geneticalgorithm2 6.5.0

Supported highly optimized and flexible genetic algorithm package for python

This is the supported advanced fork of non-supported package geneticalgorithm of Ryan (Mohammad) Solgi

• About
• Installation
• Updates information
  • 6.5.0 minor update (refactoring)
  • 6.4.1 patch (bug fix)
  • 6.4.0 minor update (refactoring)
  • 6.3.0 minor update (refactoring)
• Working process
  • Methods and Properties of model:
  • Constructor parameters
  • Genetic algorithm's parameters
• Examples for begginer
  • A minimal example
  • The simple example with integer variables
  • The simple example with Boolean variables
  • The simple example with mixed variables
  • Optimization problems with constraints
  • Select fixed count of objects from set
• U should know these features
  • Available crossovers
  • Function timeout
  • Standard GA vs. Elitist GA
  • Standard crossover vs. stud EA crossover
  • Creating better start population
    • Select best N of kN
    • Do local optimization
    • Optimization with oppositions
  • Revolutions
  • Duplicates removing
  • Cache
  • Middle callbacks
  • How to compare efficiency of several versions of GA optimization
  • Hints on how to adjust genetic algorithm's parameters (from geneticalgorithm package)
• Examples pretty collection
  • Optimization test functions
    • Sphere
    • Ackley
    • AckleyTest
    • Rosenbrock
    • Fletcher
    • Griewank
    • Penalty2
    • Quartic
    • Rastrigin
    • SchwefelDouble
    • SchwefelMax
    • SchwefelAbs
    • SchwefelSin
    • Stairs
    • Abs
    • Michalewicz
    • Scheffer
    • Eggholder
    • Weierstrass
  • Using GA in reinforcement learning
  • Using GA with image reconstruction by polygons
• Popular questions
  • How to disable autoplot?
  • How to plot population scores?
  • How to specify evaluated function for all population?
  • What about parallelism?
  • How to initialize start population? How to continue optimization with new run?

About

geneticalgorithm2 is very flexible Python library distributed on PyPI for implementing standard and elitist genetic-algorithm (GA).

This package solves continuous, combinatorial and mixed optimization problems with continuous, discrete, and mixed variables. It provides an easy optimized implementation of genetic-algorithm (GA) in Python.

Installation

Use the package manager pip to install geneticalgorithm2 in Python.

pip install geneticalgorithm2

Updates information

6.5.0 minor update (refactoring)

• another form of data object using with middle callbacks (MiddleCallbackData dataclass instead of dictionary)
• type hints for callbacks module

6.4.1 patch (bug fix)

• fix bug setting attribute to algorithm parameters (in middle callbacks)

6.4.0 minor update (refactoring)

• new valid forms for start_generation; now it's valid to use

  • None
  • str path to saved generation
  • dictionary with structure {'variables': variables/None, 'scores': scores/None}
  • Generation object: Generation(variables = variables, scores = scores)
  • np.ndarray with shape (samples, dim) for only population or (samples, dim+1) for concatenated population and score (scores is the last matrix column)
  • tuple(np.ndarray/None, np.ndarray/None) for variables and scores

  here variables is 2D numpy array with shape (samples, dim), scores is 1D numpy array with scores (function values) for each sample; here and here u can see examples of using these valid forms

6.3.0 minor update (refactoring)

• type hints for entire part of functions
• new valid forms for function parameters (now u don't need to use numpy arrays everywhere)
• AlgorithmParams class for base GA algorithm parameters (instead of dictionary)
• Generation class for saving/loading/returning generation (instead of dictionary)

All that classes are collected in file. To maintain backward compatibility, AlgorithmParams and Generation classes have dictionary-like interface for getting fields: u can use object.field or object['field'] notations.

Working process

Firstly, u should import needed packages. All available imports are:

import numpy as np

from geneticalgorithm2 import geneticalgorithm2 as ga # for creating and running optimization model

from geneticalgorithm2 import Generation, AlgorithmParams, MiddleCallbackData # classes for comfortable parameters setting and getting

from geneticalgorithm2 import Crossover, Mutations, Selection # classes for specific mutation and crossover behavior

from geneticalgorithm2 import Population_initializer # for creating better start population

from geneticalgorithm2 import np_lru_cache # for cache function (if u want)

from geneticalgorithm2 import plot_pop_scores # for plotting population scores, if u want

from geneticalgorithm2 import Callbacks # simple callbacks

from geneticalgorithm2 import Actions, ActionConditions, MiddleCallbacks # middle callbacks

Next step: define minimized function like

def function(X): # X as numpy array
    return np.sum(X**2) + X.mean() + X.min() + X[0]*X[2] # some float result

If u want to find maximum, use this idea:

f_tmp = lambda arr: -target(arr)

#
# ... find global min
#

tagret_result = -global_min

Okay, also u should create the bounds for each variable (if exist) like here:

var_bound = np.array([[0,10]]*3) # 2D numpy array with shape (dim, 2)

# also u can use Sequence of Tuples (from version 6.3.0)
var_bound = [
    (0, 10),
    (0, 10),
    (0, 10)
]

After that create a geneticalgorithm2 object:

model = ga(function, dimension = 3, 
                variable_type='real', 
                 variable_boundaries = var_bound,
                 variable_type_mixed = None, 
                 function_timeout = 10,
                 algorithm_parameters={'max_num_iteration': None,
                                       'population_size':100,
                                       'mutation_probability':0.1,
                                       'elit_ratio': 0.01,
                                       'crossover_probability': 0.5,
                                       'parents_portion': 0.3,
                                       'crossover_type':'uniform',
                                       'mutation_type': 'uniform_by_center',
                                       'selection_type': 'roulette',
                                       'max_iteration_without_improv':None}
            )

# from version 6.3.0 it is equal to

model = ga(function, dimension = 3, 
                variable_type='real', 
                 variable_boundaries = var_bound,
                 variable_type_mixed = None, 
                 function_timeout = 10,
                 algorithm_parameters=AlgorithmParams(
                     max_num_iteration = None,
                     population_size = 100,
                     mutation_probability = 0.1,
                     elit_ratio = 0.01,
                     crossover_probability = 0.5,
                     parents_portion = 0.3,
                     crossover_type = 'uniform',
                     mutation_type = 'uniform_by_center',
                     selection_type = 'roulette',
                     max_iteration_without_improv = None
                     )
            )

# or            
model = ga(function, dimension = 3, 
                variable_type='real', 
                 variable_boundaries = var_bound,
                 variable_type_mixed = None, 
                 function_timeout = 10,
                 algorithm_parameters=AlgorithmParams()
            )           

Run the search method:

model.run(
    no_plot = False, 
    disable_progress_bar = False,
    disable_printing = False,

    set_function = None, 
    apply_function_to_parents = False, 
    start_generation = {'variables':None, 'scores': None},
    studEA = False,
    mutation_indexes = None,

    init_creator = None,
    init_oppositors = None,
    duplicates_oppositor = None,
    remove_duplicates_generation_step = None,
    revolution_oppositor = None,
    revolution_after_stagnation_step = None,
    revolution_part = 0.3,
    
    population_initializer = Population_initializer(select_best_of = 1, local_optimization_step = 'never', local_optimizer = None),
    
    stop_when_reached = None,
    callbacks = [],
    middle_callbacks = [],
    time_limit_secs = None, 
    save_last_generation_as = None,
    seed = None
    )

Your best solution is computed!

Methods and Properties of model:

run(): implements the genetic algorithm (GA) with parameters:

• param no_plot (bool) - do not plot results using matplotlib by default

• param disable_progress_bar (bool) - do not show progress bar (also it can be faster by 10-20 seconds)

• param disable_printing (bool) - don't print any text (except progress bar)

• param set_function (Optional[Callable[[np.ndarray], np.ndarray]]): 2D-array -> 1D-array function, which applies to matrix of population (size (samples, dimension)) to estimate their values

• param apply_function_to_parents (bool) - apply function to parents from previous generation (if it's needed, it can be needed at working with games agents)

• param start_generation (Union[str, Dict[str, np.ndarray], Generation, np.ndarray, Tuple[Optional[np.ndarray], Optional[np.ndarray]]]) - Generation object or a dictionary with structure {'variables':2D-array of samples, 'scores': function values on samples} or path to .npz file (str) with saved generation (see example) or np.ndarray (with shape (samples, dim) or (samples, dim+1)) or tuple of np.ndarrays/None. If 'scores' value is None the scores will be compute. See this

• param studEA (bool) - using stud EA strategy (crossover with best object always). Default is false. Take a look

• param mutation_indexes (Optional[Union[Sequence[int], Set[int]]]) - indexes of dimensions where mutation can be performed (all dimensions by default). Example

• param init_creator: (Optional[Callable[[], np.ndarray]]), the function creates population samples. By default -- random uniform for real variables and random uniform for int. Example

• param init_oppositors: (Optional[Sequence[Callable[[np.ndarray], np.ndarray]]]) list, the list of oppositors creates oppositions for base population. No by default. Example

• param duplicates_oppositor: Optional[Callable[[np.ndarray], np.ndarray]], oppositor for applying after duplicates removing. By default -- using just random initializer from creator. Example

• param remove_duplicates_generation_step: None/int, step for removing duplicates (have a sense with discrete tasks). No by default. Example

• param revolution_oppositor = Optional[Callable[[np.ndarray], np.ndarray]], oppositor for revolution time. No by default. Example

• param revolution_after_stagnation_step = None/int, create revolution after this generations of stagnation. No by default. Example

• param revolution_part (float): the part of generation to being oppose. By default is 0.3. Example

• param population_initializer (Tuple[int, Callable[[np.ndarray, np.ndarray], Tuple[np.ndarray, np.ndarray]]]) - object for actions at population initialization step to create better start population. Take a look

• param stop_when_reached (Optional[float]) - stop searching after reaching this value (it can be potential minimum or something else)

• param callbacks (Optional[Sequence[Callable[[int, List[float], np.ndarray, np.ndarray], None]]]) - list of callback functions with structure:

  def callback(generation_number, report_list, last_population_as_2D_array, last_population_scores_as_1D_array):
      #
      # do some action
      #
  

  See example of using callbacks. There are several callbacks in Callbacks class, such as:

  • Callbacks.SavePopulation(folder, save_gen_step = 50, file_prefix = 'population')
  • Callbacks.PlotOptimizationProcess(folder, save_gen_step = 50, show = False, main_color = 'green', file_prefix = 'report')

• param middle_callbacks (Sequence) - list of functions made MiddleCallbacks class (please, have a look at this)

• param time_limit_secs (Optional[float]) - limit time of working (in seconds). If None, there is no time limit (limit only for count of generation and so on). See little example of using. Also there is simple conversion function for conversion some time in seconds:

  from truefalsepython import time_to_seconds
  
  total_seconds = time_to_seconds(
      days = 2, # 2 days
      hours = 13, # plus 13 hours
      minutes = 7, # plus 7 minutes
      seconds = 44 # plus 44 seconds
  )
  

• param save_last_generation_as (Optional[str]) - path to .npz file for saving last_generation as numpy dictionary like {'population': 2D-array, 'scores': 1D-array}, None if doesn't need to save in file; take a look

• param seed (Optional[int]) - random seed (None is doesn't matter)

It would be more logical to use params like studEA as an algorithm param, but run()-way can be more comfortable for real using.

output:

• output_dict: is a dictionary including the best set of variables found and the value of the given function associated to it. Structure:

output_dict = {
            'variable': best_variable, // as 1D-array
            'function': best_function_value, // a number
            'last_generation': Generation object
}

// Generation object is equal to dictionary:
// {
//     // values are sorted by scores
//     'variables':last_generation_variables, // 2D-array samples*dim
//     'scores': last_generation_function_values // 1D-array of scores
// }

• report: is a record of the progress of the algorithm over iterations. There are also report_average and report_min fields which are the average and min generation values by each generation

Constructor parameters

• param function (Callable[[np.ndarray], float]) - the given objective function to be minimized
  NOTE: This implementation minimizes the given objective function. (For maximization multiply function by a negative sign: the absolute value of the output would be the actual objective function)

• param dimension (int) - the number of decision variables

• param variable_type (str) - 'bool' if all variables are Boolean; 'int' if all variables are integer; and 'real' if all variables are real value or continuous (for mixed type see @param variable_type_mixed).

• param variable_boundaries (Optional[Union[np.ndarray, Sequence[Tuple[float, float]]]]) - Default None; leave it None if variable_type is 'bool'; otherwise provide an sequence of tuples of length two as boundaries for each variable; the length of the array must be equal dimension. For example, np.array([0,100],[0,200]) determines lower boundary 0 and upper boundary 100 for first and upper boundary 200 for second variable where dimension is 2.

• param variable_type_mixed (Optional[Sequence[str]]) - Default None; leave it None if all variables have the same type; otherwise this can be used to specify the type of each variable separately. For example if the first variable is integer but the second one is real the input is: ['int', 'real']. NOTE: it does not accept 'bool'. If variable type is Boolean use 'int' and provide a boundary as [0,1] in variable_boundaries. Also if variable_type_mixed is applied, variable_boundaries has to be defined.

• param function_timeout (float) - if the given function does not provide output before function_timeout (unit is seconds) the algorithm raise error. For example, when there is an infinite loop in the given function.

• param algorithm_parameters (Union[AlgorithmParams, Dict[str, Any]]). Dictionary or AlgorithmParams object with fields:

  • @ max_num_iteration (int/None) - stoping criteria of the genetic algorithm (GA)
  • @ population_size (int > 0)
  • @ mutation_probability (float in [0,1])
  • @ elit_ration (float in [0,1]) - part of elit objects in population; if > 0, there always will be 1 elit object at least
  • @ crossover_probability (float in [0,1])
  • @ parents_portion (float in [0,1]) - part of parents from previous population to save in next population (including elit_ration)
  • @ crossover_type (Union[str, Callable[[np.ndarray, np.ndarray], Tuple[np.ndarray, np.ndarray]]]) - Default is uniform.
  • @ mutation_type (Union[str, Callable[[float, float, float], float]]) - Default is uniform_by_center
  • @ selection_type (Union[str, Callable[[np.ndarray, int], np.ndarray]]) - Default is roulette
  • @ max_iteration_without_improv (int/None) - maximum number of successive iterations without improvement. If None it is ineffective

Genetic algorithm's parameters

The parameters of GA is defined as a dictionary or AlgorithmParams object:

AlgorithmParams(
                max_num_iteration = None,
                population_size = 100,
                mutation_probability = 0.1,
                elit_ratio = 0.01,
                crossover_probability = 0.5,
                parents_portion = 0.3,
                crossover_type = 'uniform',
                mutation_type = 'uniform_by_center',
                selection_type = 'roulette',
                max_iteration_without_improv = None
            )


# old style with dictionary
algorithm_param = {
                   'max_num_iteration': None,
                   'population_size':100,
                   'mutation_probability':0.1,
                   'elit_ratio': 0.01,
                   'crossover_probability': 0.5,
                   'parents_portion': 0.3,
                   'crossover_type':'uniform',
                   'mutation_type': 'uniform_by_center',
                   'selection_type': 'roulette',
                   'max_iteration_without_improv':None
                   }

The above dictionary refers to the default values that has been set already. One may simply copy this code from here and change the values and use the modified dictionary as the argument of geneticalgorithm2.

Another way of accessing this params object is using the command below:

import numpy as np
from geneticalgorithm2 import geneticalgorithm2 as ga

def f(X):
    return np.sum(X)
    
model=ga(function=f,dimension=3,variable_type='bool')

print(model.param)

An example of setting a new set of parameters for genetic algorithm and running geneticalgorithm2 for our first simple example again:

import numpy as np
from geneticalgorithm2 import geneticalgorithm2 as ga

def f(X):
    return np.sum(X)
    
    
varbound=[(0,10)]*3

algorithm_param = {'max_num_iteration': 3000,
                   'population_size':100,
                   'mutation_probability':0.1,
                   'elit_ratio': 0.01,
                   'crossover_probability': 0.5,
                   'parents_portion': 0.3,
                   'crossover_type':'uniform',
                   'mutation_type': 'uniform_by_center',
                   'selection_type': 'roulette',
                   'max_iteration_without_improv':None}

model=ga(function=f,
            dimension=3,
            variable_type='real',
            variable_boundaries=varbound,
            algorithm_parameters=algorithm_param)

model.run()

Important. U may use the small dictionary with only important parameters; other parameters will be default. It means the dictionary

algorithm_param = {'max_num_iteration': 150,
                   'population_size':1000}

is equal to:

algorithm_param = {'max_num_iteration': 150,
                   'population_size':1000,
                   'mutation_probability':0.1,
                   'elit_ratio': 0.01,
                   'crossover_probability': 0.5,
                   'parents_portion': 0.3,
                   'crossover_type':'uniform',
                   'mutation_type': 'uniform_by_center',
                   'selection_type': 'roulette',
                   'max_iteration_without_improv':None}

But it is better to use AlgorithmParams object instead of dictionaries.

Parameters of algorithm:

• max_num_iteration: The termination criterion of GA. If this parameter's value is None the algorithm sets maximum number of iterations automatically as a function of the dimension, boundaries, and population size. The user may enter any number of iterations that they want. It is highly recommended that the user themselves determines the max_num_iterations and not to use None. Notice that max_num_iteration has been changed to 3000 (it was already None).

• population_size: determines the number of trial solutions in each iteration. The default value is 100.

• mutation_probability: determines the chance of each gene in each individual solution to be replaced by a random value. The default is 0.1 (i.e. 10 percent).

• elit_ration: determines the number of elites in the population. The default value is 0.01 (i.e. 1 percent). For example when population size is 100 and elit_ratio is 0.01 then there is one elite in the population. If this parameter is set to be zero then geneticalgorithm2 implements a standard genetic algorithm instead of elitist GA. See example

• crossover_probability: determines the chance of an existed solution to pass its genome (aka characteristics) to new trial solutions (aka offspring); the default value is 0.5 (i.e. 50 percent)

• parents_portion: the portion of population filled by the members of the previous generation (aka parents); default is 0.3 (i.e. 30 percent of population)

• max_iteration_without_improv: if the algorithms does not improve the objective function over the number of successive iterations determined by this parameter, then GA stops and report the best found solution before the max_num_iterations to be met. The default value is None.

• crossover_type: there are several options including one_point, two_point, uniform, segment, shuffle crossover functions; default is uniform crossover. U also can use crossover functions from Crossover class:

  • Crossover.one_point()
  • Crossover.two_point()
  • Crossover.uniform()
  • Crossover.uniform_window(window = 7)
  • Crossover.shuffle()
  • Crossover.segment()
  • Crossover.mixed(alpha = 0.5) only for real variables
  • Crossover.arithmetic() only for real variables

  Have a look at crossovers' logic

  If u want, write your own crossover function using syntax:

  def my_crossover(parent_a, parent_b):
      # some code
      return child_1, child_2
  

• mutation_type: there are several options (only for real) including uniform_by_x, uniform_by_center, gauss_by_x, gauss_by_center; default is uniform_by_center. U also can use mutation functions from Mutations class:

  • Mutations.gauss_by_center(sd = 0.2)
  • Mutations.gauss_by_x(sd = 0.1)
  • Mutations.uniform_by_center()
  • Mutations.uniform_by_x()

  (If u want) write your mutation function using syntax:

  def my_mutation(current_value, left_border, right_border):
      # some code
      return new_value 
  

• selection_type: there are several options (only for real) including fully_random, roulette, stochastic, sigma_scaling, ranking, linear_ranking, tournament; default is roulette. U also can use selection functions from Selection class:

  • Selection.fully_random()
  • Selection.roulette()
  • Selection.stochastic()
  • Selection.sigma_scaling(epsilon = 0.05)
  • Selection.ranking()
  • Selection.linear_ranking(selection_pressure = 1.5)
  • Selection.tournament(tau = 2)

  If u want, write your selection function using syntax:

  def my_mutation(sorted_scores, parents_count):
      # some code
      return array_of_parents_indexes 
  

Examples for begginer

A minimal example

Assume we want to find a set of X = (x1,x2,x3) that minimizes function f(X)=x1+x2+x3 where X can be any real number in [0,10].

This is a trivial problem and we already know that the answer is X=(0,0,0) where f(X)=0.
We just use this simple example to see how to implement geneticalgorithm2. First we import geneticalgorithm2 and numpy. Next, we define function f which we want to minimize and the boundaries of the decision variables. Then simply geneticalgorithm2 is called to solve the defined optimization problem as follows:

import numpy as np
from geneticalgorithm2 import geneticalgorithm2 as ga

def f(X):
    return np.sum(X)
    
    
varbound = np.array([[0,10]]*3)

model = ga(function=f, dimension=3, variable_type='real', variable_boundaries=varbound)

model.run()

Notice that we define the function f so that its output is the objective function we want to minimize where the input is the set of X (decision variables). The boundaries for variables must be defined as a numpy array and for each variable we need a separate boundary. Here I have three variables and all of them have the same boundaries (For the case the boundaries are different see the example with mixed variables).

geneticalgorithm2 has some arguments:

1. Obviously the first argument is the function f we already defined (for more details about the argument and output see Function).
2. Our problem has three variables so we set dimension equal three.
3. Variables are real (continuous) so we use string 'real' to notify the type of variables (geneticalgorithm2 accepts other types including Boolean, Integers and Mixed; see other examples).
4. Finally, we input varbound which includes the boundaries of the variables. Note that the length of variable_boundaries must be equal to dimension.

If you run the code, you should see a progress bar that shows the progress of the genetic algorithm (GA) and then the solution, objective function value and the convergence curve as follows:

Also we can access to the best answer of the defined optimization problem found by GA as a dictionary and a report of the progress of the genetic algorithm. To do so we complete the code as follows:

convergence = model.report

solution = model.output_dict

output_dict is a dictionary including the best set of variables found and the value of the given function associated to it ({'variable': , 'function': , 'last_generation': }). report is a list including the convergence of the algorithm over iterations

The simple example with integer variables

Considering the problem given in the simple example above. Now assume all variables are integers. So x1, x2, x3 can be any integers in [0,10]. In this case the code is as the following:

import numpy as np
from geneticalgorithm2 import geneticalgorithm2 as ga

def f(X):
    return np.sum(X)
    
    
varbound = np.array([[0,10]]*3)

model = ga(function=f, dimension=3, variable_type='int', variable_boundaries=varbound)

model.run()

So, as it is seen the only difference is that for variable_type we use string 'int'.

The simple example with Boolean variables

Considering the problem given in the simple example above. Now assume all variables are Boolean instead of real or integer. So X can be either zero or one. Also instead of three let's have 30 variables. In this case the code is as the following:

import numpy as np
from geneticalgorithm2 import geneticalgorithm2 as ga

def f(X):
    return np.sum(X)
    

model = ga(function=f, dimension=30, variable_type='bool')

model.run()

Note for variable_type we use string 'bool' when all variables are Boolean.
Note that when variable_type equal 'bool' there is no need for variable_boundaries to be defined.

The simple example with mixed variables

Considering the problem given in the the simple example above where we want to minimize f(X)=x1+x2+x3. Now assume x1 is a real (continuous) variable in [0.5,1.5], x2 is an integer variable in [1,100], and x3 is a Boolean variable that can be either zero or one. We already know that the answer is X=(0.5,1,0) where f(X)=1.5 We implement geneticalgorithm2 as the following:

import numpy as np
from geneticalgorithm2 import geneticalgorithm2 as ga

def f(X):
    return np.sum(X)
    
varbound = np.array([[0.5,1.5],[1,100],[0,1]])
vartype = np.array(['real','int','int'])
model = ga(function=f, dimension=3, variable_type_mixed=vartype, variable_boundaries=varbound)

model.run()

Note that for mixed variables we need to define boundaries also we need to make a numpy array of variable types as above (vartype). Obviously the order of variables in both arrays must match. Also notice that in such a case for Boolean variables we use string 'int' and boundary [0,1].
Notice that we use argument variable_type_mixed to input a numpy array of variable types for functions with mixed variables.

Optimization problems with constraints

In all above examples, the optimization problem was unconstrained. Now consider that we want to minimize f(X)=x1+x2+x3 where X is a set of real variables in [0,10]. Also we have an extra constraint so that sum of x1 and x2 is equal or greater than 2. The minimum of f(X) is 2. In such a case, a trick is to define penalty function. Hence we use the code below:

import numpy as np
from geneticalgorithm2 import geneticalgorithm2 as ga

def f(X):
    pen=0
    if X[0]+X[1]<2:
        pen=500+1000*(2-X[0]-X[1])
    return np.sum(X)+pen
    
varbound=np.array([[0,10]]*3)

model=ga(function=f,dimension=3,variable_type='real',variable_boundaries=varbound)

model.run()

As seen above we add a penalty to the objective function whenever the constraint is not met.

Some hints about how to define a penalty function:

1. Usually you may use a constant greater than the maximum possible value of the objective function if the maximum is known or if we have a guess of that. Here the highest possible value of our function is 300 (i.e. if all variables were 10, f(X)=300). So I chose a constant of 500. So, if a trial solution is not in the feasible region even though its objective function may be small, the penalized objective function (fitness function) is worse than any feasible solution.
2. Use a coefficient big enough and multiply that by the amount of violation. This helps the algorithm learn how to approach feasible domain.
3. How to define penalty function usually influences the convergence rate of an evolutionary algorithm. In my book on metaheuristics and evolutionary algorithms you can learn more about that.
4. Finally after you solved the problem test the solution to see if boundaries are met. If the solution does not meet constraints, it shows that a bigger penalty is required. However, in problems where optimum is exactly on the boundary of the feasible region (or very close to the constraints) which is common in some kinds of problems, a very strict and big penalty may prevent the genetic algorithm to approach the optimal region. In such a case designing an appropriate penalty function might be more challenging. Actually what we have to do is to design a penalty function that let the algorithm searches unfeasible domain while finally converge to a feasible solution. Hence you may need more sophisticated penalty functions. But in most cases the above formulation work fairly well.

Select fixed count of objects from set

For some task u need think a lot and create good specific crossover or mutation functions. For example, take a look at this problem:

From set like X = {x1, x2, x3, ..., xn} u should select only k objects which get the best function value

U can do it using this code:

import numpy as np
from geneticalgorithm2 import geneticalgorithm2 as ga


subset_size = 20 # how many objects we can choose

objects_count = 100 # how many objects are in set

my_set = np.random.random(objects_count)*10 - 5 # set values

# minimized function
def f(X):
    return abs(np.mean(my_set[X==1]) - np.median(my_set[X==1]))

# initialize start generation and params

N = 1000 # size of population
start_generation = np.zeros((N, objects_count))
indexes = np.arange(0, objects_count, dtype = np.int8) # indexes of variables

for i in range(N):
    inds = np.random.choice(indexes, subset_size, replace = False)
    start_generation[i, inds] = 1 


def my_crossover(parent_a, parent_b):
    a_indexes = set(indexes[parent_a == 1])
    b_indexes = set(indexes[parent_b == 1])
    
    intersect = a_indexes.intersection(b_indexes) # elements in both parents
    a_only = a_indexes - intersect # elements only in 'a' parent
    b_only = b_indexes - intersect
    
    child_inds = np.array(list(a_only) + list(b_only), dtype = np.int8)
    np.random.shuffle(child_inds) # mix
    
    childs = np.zeros((2, parent_a.size))
    if intersect:
        childs[:, np.array(list(intersect))] = 1
    childs[0, child_inds[:int(child_inds.size/2)]] = 1
    childs[1, child_inds[int(child_inds.size/2):]] = 1
    
    return childs[0,:], childs[1,:]
    

model = ga(function=f, 
           dimension=objects_count, 
           variable_type='bool',
           algorithm_parameters={
                       'max_num_iteration': 500,
                       'mutation_probability': 0, # no mutation, just crossover
                       'elit_ratio': 0.05,
                       'crossover_probability': 0.5,
                       'parents_portion': 0.3,
                       'crossover_type': my_crossover,
                       'max_iteration_without_improv': 20
               }
           )

model.run(no_plot = False, start_generation={'variables': start_generation, 'scores': None})

U should know these features

Available crossovers

For two example parents (one with ones and one with zeros) next crossovers will give same children (examples):

• one_point:

0	0	0	0	0	0	0	0	1	1	1	1	1	1	1
1	1	1	1	1	1	1	1	0	0	0	0	0	0	0

• two_point:

1	1	0	0	0	0	0	0	0	0	0	0	1	1	1
0	0	1	1	1	1	1	1	1	1	1	1	0	0	0

• uniform:

1	1	1	0	1	1	0	0	0	0	0	1	0	0	0
0	0	0	1	0	0	1	1	1	1	1	0	1	1	1

• uniform_window:

1	1	1	1	1	1	1	1	1	0	0	0	1	1	1
0	0	0	0	0	0	0	0	0	1	1	1	0	0	0

• shuffle:

0	0	0	1	1	1	1	0	0	1	1	1	0	1	0
1	1	1	0	0	0	0	1	1	0	0	0	1	0	1

• segment:

0	1	1	0	0	1	0	1	0	0	1	0	0	1	1
1	0	0	1	1	0	1	0	1	1	0	1	1	0	0

• arithmetic:

0.13	0.13	0.13	0.13	0.13	0.13	0.13	0.13	0.13	0.13	0.13	0.13	0.13	0.13	0.13
0.87	0.87	0.87	0.87	0.87	0.87	0.87	0.87	0.87	0.87	0.87	0.87	0.87	0.87	0.87

• mixed:

0.63	0.84	1.1	0.73	0.67	-0.19	0.3	0.72	-0.18	0.61	0.84	1.14	1.36	-0.37	-0.19
0.51	0.58	0.43	0.42	0.55	0.49	0.57	0.48	0.46	0.56	0.56	0.54	0.44	0.51	0.4

Function timeout

geneticalgorithm2 is designed such that if the given function does not provide any output before timeout (the default value is 10 seconds), the algorithm would be terminated and raise the appropriate error. In such a case make sure the given function works correctly (i.e. there is no infinite loop in the given function). Also if the given function takes more than 10 seconds to complete the work make sure to increase function_timeout in arguments.

Standard GA vs. Elitist GA

The convergence curve of an elitist genetic algorithm is always non-increasing. So, the best ever found solution is equal to the best solution of the last iteration. However, the convergence curve of a standard genetic algorithm is different. If elit_ratio is zero geneticalgroithm2 implements a standard GA. The output of geneticalgorithm2 for standard GA is the best ever found solution not the solution of the last iteration. The difference between the convergence curve of standard GA and elitist GA is shown below:

Standard crossover vs. stud EA crossover

Stud EA is the idea of using crossover always with best object. So one of two parents is always the best object of population. It can help us in a lot of tasks!

Creating better start population

There is Population_initializer(select_best_of = 4, local_optimization_step = 'never', local_optimizer = None) object for creating better start population. It has next arguments:

• select_best_of (int) -- select 1/select_best_of best part of start population. For example, for select_best_of = 4 and population_size = N will be selected N best objects from 5N generated objects (if start_generation = None dictionary). If start_generation is not None dictionary, it will be selected best size(start_generation)/N objects

• local_optimization_step (str) -- when should we do local optimization? Available values:

  • 'never' -- don't do local optimization
  • 'before_select' -- before selection best N objects (example: do local optimization for 5N objects and select N best results)
  • 'after_select' -- do local optimization on best selected N objects

• local_optimizer (function) -- local optimization function like:

  def loc_opt(object_as_array, current_score):
      # some code
      return better_object_as_array, better_score
  

Select best N of kN

This little option can help u especially with multimodal tasks.

Do local optimization

We can apply some local optimization on start generation before starting GA search. It can be some gradient descent or hill climbing and so on. Also we can apply it before selection best objects (on entire population) or after (on best part of population) and so forth.

In next example I'm using my DiscreteHillClimbing algorithm for local optimization my discrete task:

import numpy as np
import matplotlib.pyplot as plt

from DiscreteHillClimbing import Hill_Climbing_descent

from geneticalgorithm2 import geneticalgorithm2 as ga
from geneticalgorithm2 import Population_initializer


def f(arr):
    arr2 = arr/25
    return -np.sum(arr2*np.sin(np.sqrt(np.abs(arr2))))**5 + np.sum(np.abs(arr2))**2

iterations = 100    
    
varbound = np.array([[-100, 100]]*15)

available_values = [np.arange(-100, 101)]*15


my_local_optimizer = lambda arr, score: Hill_Climbing_descent(function = f, available_predictors_values=available_values, max_function_evals=50, start_solution=arr )


model = ga(function=f, dimension=varbound.shape[0], 
           variable_type='int', 
           variable_boundaries = varbound,
           algorithm_parameters={
               'max_num_iteration': iterations,
               'population_size': 400
               })


for time in ('before_select', 'after_select', 'never'):
    model.run(no_plot = True,
                  population_initializer = Population_initializer(
                      select_best_of = 3,
                      local_optimization_step = time,
                      local_optimizer = my_local_optimizer
                      )
                  )

    plt.plot(model.report, label = f"local optimization time = '{time}'")


plt.xlabel('Generation')
plt.ylabel('Minimized function (40 simulations average)')
plt.title('Selection best N object before running GA')
plt.legend()

Optimization with oppositions

Also u can create start population with oppositions. See example of code

Revolutions

U can create revolutions in your population after some stagnation steps. It really can help u for some tasks. See example

Duplicates removing

If u remove duplicates each k generations, u can speed up the optimization process (example)

Cache

It can be useful for run-speed to use cache with some discrete tasks. For this u can import np_lru_cache decorator and use it like here:

import np_lru_cache

@np_lru_cache(maxsize = some_size)
def minimized_func(arr):
    # code
    return result

#
# run
#    algorithm
#


# don't forget to clear cache
minimized_func.cache_clear()

Middle callbacks

There is an amazing way to control optimization process using MiddleCallbacks class. Just learn next logic:

1. u can use several MiddleCallbacks callbacks as list at middle_callbacks parameter in run() method
2. each middle callback is the pair of action and condition functions
3. condition(data) (Callable[[MiddleCallbackData], bool]) function gets data object (dataclass MiddleCallbackData from version 6.5.0) about primary model parameters and makes logical decision about applying action function
4. action(data) (Callable[[MiddleCallbackData],MiddleCallbackData]) function modifies data objects as u need -- and model will be modified by new data
5. data object is the structure with several parameters u can modify:

    data = MiddleCallbackData(
        last_generation=Generation.from_pop_matrix(pop),
        current_generation=t,
        report_list=self.report,
   
        mutation_prob=self.prob_mut,
        crossover_prob=self.prob_cross,
        mutation=self.real_mutation,
        crossover=self.crossover,
        selection=self.selection,
   
        current_stagnation=counter,
        max_stagnation=self.mniwi,
   
        parents_portion=self.param.parents_portion,
        elit_ratio=self.param.elit_ratio,
   
        set_function=self.set_function
    )
   

   So, the action function gets data objects and returns data object.

It's very simple to create your own action and condition functions. But there are several popular functions contained in Actions and ActionConditions classes:

• actions:
  • Stop() -- just stop optimization process
  • ReduceMutationProb(reduce_coef = 0.9) -- reduce mutation probability
  • ChangeRandomCrossover(available_crossovers: Sequence[Callable[[np.ndarray, np.ndarray], Tuple[np.ndarray, np.ndarray]]]) -- change another (random) crossover from list of crossovers
  • ChangeRandomSelection(available_selections: Sequence[Callable[[np.ndarray, int], np.ndarray]])
  • ChangeRandomMutation(available_mutations: Sequence[Callable[[float, float, float], float]])
  • RemoveDuplicates(oppositor = None, creator = None, converter = None); see doc
  • CopyBest(by_indexes) -- copies best population object values (from dimensions in by_indexes) to all population
  • PlotPopulationScores(title_pattern = lambda data: f"Generation {data['current_generation']}", save_as_name_pattern = None) -- plot population scores; needs 2 functions like data->string for title and file name (to save)
• conditions:
  • ActionConditions.EachGen(generation_step = 10) -- do action each generation_step generations
  • ActionConditions.Always() do action each generations, equals to ActionConditions.EachGen(1)
  • ActionConditions.AfterStagnation(stagnation_generations = 50) -- do action after stagnation_generations stagnation generations
  • ActionConditions.Several(list_of_conditions) -- do action if all conditions in list are true

To combine action and condition to callback, just use MiddleCallbacks.UniversalCallback(action, condition) methods.

There are also next high-level useful callbacks:

• MiddleCallbacks.ReduceMutationGen(reduce_coef = 0.9, min_mutation = 0.005, reduce_each_generation = 50, reload_each_generation = 500)
• MiddleCallbacks.GeneDiversityStats(step_generations_for_plotting:int = 10) -- plots some duplicates statistics each gen (example)

See code example

How to compare efficiency of several versions of GA optimization

To compare efficiency of several versions of GA optimization (such as several values of several hyperparamenters or including/excepting some actions like oppositions) u should make some count of simulations and compare results using some statistical test. I have realized this logic here

Hints on how to adjust genetic algorithm's parameters (from geneticalgorithm package)

In general the performance of a genetic algorithm or any evolutionary algorithm depends on its parameters. Parameter setting of an evolutionary algorithm is important. Usually these parameters are adjusted based on experience and by conducting a sensitivity analysis. It is impossible to provide a general guideline to parameter setting but the suggestions provided below may help:

• Number of iterations: Select a max_num_iterations sufficiently large; otherwise the reported solution may not be satisfactory. On the other hand selecting a very large number of iterations increases the run time significantly. So this is actually a compromise between the accuracy you want and the time and computational cost you spend.

• Population size: Given a constant number of functional evaluations (max_num_iterations times population_size) I would select smaller population size and greater iterations. However, a very small choice of population size is also deteriorative. For most problems I would select a population size of 100 unless the dimension of the problem is very large that needs a bigger population size.

• elit_ratio: Although having few elites is usually a good idea and may increase the rate of convergence in some problems, having too many elites in the population may cause the algorithm to easily trap in a local optima. I would usually select only one elite in most cases. Elitism is not always necessary and in some problems may even be deteriorative.

• mutation_probability: This is a parameter you may need to adjust more than the other ones. Its appropriate value heavily depends on the problem. Sometimes we may select mutation_probability as small as 0.01 (i.e. 1 percent) and sometimes even as large as 0.5 (i.e. 50 percent) or even larger. In general if the genetic algorithm trapped in a local optimum increasing the mutation probability may help. On the other hand if the algorithm suffers from stagnation reducing the mutation probability may be effective. However, this rule of thumb is not always true.

• parents_portion: If parents_portion set zero, it means that the whole of the population is filled with the newly generated solutions. On the other hand having this parameter equals 1 (i.e. 100 percent) means no new solution is generated and the algorithm would just repeat the previous values without any change which is not meaningful and effective obviously. Anything between these two may work. The exact value depends on the problem.

• crossover_type: Depends on the problem. I would usually use uniform crossover. But testing the other ones in your problem is recommended.

• max_iteration_without_improv: This is a parameter that I recommend being used cautiously. If this parameter is too small then the algorithm may stop while it trapped in a local optimum. So make sure you select a sufficiently large criteria to provide enough time for the algorithm to progress and to avoid immature convergence.

Finally to make sure that the parameter setting is fine, we usually should run the algorithm for several times and if convergence curves of all runs converged to the same objective function value we may accept that solution as the optimum. The number of runs depends but usually five or ten runs is prevalent. Notice that in some problems several possible set of variables produces the same objective function value. When we study the convergence of a genetic algorithm we compare the objective function values not the decision variables.

Examples pretty collection

Optimization test functions

Here there is the implementation of geneticalgorithm2 for some benchmark problems. Test functions are got from my OptimizationTestFunctions package.

The code for optimizations process is same for each function and is contained in file.

Sphere

Ackley

AckleyTest

Rosenbrock

Fletcher

Griewank

Penalty2

Quartic

Rastrigin

SchwefelDouble

SchwefelMax

SchwefelAbs

SchwefelSin

Stairs

Abs

Michalewicz

Scheffer

Eggholder

Weierstrass

Using GA in reinforcement learning

See example of using GA optimization with keras neural networks for solving OpenGym tasks.

Better example is OpenGym using cost2fitness and geneticalgorithm2 where I use also my cost2fitness package for fast forward propagation

Using GA with image reconstruction by polygons

Links:

1. https://www.kaggle.com/demetrypascal/fork-of-imagereconstruction-with-geneticalgorithm2
2. https://www.kaggle.com/demetrypascal/imagereconstructionpolygons-with-geneticalgorithm2

Popular questions

How to disable autoplot?

Just use no_plot = True param in run method:

model.run(no_plot = True)

If u want, u can plot results later by using

model.plot_results()

Also u can create your pretty plots using model.report object (it's a list of values):

re = np.array(model.report)

plt.plot(re)
plt.xlabel('Iteration')
plt.ylabel('Objective function')
plt.title('Genetic Algorithm')
plt.show()

How to plot population scores?

There are 2 ways to plot of scores of population:

• use plot_pop_scores(scores, title = 'Population scores', save_as = None) function from geneticalgorithm2 environment
• use plot_generation_scores(self, title = 'Last generation scores', save_as = None) method of ga object for plotting scores of last generation (yes, it's wrapper of previous function)

Let's check example:

import numpy as np

from geneticalgorithm2 import geneticalgorithm2 as ga

from geneticalgorithm2 import plot_pop_scores # for plotting scores without ga object

def f(X):
    return 50*np.sum(X) - np.sum(np.sqrt(X)*np.sin(X))
    
dim = 25
varbound = np.array([[0,10]]*dim)

# create start population
start_pop = np.random.uniform(0, 10, (50, dim))
# eval scores of start population
start_scores = np.array([f(start_pop[i]) for i in range(start_pop.shape[0])])

# plot start scores using plot_pop_scores function
plot_pop_scores(start_scores, title = 'Population scores before beggining of searching', save_as= 'plot_scores_start.png')


model = ga(function=f, dimension=dim, variable_type='real', variable_boundaries=varbound)
# run optimization process
model.run(no_plot = True,
          start_generation={
              'variables': start_pop,
              'scores': start_scores
              })
# plot and save optimization process plot
model.plot_results(save_as = 'plot_scores_process.png')

# plot scores of last population
model.plot_generation_scores(title = 'Population scores after ending of searching', save_as= 'plot_scores_end.png')

How to specify evaluated function for all population?

U can do it using set_function parameter into run() method.

This function should get numpy 2D-array (samples x dimension) and return 1D-array with results.

By default it uses set_function = geneticalgorithm2.default_set_function(function), where

    def default_set_function(function_for_set):
        def func(matrix):
            return np.array([function_for_set(matrix[i,:]) for i in range(matrix.shape[0])])
        return func

U may want to use it for creating some specific or fast-vectorized evaluations like here:

def sigmoid(z):
    return 1/(1+np.exp(-z))

matrix = np.random.random((1000,100))

def vectorised(X):
    return sigmoid(matrix.dot(X))

model.run(set_function = vectorised)

What about parallelism?

By using set_function u can determine your own behavior for parallelism or u can use geneticalgorithm2.set_function_multiprocess(f, n_jobs = -1) for using just parallelism (recommended for heavy functions and big populations, not recommended for fast functions and small populations).

For example:

import numpy as np
from geneticalgorithm2 import geneticalgorithm2 as ga

def f(X):
    import math
    a = X[0]
    b = X[1]
    c = X[2]
    s = 0
    for i in range(10000):
        s += math.sin(a*i) + math.sin(b*i) + math.cos(c*i)

    return s
 

algorithm_param = {'max_num_iteration': 50,
                   'population_size':100,
                   'mutation_probability':0.1,
                   'elit_ratio': 0.01,
                   'crossover_probability': 0.5,
                   'parents_portion': 0.3,
                   'crossover_type':'uniform',
                   'mutation_type': 'uniform_by_center',
                   'selection_type': 'roulette',
                   'max_iteration_without_improv':None}   
    
varbound = np.array([[-10,10]]*3)

model = ga(function=f, dimension=3, 
    variable_type='real',           
    variable_boundaries=varbound, 
    algorithm_parameters = algorithm_param)

########

%time model.run()
# Wall time: 1min 52s

%time model.run(set_function= ga.set_function_multiprocess(f, n_jobs = 6))
# Wall time: 31.7 s

How to initialize start population? How to continue optimization with new run?

For this there is start_generation parameter in run() method. It's the dictionary with structure like returned model.output_dict['last_generation']. Let's see example how can u to use it:

import numpy as np
from geneticalgorithm2 import geneticalgorithm2 as ga

def f(X):
    return np.sum(X)
    
dim = 6
    
varbound = [(0,10)]*dim


algorithm_param = {'max_num_iteration': 500,
                   'population_size':100,
                   'mutation_probability':0.1,
                   'elit_ratio': 0.01,
                   'crossover_probability': 0.5,
                   'parents_portion': 0.3,
                   'crossover_type':'uniform',
                   'max_iteration_without_improv':None}

model = ga(function=f, 
           dimension=dim, 
           variable_type='real', 
           variable_boundaries=varbound,
           algorithm_parameters = algorithm_param)

# start generation
# as u see u can use any values been valid for ur function
samples = np.random.uniform(0, 50, (300, dim)) # 300 is the new size of your generation



model.run(no_plot = False, start_generation={'variables':samples, 'scores': None}) 
# it's not necessary to evaluate scores before
# but u can do it if u have evaluated scores and don't wanna repeat calcucations



# from version 6.3.0 it's recommended to use this form
from geneticalgorithm2 import Generation
model.run(no_plot = False, start_generation=Generation(variables = samples, scores = None))


# from version 6.4.0 u also can use these forms
model.run(no_plot = False, start_generation= samples)
model.run(no_plot = False, start_generation= (samples, None))


# if u have scores array, u can put it too
scores = np.array([f(sample) for sample in samples])
model.run(no_plot = False, start_generation= (samples, scores))


##
## after first run
## best value = 0.10426190111045064
##

# okay, let's continue optimization using saved last generation
model.run(no_plot = True, start_generation=model.output_dict['last_generation']) 

##
## after second run
## best value = 0.06128462776296528
##

Also u can save and load populations using likely code:

import numpy as np

from geneticalgorithm2 import geneticalgorithm2 as ga

from OptimizationTestFunctions import Eggholder


dim = 2*15

f =  Eggholder(dim)

xmin, xmax, ymin, ymax = f.bounds
        
varbound = np.array([[xmin, xmax], [ymin, ymax]]*15)
    
model = ga(function=f,
               dimension = dim,
               variable_type='real',
               variable_boundaries=varbound,
               algorithm_parameters = {
                       'max_num_iteration': 300,
                       'population_size': 100
                       })

# first run and save last generation to file
filename = "eggholder_lastgen.npz"
model.run(save_last_generation_as = filename)


# load start generation from file and run again (continue optimization)
model.run(start_generation=filename)