Mutation Algorithms for Real-Valued Parameters (GA)

Genetic Algorithms(GAs) are adaptive heuristic search algorithms that belong to the larger part of evolutionary algorithms. In each generation chromosomes(our solution candidates) undergo mutation and crossover and selection to produce a better population whose chromosomes are nearer to our desired solution. Mutation Operator is a unary operator and it needs only one parent to work on. It does so by selecting a few genes from our selected chromosome (parent) and then by applying the desired mutation operator on them.

In this article, I will be talking about four Mutation Algorithms for real-valued parameters –
1) Uniform Mutation
2) Non-Uniform
3) Boundary Mutation
4) Gaussian Mutation

Here ,we are considering a chromosome with n real numbers (which are our genes) and x_i represents a gene and i belongs to [1,n].

Uniform Mutation –

In uniform mutation we select a random gene from our chromosome, let’s say x_i and assign a uniform random value to it.
Let x_i be within the range [a_i,b_i] then we assign U(a_i,b_i) to x_i
U(a_i,b_i) denotes a uniform random number from within the range [a_i,b_i].

Algorithm –
1.    Select a random integer number i from [1,n]
2.    Set xi to U(ai,bi).

Boundary Mutation –

In boundary mutation we select a random gene from our chromosome , let’s say x_i and assign the upper bound or the lower bound of x_i to it.
Let x_i be within the range [a_i,b_i] then we assign either a_i or b_i to x_i.
We also select a variable r= U(0,1) ( r is a number between 0 and 1).
If r is greater than or equal to 0.5 , assign b_i to x_i else assign a_i to x_i.

Algorithm –
1.    select a random integer number i form [1,n]
2.    select a random real value r from (0,1).
3.    If(r >= 0.5)
             Set xi to bi
       else
             Set xi to ai

Non-Uniform Mutation –

In non-uniform mutation we select a random gene from our chromosome, let’s say x_i and assign a non-uniform random value to it.
Let x_i be within the range [a_i,b_i] then we assign a non-uniform random value to it.

We use a function,
f(G)=(r2*(1-G/Gmax))b ,
where r2 = a uniform random number between (0,1)
G = the current generation number
Gmax = the maximum number of generations
b = a shape parameter
Here we select a uniform random number r1 between (0,1).
If r greater than or equal to 0.5 we assign (b_i-x_i) * f(G) to x_i else we assign (a_i+ x_i) * f(G).

Algorithm –
1.    Select a random integer i within [1,n]
2.    select two random real values r1 ,r2 from (0,1).
3.    If(r1 >= 0.5)
             Set xi to (bi-xi) * f(G)
       else
             Set xi to (ai+ xi) * f(G)

Gaussian Mutation –

Gaussian Mutation makes use of the Gauss error function . It is far more efficient in converging than the previously mentioned algorithms. We select a random gene let’s say x_i which belongs to the range [a_i,b_i]. Let the mutated off spring be x’_i. Every variable has a mutation strength operator (σ_i). We use σ= σ_i/(b_i-a_i) as a fixed non-dimensionalized parameter for all n variables;
Thus the offspring x’_i is given by —

x’i= xi + √2 * σ * (bi-ai)erf-1(u’i)

Here erf() denotes the Gaussian error function.

erf(y)=2⁄√π ∫y0 e-t2 dt

For calculation u_i’ we first select a random value u_i from within the range (0,1) and then use the following formula

if(ui>=0.5)
    u’i=2*uL*(1-2*ui)
else
    u’i=2*uR*(2*ui-1)

Again u_L and u_R are given by the formula

uL=0.5(erf( (ai-xi)⁄(√2(bi-ai)σ) )+1)
uR=0.5(erf( (bi-xi)⁄(√2(bi-ai)σ) )+1)