A ligthweight genetic algorithm for solving real-encoded optimization problems, with constraints too.
Inspired by the natural selection concept, originally developed by Charles Darwin, a genetic algorithm:
- Creates a random initial population of possible solutions (individuals).
- Generates a number of new populations, where each generation is created by performing:
- selection based on fitness evaluation,
- crossover between parents,
- mutation of individuals.
When the generation process stops, best individual is chosen as the solution to given problem.
- The optimization problem is handled as a minimization problem, that is Meltingpot tries to find the minimum of given objective function.
- Unlike many common implementations, here fitness function is equal to the objective function: therefore, best individuals have lowest score and vice versa.
from meltingpot import GeneticAlgorithm # define the objective function - mandatory f = lambda x : (x-1)**2+(x-1)**2 # define number of variables - mandatory too nvars = 2 # init GA ga = GeneticAlgorithm(f,nvars) # run and get solution with score sol,score = ga.run()
Functions can be anonymous (lambda) or normal function objects. They must accept a vector of length nvars and return a scalar value.
Minimize Rosenbrock function with a global minimum at (1,1). Population set to 100 individuals, fixed mutation operator.
from meltingpot import GeneticAlgorithm, Mutation # Rosenbrock function a = 1 b = 100 f = lambda x : (a-x)**2 + b*(x-x**2)**2 nvars = 2 # define mutation parameters mutation = Mutation(shrink=0.5,sigma=1) # define lower/upper boundaries for point coordinates LB=[-5,-5] UB=[10,10] ga = GeneticAlgorithm(f,nvars,LB=LB,UB=UB,pop_size=100,mutation=mutation) sol,score = ga.run()
Run and display results:
>>> sol,score = ga.run() >>> print(sol) [1.00057427 1.00115261] >>> print(score) 3.3118979646719515e-07
Minimize the constrained G06 function, which has a minimum at (14.095, 0.84296) and is subject to following inequality constraints:
- -(x-5)^2 - (x-5)^2 + 100 <= 0
- (x-6)^2 + (x-5)^2 - 82.81 <= 0
# objective function - G06 f = lambda x : (x-10)**3 + (x-20)**3 nvars = 2 # inequality constraints g1 = lambda x: -(x-5)**2 - (x-5)**2 + 100 g2 = lambda x: (x-6)**2 + (x-5)**2 - 82.81 ics = [g1,g2] # define penalty schema penalty = Penalty(alpha=5,beta=5,C=1000) # define mutation operator mutation = Mutation(shrink=1,sigma=5) # init and run ga = GeneticAlgorithm(f,nvars,ics=ics,LB=[13,0],UB=[100,100],pop_size=1000,num_iters=100,mutation=mutation,penalty=penalty) sol,score = ga.run()
Lower and upper boundaries are defined by
UB parameters respectively. They must be a list with length equal to nvars.
Boundaries are hard constraints, i.e. the algorithm admits only individuals inside boundaries.
Selection is based on Stochastic Universal Sampling (SUS) technique and scales values according to their rank instead of their objective raw score, in order to mitigate effect of score variance. Given ranking of N individuals (where rank 1 is the most fit and N the least), scaled fitness holds 3 basic properties:
- Given an individual with rank k, it is proportional to 1/sqrt(k).
- Sum of scaled fitness is equal to the number of candidates needed for the new generation.
- Scaled values are inversely proportional to raw score, i.e. best individual has the highest scaled value and vice versa.
Define number of elite members to keep with
ga = GeneticAlgorithm(f,nvars,elites=2)
Default value is
Crossover rate sets the fraction of population which generated by crossover and can be set with
ga = GeneticAlgorithm(f,nvars,crossover=0.75)
Default value is
Each pair of parents generates two different individuals with intermediate point method: given parents p1 and p2, child is p1 + rand*(p2-p1), being rand uniformly distributed in [0,1].
The total fraction of mutation children is equal to 1-crossover.
Mutation changes individual vectors by adding a zero-mean gaussian random value to its entries; resulting value is clipped to lower/upper bounds. At each iteration sigma is updated according to a
shrink value and the previous
sigma value, so that sigma_k = sigma_k-1 * (1-shrink * k/num_iters) where k is current iteration. Setting shrink=0 let
sigma be constant.
Default values are
Inequality and equality constraints can be passed as function objects using, respectively,
# objective f = lambda x: x*x # equality constraint: x+y=6 g = lambda x: x+x-6 ecs = [g] # customize the penalty policy penalty = Penalty(alpha=3,beta=3,C=100) # set boundaries LB = [-10,-10] UB = [10,10] # init and run ga = GeneticAlgorithm(f,nvars,ecs=ecs,LB=LB,UB=UB,penalty=penalty) sol,score = ga.run()
Meltingpot handles constraints (both linear and nonlinear) through dynamic penalty functions. Penalty functions aim to replace the constrained optimization problem with an unconstrained problem, which is formed by adding cost terms - penalty functions - to the objective function. A penalty function is equal to zero if the constraint is satisfied, or a positive number if it is violated. Since individuals with lowest score are most fit, adding a positive term decreases their fitness.
The penalty evaluated at the ii-th iteration for individual x respect to constraint cs is P = (ii*C)^a + v^b, where C, a and b are constant values. For inequality constraints, value v is equal to f(x) if cs is violated or 0 otherwise. For equality constraints, v is equal to abs(f(x)) is cs is violated or 0 otherwise.
Default values for penalty are:
To detect a premature convergence of individuals towards local solutions, a stall check is performed at each iteration. If stall conditions are verified, a catastrophic event is triggered. Cataclysms keep alive only a small fraction of most fitting individuals and re-generate the remainder of population.
A stall condition is declared if average improvement in best score values over
stall_generations is less than or equal to
Counters for mutation and penalty functions are reset, as a new evolution stage would actually imply.
from meltingpot import Stall # define a customized stall function stall = Stall(tol_value=0.1,stall_generations=10) # pass stall function ga = GeneticAlgorithm(f,nvars,stall=stall) # set fraction of cataclism survivors ga.survivors = 0.2
Default value are