, monitor = FALSE, maxiter = 500) ## End(Not run) #Example 3 #Two Objectives - Real Valued with Preference-guided zdt2 <- function (x) { if (is.null(dim(x))) { x <- matrix(x, nrow = 1) } n <- ncol(x) g <- 1 + rowSums(x[, 2:n, drop = FALSE]) * 9/(n - 1) return(cbind(x[, 1], g * (1 - (x[, 1]/g)^2))) } #Define uniformly distributed reference points. ref_points <- rbind(c(1.0, 0.0), c(0.0, 1.0), c(0.5, 0.5)) #Not run ## Not run: result <- rmoo(type = "real-valued", fitness = zdt2, algorithm = "R-NSGA-II", lower = c(0,0), upper = c(1,1), reference_dirs = ref_points, popSize = 92, nObj = 2, monitor = FALSE, maxiter = 500) ## End(Not run)#Example 1 #Two Objectives - Real Valued zdt1 <- function (x) { if (is.null(dim(x))) { x <- matrix(x, nrow = 1) } n <- ncol(x) g <- 1 + rowSums(x[, 2:n, drop = FALSE]) * 9/(n - 1) return(cbind(x[, 1], g * (1 - sqrt(x[, 1]/g)))) } #Not run: ## Not run: result <- rmoo(type = "real-valued", fitness = zdt1, algorithm = "NSGA-II", lower = c(0,0), upper = c(1,1), popSize = 100, nObj = 2, monitor = FALSE, maxiter = 500) ## End(Not run) #Example 2 #Three Objectives - Real Valued dtlz1 <- function (x, nobj = 3){ if (is.null(dim(x))) { x <- matrix(x, 1) } n <- ncol(x) y <- matrix(x[, 1:(nobj - 1)], nrow(x)) z <- matrix(x[, nobj:n], nrow(x)) g <- 100 * (n - nobj + 1 + rowSums((z - 0.5)^2 - cos(20 * pi * (z - 0.5)))) tmp <- t(apply(y, 1, cumprod)) tmp <- cbind(t(apply(tmp, 1, rev)), 1) tmp2 <- cbind(1, t(apply(1 - y, 1, rev))) f <- tmp * tmp2 * 0.5 * (1 + g) return(f) } #Define uniformly distributed reference points. ref_points <- generate_reference_points(3,12) #Not Run ## Not run: result <- rmoo(type = "real-valued", fitness = dtlz1, algorithm = "NSGA-III", lower = c(0,0,0), upper = c(1,1,1), popSize = 92, nObj = 3, reference_dirs = ref_points, monitor = FALSE, maxiter = 500) ## End(Not run) #Example 3 #Two Objectives - Real Valued with Preference-guided zdt2 <- function (x) { if (is.null(dim(x))) { x <- matrix(x, nrow = 1) } n <- ncol(x) g <- 1 + rowSums(x[, 2:n, drop = FALSE]) * 9/(n - 1) return(cbind(x[, 1], g * (1 - (x[, 1]/g)^2))) } #Define uniformly distributed reference points. ref_points <- rbind(c(1.0, 0.0), c(0.0, 1.0), c(0.5, 0.5)) #Not run ## Not run: result <- rmoo(type = "real-valued", fitness = zdt2, algorithm = "R-NSGA-II", lower = c(0,0), upper = c(1,1), reference_dirs = ref_points, popSize = 92, nObj = 2, monitor = FALSE, maxiter = 500) ## End(Not run)
Functions implementing mutation non-dominated genetic operator.
rmoobin_raMutation(object, parent) rmooreal_raMutation(object, parent) rmooreal_polMutation(object, parent, eta = 20, indpb = 0.5) rmooperm_simMutation(object, parent)rmoobin_raMutation(object, parent) rmooreal_raMutation(object, parent) rmooreal_polMutation(object, parent, eta = 20, indpb = 0.5) rmooperm_simMutation(object, parent)
object |
An object of class |
parent |
A vector of values for the parent from the current population where mutation should occur. |
eta |
Distribution parameter for mutation operator. |
indpb |
Independent mutation probability. |
Return a vector of values containing the mutated string.
Francisco Benitez
Scrucca, L. (2017) On some extensions to 'GA' package: hybrid optimisation, parallelisation and islands evolution. The R Journal, 9/1, 187-206, doi: 10.32614/RJ-2017-008.
Functions for creating a random initial population to be used in non-dominated genetic algorithms.
rmoobin_Population(object) rmooreal_Population(object) rmooperm_Population(object)rmoobin_Population(object) rmooreal_Population(object) rmooperm_Population(object)
object |
An object of class |
rmoobin_Population generates a random population of object@nBits binary values;
rmooreal_Population generates a random (uniform) population of real values in the range [object@lower, object@upper];
rmooperm_Population generates a random (uniform) population of integer values in the range [object@lower, object@upper].
Return a matrix of dimension object@popSize times the number of decision variables.
Francisco Benitez
Scrucca, L. (2017) On some extensions to 'GA' package: hybrid optimisation, parallelisation and islands evolution. The R Journal, 9/1, 187-206, doi: 10.32614/RJ-2017-008.
Functions implementing selection non-dominated genetic operator.
rmoo_lrSelection(object, r, q) rmoo_tourSelection(object, k = 2, ...) rmoobin_lrSelection(object, r, q) rmoobin_tourSelection(object, k = 2, ...) rmooreal_lrSelection(object, r, q) rmooreal_tourSelection(object, k = 2, ...) rmooperm_lrSelection(object, r, q) rmooperm_tourSelection(object, k = 2, ...)rmoo_lrSelection(object, r, q) rmoo_tourSelection(object, k = 2, ...) rmoobin_lrSelection(object, r, q) rmoobin_tourSelection(object, k = 2, ...) rmooreal_lrSelection(object, r, q) rmooreal_tourSelection(object, k = 2, ...) rmooperm_lrSelection(object, r, q) rmooperm_tourSelection(object, k = 2, ...)
object |
An object of class |
r |
A tuning parameter for the specific selection operator. |
q |
A tuning parameter for the specific selection operator. |
k |
A tuning parameter for the specific selection operator. |
... |
Further arguments passed to or from other methods. |
Return a list with two elements:
population |
a matrix of dimension |
fitness |
a vector of length |
Francisco Benitez
Scrucca, L. (2017) On some extensions to 'GA' package: hybrid optimisation, parallelisation and islands evolution. The R Journal, 9/1, 187-206, doi: 10.32614/RJ-2017-008.
Default settings for non-dominated genetic operators used in the 'rmoo' package.
rmooControl(...)rmooControl(...)
... |
no arguments, a single character vector, or a named list with components. |
If the function is called with no arguments returns the current default settings, i.e., a list with the following default components:
"binary"population = "rmoobin_Population"
selection = "rmoobin_tourSelection"
crossover = "rmoobin_spCrossover"
mutation = "rmoobin_raMutation"
"real-valued"population = "rmooreal_Population"
selection = "rmooreal_tourSelection"
crossover = "rmooreal_sbxCrossover"
mutation = "rmooreal_polMutation"
"permutation"population = "rmooperm_Population"
selection = "rmooperm_tourSelection"
crossover = "rmooperm_oxCrossover"
mutation = "rmooperm_simMutation"
"discrete"population = "rmooint_Population"
selection = "rmooint_tourSelection"
crossover = "rmooint_uxCrossover"
mutation = "rmooint_uxMutation"
"eps" = the tolerance value used by the package functions. By default set at sqrt(.Machine$double.eps).
The function may be called with a single string specifying the name of the component. In this case the function returns the current default settings.
To change the default values, a named component must be followed by a single value (in case of "eps") or a list of component(s) specifying the name of the function for a genetic operator. See the Examples section.
If the argument list is empty the function returns the current list of values. If the argument list is not empty, the returned list is invisible.
The parameter values set via a call to this function will remain in effect for the rest of the session, affecting the subsequent behaviour of the functions for which the given parameters are relevant.
Francisco Benitez
Scrucca, L. (2017) On some extensions to 'GA' package: hybrid optimisation, parallelisation and islands evolution. The R Journal, 9/1, 187-206, doi: 10.32614/RJ-2017-008.
# get and save defaults defaultControl <- rmooControl() print(defaultControl) # get current defaults only for real-valued search rmooControl("real-valued") # set defaults for selection operator of real-valued search rmooControl("real-valued" = list(selection = "rmooreal_lrSelection")) rmooControl("real-valued") # set defaults for selection and crossover operators of real-valued search rmooControl("real-valued" = list(selection = "rmooreal_lrSelection", crossover = "rmooreal_spCrossover")) rmooControl("real-valued") # restore defaults rmooControl(defaultControl) rmooControl()# get and save defaults defaultControl <- rmooControl() print(defaultControl) # get current defaults only for real-valued search rmooControl("real-valued") # set defaults for selection operator of real-valued search rmooControl("real-valued" = list(selection = "rmooreal_lrSelection")) rmooControl("real-valued") # set defaults for selection and crossover operators of real-valued search rmooControl("real-valued" = list(selection = "rmooreal_lrSelection", crossover = "rmooreal_spCrossover")) rmooControl("real-valued") # restore defaults rmooControl(defaultControl) rmooControl()
Functions to plotting fitness values at each iteration of a search for the 'rmoo' package.
rmooMonitor(object, ...)rmooMonitor(object, ...)
object |
an object of class |
... |
further arguments passed to or from other methods. |
These functions plot the fitness values of the current step of the nsga3 on the console.
By default, rmooMonitor is called in interactive sessions by nsga, nsga2, or nsga3.
The function can be modified by the user to plot or print the values it considers by iteration.
Francisco Benitez
Scrucca, L. (2017) On some extensions to 'GA' package: hybrid optimisation, parallelisation and islands evolution. The R Journal, 9/1, 187-206, doi: 10.32614/RJ-2017-008.
Minimization of a fitness function using reference point based non-dominated sorting genetic algorithms - II (R-NSGA-IIs). Multiobjective evolutionary algorithms
rnsga2( type = c("binary", "real-valued", "permutation"), fitness, ..., lower, upper, nBits, population = rmooControl(type)$population, selection = rmooControl(type)$selection, crossover = rmooControl(type)$crossover, mutation = rmooControl(type)$mutation, reference_dirs = NULL, epsilon = 0.001, normalization = c("ever", "front", "no"), extreme_points_as_ref_dirs = FALSE, weights = NULL, popSize = 50, nObj = NULL, pcrossover = 0.8, pmutation = 0.1, maxiter = 100, run = maxiter, maxFitness = Inf, names = NULL, suggestions = NULL, parallel = FALSE, monitor = if (interactive()) rmooMonitor else FALSE, summary = FALSE, seed = NULL )rnsga2( type = c("binary", "real-valued", "permutation"), fitness, ..., lower, upper, nBits, population = rmooControl(type)$population, selection = rmooControl(type)$selection, crossover = rmooControl(type)$crossover, mutation = rmooControl(type)$mutation, reference_dirs = NULL, epsilon = 0.001, normalization = c("ever", "front", "no"), extreme_points_as_ref_dirs = FALSE, weights = NULL, popSize = 50, nObj = NULL, pcrossover = 0.8, pmutation = 0.1, maxiter = 100, run = maxiter, maxFitness = Inf, names = NULL, suggestions = NULL, parallel = FALSE, monitor = if (interactive()) rmooMonitor else FALSE, summary = FALSE, seed = NULL )
type |
the type of genetic algorithm to be run depending on the nature of decision variables. Possible values are:
|
fitness |
the fitness function, any allowable R function which takes as input an individual string representing a potential solution, and returns a numerical value describing its 'fitness'. |
... |
additional arguments to be passed to the fitness function. This allows to write fitness functions that keep some variables fixed during the search |
lower |
a vector of length equal to the decision variables providing the lower bounds of the search space in case of real-valued or permutation encoded optimizations. |
upper |
a vector of length equal to the decision variables providing the upper bounds of the search space in case of real-valued or permutation encoded optimizations. |
nBits |
a value specifying the number of bits to be used in binary encoded optimizations |
population |
an R function for randomly generating an initial population.
See |
selection |
an R function performing selection, i.e. a function which
generates a new population of individuals from the current population
probabilistically according to individual fitness. See |
crossover |
an R function performing crossover, i.e. a function which
forms offsprings by combining part of the genetic information from their
parents. See |
mutation |
an R function performing mutation, i.e. a function which
randomly alters the values of some genes in a parent chromosome.
See |
reference_dirs |
Function to generate reference points using Das and Dennis approach or matrix with supplied reference points. |
epsilon |
controls the extent of obtained solutions by grouping all solutions that have a normalized difference sum in objective values of epsilon or less. |
normalization |
of the ideal points and nadir. They can be:
|
extreme_points_as_ref_dirs |
flag to use extreme points as reference points. |
weights |
vector specifies the importance of one objective function over the other, by default all objectives have equal weights. |
popSize |
the population size. |
nObj |
number of objective in the fitness function. |
pcrossover |
the probability of crossover between pairs of chromosomes. Typically this is a large value and by default is set to 0.8. |
pmutation |
the probability of mutation in a parent chromosome. Usually mutation occurs with a small probability, and by default is set to 0.1. |
maxiter |
the maximum number of iterations to run before the NSGA search is halted. |
run |
the number of consecutive generations without any improvement in the best fitness value before the NSGA is stopped |
maxFitness |
the upper bound on the fitness function after that the NSGA search is interrupted. |
names |
a vector of character strings providing the names of decision variables. |
suggestions |
a matrix of solutions strings to be included in the initial population. If provided the number of columns must match the number of decision variables. |
parallel |
An optional argument which allows to specify if the NSGA-II should be run sequentially or in parallel. |
monitor |
a logical or an R function which takes as input the current state of the nsga-class object and show the evolution of the search. By default, for interactive sessions the function rmooMonitor prints the average and best fitness values at each iteration. If set to plot these information are plotted on a graphical device. Other functions can be written by the user and supplied as argument. In non interactive sessions, by default monitor = FALSE so any output is suppressed. |
summary |
If there will be a summary generation after generation. |
seed |
an integer value containing the random number generator state. This argument can be used to replicate the results of a NSGA search. Note that if parallel computing is required, the doRNG package must be installed. |
R-NSGA-II is a meta-heuristic proposed by K. Deb and J. Sundar in 2006. It is a modification of NSGA-II based on reference points in which the decision-maker supplies one or more preference points and a weight vector that will guide the solutions towards regions desired by the user.
Returns an object of class rnsga2-class. See rnsga2 for a description of available slots information.
Francisco Benitez [email protected]
Kalyanmoy Deb and J. Sundar. 2006. Reference point based multi-objective optimization using evolutionary algorithms. In Proceedings of the 8th annual conference on Genetic and evolutionary computation (GECCO '06). Association for Computing Machinery, New York, NY, USA, 635–642. doi: 10.1145/1143997.1144112
#Example #Two Objectives - Real Valued zdt1 <- function (x) { if (is.null(dim(x))) { x <- matrix(x, nrow = 1) } n <- ncol(x) g <- 1 + rowSums(x[, 2:n, drop = FALSE]) * 9/(n - 1) return(cbind(x[, 1], g * (1 - sqrt(x[, 1]/g)))) } #Define the reference points reference_points = rbind(c(0.2, 0.8), c(0.8, 0.2), c(0.4, 0.5)) #Not run: ## Not run: result <- rnsga2(type = "real-valued", fitness = zdt1, lower = c(0,0), upper = c(1,1), reference_dirs = reference_points, popSize = 100, nObj = 2, monitor = FALSE, maxiter = 500, seed = 45) ## End(Not run) #Example 2 #Three Objectives - Real Valued dtlz1 <- function (x, nobj = 3){ if (is.null(dim(x))) { x <- matrix(x, 1) } n <- ncol(x) y <- matrix(x[, 1:(nobj - 1)], nrow(x)) z <- matrix(x[, nobj:n], nrow(x)) g <- 100 * (n - nobj + 1 + rowSums((z - 0.5)^2 - cos(20 * pi * (z - 0.5)))) tmp <- t(apply(y, 1, cumprod)) tmp <- cbind(t(apply(tmp, 1, rev)), 1) tmp2 <- cbind(1, t(apply(1 - y, 1, rev))) f <- tmp * tmp2 * 0.5 * (1 + g) return(f) } #Define the reference points reference_points <- rbind(c(1.0, 0.5, 0.0), c(0.0, 0.5, 1.0), c(0.5, 0.5, 0.5)) #Not run: ## Not run: result <- rnsga2(type = "real-valued", fitness = dtlz1, lower = c(0,0,0), upper = c(1,1,1), reference_dirs = reference_points, popSize = 92, nObj = 3, monitor = FALSE, maxiter = 500) ## End(Not run)#Example #Two Objectives - Real Valued zdt1 <- function (x) { if (is.null(dim(x))) { x <- matrix(x, nrow = 1) } n <- ncol(x) g <- 1 + rowSums(x[, 2:n, drop = FALSE]) * 9/(n - 1) return(cbind(x[, 1], g * (1 - sqrt(x[, 1]/g)))) } #Define the reference points reference_points = rbind(c(0.2, 0.8), c(0.8, 0.2), c(0.4, 0.5)) #Not run: ## Not run: result <- rnsga2(type = "real-valued", fitness = zdt1, lower = c(0,0), upper = c(1,1), reference_dirs = reference_points, popSize = 100, nObj = 2, monitor = FALSE, maxiter = 500, seed = 45) ## End(Not run) #Example 2 #Three Objectives - Real Valued dtlz1 <- function (x, nobj = 3){ if (is.null(dim(x))) { x <- matrix(x, 1) } n <- ncol(x) y <- matrix(x[, 1:(nobj - 1)], nrow(x)) z <- matrix(x[, nobj:n], nrow(x)) g <- 100 * (n - nobj + 1 + rowSums((z - 0.5)^2 - cos(20 * pi * (z - 0.5)))) tmp <- t(apply(y, 1, cumprod)) tmp <- cbind(t(apply(tmp, 1, rev)), 1) tmp2 <- cbind(1, t(apply(1 - y, 1, rev))) f <- tmp * tmp2 * 0.5 * (1 + g) return(f) } #Define the reference points reference_points <- rbind(c(1.0, 0.5, 0.0), c(0.0, 0.5, 1.0), c(0.5, 0.5, 0.5)) #Not run: ## Not run: result <- rnsga2(type = "real-valued", fitness = dtlz1, lower = c(0,0,0), upper = c(1,1,1), reference_dirs = reference_points, popSize = 92, nObj = 3, monitor = FALSE, maxiter = 500) ## End(Not run)
The class 'rnsga2' is instantiated within the execution of rmoo and will be returned as a result of it. All data generated during execution will be stored in it.
crowdingDistanceCrowding-comparison approach to estimate of the perimeter of the cuboid formed by using the nearest neighbors as the vertices.
reference_pointsR-NSGA-II uses a set of reference points defined by the user to ensure diversity in obtained solutions.
extreme_pointsare selected using the ASF in the (PerformScalarizing()).
Necessary in the nadir point generation.
sminIndex used to obtain the extreme points.
showClass('rnsga2')showClass('rnsga2')
A implementation of Das and Dennis's Reference Points Generation.
scale_reference_directions(ref_dirs, scaling)scale_reference_directions(ref_dirs, scaling)
ref_dirs, scaling
|
where 'ref_dirs' are the reference points generated and 'scaling' are the scale on which the points are distributed. |
The implemented Reference Point Generation is based on the Das and Dennis's systematic approach that places points on a normalized hyper-plane which is equally inclined to all objective axes and has an intercept of one on each axis.
A matrix with rescaled reference points uniformly distributed.
Francisco Benitez [email protected]
J. Blank and K. Deb, "Pymoo: Multi-Objective Optimization in Python," in IEEE Access, vol. 8, pp. 89497-89509, 2020, doi: 10.1109/ACCESS.2020.2990567.
generate_reference_points() and get_fixed_rowsum_integer_matrix()
Calculate of sharing distance and dummy fitness
sharing(object)sharing(object)
object |
An object of class 'nsga', usually resulting from a call to function nsga. Fitness Function Objective Numbers. |
The sharing distance operator guides the selection process at the various stages of the algorithm toward a uniformly spread-out Pareto-optimal front
A vector with the dummy fitness.
Francisco Benitez [email protected]
N. Srinivas and K. Deb, 'Multiobjective Optimization Using Nondominated Sorting in Genetic Algorithms,' in Evolutionary Computation, vol. 2, no. 3, pp. 221-248, Sept. 1994, doi: 10.1162/evco.1994.2.3.221.
Method used to summarize the results of the evaluations, passing additional arguments in the summary method the performance metrics is evaluated.
summary(object, ...) ## S4 method for signature 'nsga' summary(object, ...) ## S4 method for signature 'nsga1' summary(object, ...) ## S4 method for signature 'nsga2' summary(object, ...) ## S4 method for signature 'nsga3' summary(object, ...)summary(object, ...) ## S4 method for signature 'nsga' summary(object, ...) ## S4 method for signature 'nsga1' summary(object, ...) ## S4 method for signature 'nsga2' summary(object, ...) ## S4 method for signature 'nsga3' summary(object, ...)
object |
|
... |
other arguments passed on to methods. Passing |
A summary of the values resulting from the execution of an algorithm.
Francisco Benitez [email protected]
# Where 'out' is an object of class nsga1, nsga2, or nsga3 # # summary(out) # # For the evaluation of the metrics, pass the reference point # # ref_points <- generate_reference_points(3,12) # summary(out, reference_dirs = ref_points)# Where 'out' is an object of class nsga1, nsga2, or nsga3 # # summary(out) # # For the evaluation of the metrics, pass the reference point # # ref_points <- generate_reference_points(3,12) # summary(out, reference_dirs = ref_points)
Functions to scalarize the members of the population to locate them in a normalized hyperplane, finding the ideal point, nadir point, worst point and the extreme points.
UpdateIdealPoint(object, nObj) UpdateWorstPoint(object, nObj) PerformScalarizing(population, fitness, smin, extreme_points, ideal_point) get_nadir_point(object)UpdateIdealPoint(object, nObj) UpdateWorstPoint(object, nObj) PerformScalarizing(population, fitness, smin, extreme_points, ideal_point) get_nadir_point(object)
object |
An object of class |
nObj |
numbers of objective values of the function to evaluate. |
population |
individuals of the population until last front. |
fitness |
objective values of the population until last front. |
smin |
Achievement Escalation Function Index. |
extreme_points |
Extreme points of the previous generation to upgrade. |
ideal_point |
Ideal point of the current generation to translate objectives. |
Return scalarized objective values in a normalized hyperplane.
Francisco Benitez
J. Blank and K. Deb, "Pymoo: Multi-Objective Optimization in Python," in IEEE Access, vol. 8, pp. 89497-89509, 2020, doi: 10.1109/ACCESS.2020.2990567.
K. Deb and H. Jain, "An Evolutionary Many-Objective Optimization Algorithm Using Reference-Point-Based Nondominated Sorting Approach, Part I: Solving Problems With Box Constraints," in IEEE Transactions on Evolutionary Computation, vol. 18, no. 4, pp. 577-601, Aug. 2014, doi: 10.1109/TEVC.2013.2281535.