Data set for example G01

Description

This data set contains the initial positions for a population of size 20 to be used with the example G01.

EmiR optimization options

Description

A S4 class storing the options for the optimization algorithms in EmiR.

Slots

maximize

if TRUE the objective function is maximized instead of being minimized. Default is FALSE.

silent_mode

if TRUE no output to console is generated. Default is FALSE.

save_pop_history

if TRUE the position of all individuals in the population at each iteration is stored. This is necessary for functions like plot_population and animate_population to work. Default is FALSE.

constrained_method

method for constrained optimization. Possible values are:

• "PENALTY" - Penalty Method: the constrained problem is converted to an unconstrained one, by adding a penalty function to the objective function. The penalty function consists of a penalty parameter multiplied by a measure of violation of the constraints. The penalty parameter is multiplied by a scale factor (see penalty_scale) at every iteration;

• "BARRIER" - Barrier Method: the value of the objective function is set equal to an arbitrary large positive (or negative in case of maximization) number if any of the constraints is violated;

• "ACCREJ" - Acceptance-Rejection method: a solution violating any of the constraints is replaced by a randomly generated new one in the feasible region. Default is "PENALTY".

penalty_scale

scale factor for the penalty parameter at each iteration. It should be greater than 1. Default is 10.

start_penalty_param

initial value of the penalty parameter. It should be greater than 0. Default is 2.

max_penalty_param

maximum value for the penalty parameter. It should be greater than 0. Default is 1.e+10.

constr_init_pop

if TRUE the initial population is generated in the feasible region only. Default is TRUE.

oob_solutions

strategy to treat out-of-bound solutions. Possible values are:

• "RBC" - Reflective Boundary Condition: the solution is placed back inside the search domain at a position which is distanced from the boundary as the out-of-bound excess. Depending on the optimization algorithm, the velocity of the corresponding individual of the population could be also inverted;

• "PBC" - Periodic Boundary Condition: the solution is placed back inside the search domain at a position which is distanced from the opposite boundary as the out-of-bound excess;

• "BAB" - Back At Boundary: the solution is placed back at the boundaries for the out-of-bound dimensions;

• "DIS" - Disregard the solution: the solution is replaced by a new one, which is randomly generated in the search space. Default is "DIS".

seed

seed for the internal random number generator. Accepted values are strictly positive integers. If NULL a random seed at each execution is used. Default is NULL.

initial_population

manually specify the position of the initial population. A \(n \times d\) matrix has to be provided, where \(n\) is the population size and \(d\) is the number of parameters the objective function is minimized with respect to.

generation_function

manually specify the function to generate to new solutions.

EmiR optimization results

Description

A S4 class storing all relevant data from an optimization with EmiR.

Slots

algorithm

the name of the algorithm.

iterations

the number of iterations.

population_size

the number of individuals in the population.

obj_function

the minimized/maximized objective function.

constraints

the constraints the objective function is subjected to.

best_cost

the best value of the objective function found.

best_parameters

the parameter values for which the best cost was obtained.

parameter_range

the range on the parameters.

pop_history

list containing the positions of all individuals in the population at each iteration. The list is filled only if save_pop_history is TRUE in the options of the minimizer (see MinimizerOpts).

cost_history

the vector storing the best value of the objective function at each iteration.

exec_time_sec

the execution time in seconds.

is_maximization

if TRUE the objective function has been maximized insted of being minimized.

Ackley Function

Description

Implementation of n-dimensional Ackley function, with \(a=20\), \(b=0.2\) and \(c=2\pi\) (see definition below).

Usage

ackley_func(x)

Arguments

Details

On an n-dimensional domain it is defined by

and is usually evaluated on \(x_{i} \in [ -32.768, 32.768 ]\), for all \(i=1,...,n\). The function has one global minimum at \(f(\vec{x})=0\) for \(x_{i}=0\) for all \(i=1,...,n\).

Value

The value of the function.

References

Ackley DH (1987). A Connectionist Machine for Genetic Hillclimbing. Springer US. doi:10.1007/978-1-4613-1997-9.

Animation of population motion

Description

Create an animation of the population motion for the minimization of 1D and 2D functions. The animation can be produced only if save_pop_history is TRUE in the options of the minimizer (see MinimizerOpts).

Usage

animate_population(minimizer_result, n_points = 100)

Arguments

Bohachevsky Function

Description

Implementation of 2-dimensional Bohachevsky function.

Usage

bohachevsky_func(x)

Arguments

Details

On an 2-dimensional domain it is defined by

and is usually evaluated on \(x_{i} \in [ -100, 100 ]\), for all \(i=1,2\). The function has one global minimum at \(f(\vec{x}) = 0\) for \(\vec{x} = [ 0, 0 ]\).

Value

The value of the function.

References

Bohachevsky IO, Johnson ME, Stein ML (1986). “Generalized simulated annealing for function optimization.” Technometrics, 28(3), 209–217.

Configuration object for the Artificial Bee Colony Algorithm

Description

Create a configuration object for the Artificial Bee Colony Algorithm (ABC). At minimum the number of iterations (parameter iterations) and the number of bees (parameter population_size) have to be provided.

Usage

config_abc(
  iterations,
  population_size,
  iterations_same_cost = NULL,
  absolute_tol = NULL,
  employed_frac = 0.5,
  n_scout = 1
)

Arguments

Value

config_abc returns an object of class ABCConfig.

References

Karaboga D, Basturk B (2007). “A powerful and efficient algorithm for numerical function optimization: artificial bee colony (ABC) algorithm.” Journal of Global Optimization, 39(3), 459–471. doi:10.1007/s10898-007-9149-x.

Examples

conf <- config_abc(iterations = 100, population_size = 50, iterations_same_cost = NULL,
absolute_tol = NULL, employed_frac = 0.5, n_scout = 1)

Configuration object for algorithms

Description

Create a configuration object for one of the algorithms available in EmiR. At minimum the id of the algorithm (parameter algorithm_id), the number of iterations (parameter iterations) and the number of individuals in the population (parameter population_size) have to be provided.

Usage

config_algo(
  algorithm_id,
  iterations,
  population_size,
  iterations_same_cost = NULL,
  absolute_tol = NULL,
  ...
)

Arguments

Value

config_algo returns a configuration object specific for the specified algorithm.

Examples

conf <- config_algo(algorithm_id = "PS", population_size = 200, iterations = 10000)

Configuration object for the Bat Algorithm

Description

Create a configuration object for the Bat Algorithm (BAT). At minimum the number of iterations (parameter iterations) and the number of bats (parameter population_size) have to be provided.

Usage

config_bat(
  iterations,
  population_size,
  iterations_same_cost = NULL,
  absolute_tol = NULL,
  initial_loudness = 1.5,
  alpha = 0.9,
  initial_pulse_rate = 0.5,
  gamma = 0.9,
  freq_min = 0,
  freq_max = 2
)

Arguments

Value

config_bat returns an object of class BATConfig.

References

Yang X (2010). “A new metaheuristic bat-inspired algorithm.” In Nature inspired cooperative strategies for optimization (NICSO 2010), 65–74. Springer.

Examples

conf <- config_bat(iterations = 100, population_size = 50, iterations_same_cost = NULL,
absolute_tol = NULL, initial_loudness = 1.5, alpha = 0.9,
initial_pulse_rate = 0.5, gamma = 0.9,
freq_min = 0., freq_max = 2.)

Configuration object for the Cuckoo Search Algorithm

Description

Create a configuration object for the Cuckoo Search Algorithm (CS). At minimum the number of iterations (parameter iterations) and the number of host nests (parameter population_size) have to be provided.

Usage

config_cs(
  iterations,
  population_size,
  iterations_same_cost = NULL,
  absolute_tol = NULL,
  discovery_rate = 0.25,
  step_size = 1
)

Arguments

Value

config_cs returns an object of class CSConfig.

References

Yang X, Deb S (2009). “Cuckoo Search via Lèvy flights.” In 2009 World Congress on Nature & Biologically Inspired Computing (NaBIC). doi:10.1109/nabic.2009.5393690.

Examples

conf <- config_cs(iterations = 100, population_size = 50, iterations_same_cost = NULL,
absolute_tol = NULL, discovery_rate = 0.25, step_size = 1.0)

Configuration object for the Genetic Algorithm

Description

Create a configuration object for the Genetic Algorithm (GA). At minimum the number of iterations (parameter iterations) and the number of chromosomes (parameter population_size) have to be provided.

Usage

config_ga(
  iterations,
  population_size,
  iterations_same_cost = NULL,
  absolute_tol = NULL,
  keep_fraction = 0.4,
  mutation_rate = 0.1
)

Arguments

Value

config_ga returns an object of class GAConfig.

References

Holland JH (1992). Adaptation in Natural and Artificial Systems: An Introductory Analysis with Applications to Biology, Control and Artificial Intelligence. MIT Press, Cambridge, MA, USA. ISBN 0262082136.

Examples

conf <- config_ga(iterations = 100, population_size = 50, iterations_same_cost = NULL,
absolute_tol = NULL,keep_fraction = 0.4, mutation_rate = 0.1)

Configuration object for the Gravitational Search Algorithm

Description

Create a configuration object for the Gravitational Search Algorithm (GSA). At minimum the number of iterations (parameter iterations) and the number of planets (parameter population_size) have to be provided.

Usage

config_gsa(
  iterations,
  population_size,
  iterations_same_cost = NULL,
  absolute_tol = NULL,
  grav = 1000,
  grav_evolution = 20
)

Arguments

Value

config_gsa returns an object of class GSAConfig.

References

Rashedi E, Nezamabadi-pour H, Saryazdi S (2009). “GSA: A Gravitational Search Algorithm.” Information Sciences, 179(13), 2232–2248. doi:10.1016/j.ins.2009.03.004.

Examples

conf <- config_gsa(iterations = 100, population_size = 50, iterations_same_cost = NULL,
absolute_tol = NULL, grav = 1000, grav_evolution = 20.)

Configuration object for the Grey Wolf Optimizer Algorithm

Description

Create a configuration object for the Grey Wolf Optimizer Algorithm (GWO). At minimum the number of iterations (parameter iterations) and the number of wolves (parameter population_size) have to be provided.

Usage

config_gwo(
  iterations,
  population_size,
  iterations_same_cost = NULL,
  absolute_tol = NULL
)

Arguments

Value

config_gwo returns an object of class GWOConfig.

References

Mirjalili S, Mirjalili SM, Lewis A (2014). “Grey Wolf Optimizer.” Advances in Engineering Software, 69, 46–61. doi:10.1016/j.advengsoft.2013.12.007.

Examples

conf <- config_gwo(iterations = 100, population_size = 50, iterations_same_cost = NULL,
absolute_tol = NULL)

Configuration object for the Harmony Search Algorithm

Description

Create a configuration object for the Harmony Search Algorithm (HS). At minimum the number of iterations (parameter iterations) and the number of solutions in the harmony memory (parameter population_size) have to be provided.

Usage

config_hs(
  iterations,
  population_size,
  iterations_same_cost = NULL,
  absolute_tol = NULL,
  considering_rate = 0.5,
  adjusting_rate = 0.5,
  distance_bandwidth = 0.1
)

Arguments

Value

config_hs returns an object of class HSConfig.

References

Lee KS, Geem ZW (2004). “A new structural optimization method based on the harmony search algorithm.” Computers & Structures, 82(9-10), 781–798. doi:10.1016/j.compstruc.2004.01.002.

Examples

conf <- config_hs(iterations = 100, population_size = 50, iterations_same_cost = NULL,
absolute_tol = NULL, considering_rate = 0.5, adjusting_rate = 0.5,
distance_bandwidth = 0.1)

Configuration object for the Improved Harmony Search Algorithm

Description

Create a configuration object for the Improved Harmony Search Algorithm (IHS). At minimum the number of iterations (parameter iterations) and the number of solutions in the harmony memory (parameter population_size) have to be provided.

Usage

config_ihs(
  iterations,
  population_size,
  iterations_same_cost = NULL,
  absolute_tol = NULL,
  considering_rate = 0.5,
  min_adjusting_rate = 0.3,
  max_adjusting_rate = 0.99,
  min_distance_bandwidth = 1e-04,
  max_distance_bandwidth = 1
)

Arguments

Value

config_ihs returns an object of class IHSConfig.

References

Mahdavi M, Fesanghary M, Damangir E (2007). “An improved harmony search algorithm for solving optimization problems.” Applied Mathematics and Computation, 188(2), 1567–1579. doi:10.1016/j.amc.2006.11.033.

Examples

conf <- config_ihs(iterations = 100, population_size = 50, iterations_same_cost = NULL,
absolute_tol = NULL,considering_rate = 0.5, min_adjusting_rate = 0.3,
max_adjusting_rate = 0.99, min_distance_bandwidth = 0.0001, max_distance_bandwidth = 1)

Configuration object for the Moth-flame Optimization Algorithm

Description

Create a configuration object for the Moth-flame Optimization Algorithm (MFO). At minimum the number of iterations (parameter iterations) and the number of moths (parameter population_size) have to be provided.

Usage

config_mfo(
  iterations,
  population_size,
  iterations_same_cost = NULL,
  absolute_tol = NULL
)

Arguments

Value

config_mfo returns an object of class MFOConfig.

References

Mirjalili S (2015). “Moth-flame optimization algorithm: A novel nature-inspired heuristic paradigm.” Knowledge-Based Systems, 89, 228–249. doi:10.1016/j.knosys.2015.07.006.

Examples

conf <- config_mfo(iterations = 100, population_size = 50, iterations_same_cost = NULL,
absolute_tol = NULL)

Configuration object for the Particle Swarm Algorithm

Description

Create a configuration object for the Particle Swarm Algorithm (PS). At minimum the number of iterations (parameter iterations) and the number of particles (parameter population_size) have to be provided.

Usage

config_ps(
  iterations,
  population_size,
  iterations_same_cost = NULL,
  absolute_tol = NULL,
  alpha_vel = 0.5,
  alpha_evolution = 1,
  cognitive = 2,
  social = 2,
  inertia = 0.9
)

Arguments

Value

config_ps returns an object of class PSConfig.

References

Eberhart R, Kennedy J (1995). “A new optimizer using particle swarm theory.” In MHS'95. Proceedings of the Sixth International Symposium on Micro Machine and Human Science, 39–43. Ieee.

Examples

conf <- config_ps(iterations = 100, population_size = 50, iterations_same_cost = NULL,
absolute_tol = NULL,alpha_vel = 0.5, alpha_evolution = 1.0, cognitive = 2.0,
social = 2.0, inertia = 0.9)

Configuration object for the Simulated Annealing Algorithm

Description

Create a configuration object for the Simulated Annealing algorithm (SA). At minimum the number of iterations (parameter iterations) and the number of particles (parameter population_size) have to be provided.

Usage

config_sa(
  iterations,
  population_size,
  iterations_same_cost = NULL,
  absolute_tol = NULL,
  T0 = 50,
  Ns = 3,
  Nt = 3,
  c_step = 2,
  Rt = 0.85,
  Wmin = 0.25,
  Wmax = 1.25
)

Arguments

Value

config_sa returns an object of class SAConfig.

References

Kirkpatrick S, Gelatt CD, Vecchi MP (1983). “Optimization by Simulated Annealing.” Science, 220(4598), 671–680. doi:10.1126/science.220.4598.671.

Examples

conf <- config_sa(iterations = 100, population_size = 50, iterations_same_cost = NULL,
absolute_tol = NULL, T0 = 50., Ns = 3., Nt = 3., c_step = 2., Rt = 0.85, Wmin = 0.25,
 Wmax = 1.25)

Configuration object for the Whale Optimization Algorithm

Description

Create a configuration object for the Whale Optimization Algorithm (WOA). At minimum the number of iterations (parameter iterations) and the number of whales (parameter population_size) have to be provided.

Usage

config_woa(
  iterations,
  population_size,
  iterations_same_cost = NULL,
  absolute_tol = NULL
)

Arguments

Value

config_woa returns an object of class WOAConfig.

References

Mirjalili S, Lewis A (2016). “The Whale Optimization Algorithm.” Advances in Engineering Software, 95, 51-67. ISSN 0965-9978, doi:10.1016/j.advengsoft.2016.01.008.

Examples

conf <- config_woa(iterations = 100, population_size = 50, iterations_same_cost = NULL,
absolute_tol = NULL)

Constrained function for minimization

Description

Create a constrained function for minimization.

Usage

constrained_function(func, ...)

Arguments

Value

constrained_function returns an object of class ConstrainedFunction.

Constraint for minimization

Description

Create a constraint function for constrained optimization. Only inequality constraints are supported.

Usage

constraint(func, inequality)

Arguments

Value

constraint returns an object of class Constraint.

Examples

g1 <- function(x) 0.0193*x[3] - (x[1]*0.0625)
c1 <- constraint(g1, "<=")

Freudenstein Roth Function

Description

Implementation of 2-dimensional Freudenstein Roth function.

Usage

freudenstein_roth_func(x)

Arguments

Details

On an 2-dimensional domain it is defined by

and is usually evaluated on \(x_{i} \in [ -10, 10 ]\), for all \(i=1,2\). The function has one global minimum at \(f(\vec{x}) = 0\) for \(\vec{x} = [ 5, 4 ]\).

Value

The value of the function.

References

Rao S (2019). Engineering optimization : theory and practice. John Wiley and Sons, Ltd, Hoboken, NJ, USA. ISBN 978-1-119-45479-3.

Get population positions

Description

Return a data.frame with the position of all individuals in the population at the specified iteration, from an object of class OptimizationResults produced with the option save_pop_history set to TRUE (see MinimizerOpts).

Usage

get_population(minimizer_result, iteration)

Arguments

Value

An object of class data.frame.

Griewank Function

Description

Implementation of n-dimensional Griewank function.

Usage

griewank_func(x)

Arguments

Details

On an n-dimensional domain it is defined by

and is usually evaluated on \(x_{i} \in [ -600, 600 ]\), for all \(i=1,...,n\). The function has global minima at \(f(\vec{x}) = 0\) for \(x_{i}=0\) for all \(i=1,...,n\).

Value

The value of the function.

References

Griewank AO (1981). “Generalized descent for global optimization.” Journal of optimization theory and applications, 34(1), 11–39.

Return the list of algorithms in EmiR

Description

Return a data.frame with the ID, description and configuration function name of all the algorithms implemented in EmiR.

Usage

list_of_algorithms()

Value

An object of class data.frame.

Return the list of pre-defined functions in EmiR

Description

Return a data.frame with function name, full name, and minimum and maximum number of parameters accepted for all the pre-defined functions in EmiR.

Usage

list_of_functions()

Value

An object of class data.frame.

Miele Cantrell Function

Description

Implementation of 4-dimensional Miele Cantrell Function.

Usage

miele_cantrell_func(x)

Arguments

Details

On an 4-dimensional domain it is defined by

and is usually evaluated on \(x_{i} \in [ -2, 2 ]\), for all \(i=1,...,4\). The function has one global minimum at \(f(\vec{x}) = 0\) for \(\vec{x} = [ 0, 1, 1, 1 ]\).

Value

The value of the function.

References

Cragg EE, Levy AV (1969). “Study on a supermemory gradient method for the minimization of functions.” Journal of Optimization Theory and Applications, 4(3), 191–205.

Minimize an Objective Function

Description

Minimize (or maximize) an objective function, possibly subjected to inequality constraints, using any of the algorithms available in EmiR.

Usage

minimize(algorithm_id, obj_func, parameters, config, constraints = NULL, ...)

Arguments

Value

minimize returns an object of class OptimizationResults (see OptimizationResults).

Examples

## Not run: 
 results <- minimize(algorithm_id = "BAT", obj_func = ob, config = conf,
 parameters = list(p1,p2, p3, p4), constraints = list(c1,c2,c3),
    save_pop_history = TRUE, constrained_method = "BARRIER",
    constr_init_pop = TRUE, oob_solutions = "RBC", seed = 1)

## End(Not run)

Parameter for minimization

Description

Create a parameter the objective function is minimized with respect to.

Usage

parameter(name, min_val, max_val, integer = FALSE)

Arguments

Value

parameter returns an object of class Parameter.

Examples

p1 <- parameter("x1", 18, 32, integer = TRUE)

Set of parameters for minimization

Description

Create the set of parameters the objective function is minimized with respect to. A \(2 \times n\) matrix or a \(3 \times n\) matrix, where the first row is for the lower limits, the second one is for the upper limits, and the (optional) third one is to specify if a parameter is constrained to be integer. In case the third row is not provided, all the parameters are treated as continuous. The name of each of the \(n\) parameters is automatically generated and it is of the form \(xi\), where \(i=1,...,n\).

Usage

parameters(values)

Arguments

Value

parameters returns a list of objects of class Parameter.

Plot minimization history

Description

Plot the minimization history as a function of the number of iterations.

Usage

plot_history(minimizer_result, ...)

Arguments

Plot the population position

Description

Plot the position of all individuals in the population, at a given iteration, for 1D and 2D functions. The plot can be produced only if save_pop_history is TRUE in the options of the minimizer (see MinimizerOpts).

Usage

plot_population(minimizer_result, iteration, n_points = 100)

Arguments

Rastrigin Function

Description

Implementation of n-dimensional Rastrigin function.

Usage

rastrigin_func(x)

Arguments

Details

On an n-dimensional domain it is defined by:

and is usually evaluated on \(x_{i} \in [ -5.12, 5.12 ]\), for all \(i=1,...,n\). The function has one global minimum at \(f(\vec{x})=0\) for \(x_{i}=0\) for all \(i=1,...,n\).

Value

The value of the function.

References

Rastrigin LA (1974). “Systems of extremal control.” Nauka.

Rosenbrock Function

Description

Implementation of n-dimensional Rosenbrock function, with \(n \geq 2\).

Usage

rosenbrock_func(x)

Arguments

Details

On an n-dimensional domain it is defined by

and is usually evaluated on \(x_{i} \in [ -5, 10 ]\), for all \(i=1,...,n\). The function has one global minimum at \(f(\vec{x})=0\) for \(x_{i}=1\) for all \(i=1,...,n\).

Value

The value of the function.

References

Rosenbrock HH (1960). “An Automatic Method for Finding the Greatest or Least Value of a Function.” The Computer Journal, 3(3), 175–184. doi:10.1093/comjnl/3.3.175.

Schwefel Function

Description

Implementation of n-dimensional Schwefel function.

Usage

schwefel_func(x)

Arguments

Details

On an n-dimensional domain it is defined by

and is usually evaluated on \(x_{i} \in [ -500, 500 ]\), for all \(i=1,...,n\). The function has one global minimum at \(f(\vec{x}) = -418.9829n\) for \(x_{i}=420.9687\) for all \(i=1,...,n\).

Value

The value of the function.

References

Schwefel H (1981). Numerical optimization of computer models. John Wiley & Sons, Inc.

Styblinski-Tang Function

Description

Implementation of n-dimensional Styblinski-Tang function.

Usage

styblinski_tang_func(x)

Arguments

Details

On an n-dimensional domain it is defined by

and is usually evaluated on \(x_{i} \in [ -5, 5 ]\), for all \(i=1,...,n\). The function has one global minimum at \(f(\vec{x}) = -39.16599n\) for \(x_{i}=-2.903534\) for all \(i=1,...,n\).

Value

The value of the function.

References

Styblinski MA, Tang T (1990). “Experiments in nonconvex optimization: Stochastic approximation with function smoothing and simulated annealing.” Neural Networks, 3(4), 467–483. doi:10.1016/0893-6080(90)90029-k.