Evolutionary strategy

In computing, evolutionary strategies are a type of evolutionary algorithms that are mainly characterized by: The selection of individuals for recombination is impartial and is a deterministic process, they differ from the rest of the Algorithms Evolutionary mainly due to the shape of the mutation operator and are mainly applied in continuous optimization problems where the representation is through vectors of real numbers. They were originally created at the Technical University of Berlin in 1964.

The general form of Evolutionary Strategies algorithms has the following notation: (μ μ /ρ ρ ,λ λ )− − ES{displaystyle (mu /rholambda)-ES}

Where

• μ: Population size
• ρ: Number of parents selected to be recombined
• λ: Number of individuals in descent

A pseudocode for the general algorithm might be as follows:

0 given ρ, μ, λ ε N+
1 initialize P = {(xk; f(xk)) Δ 1 ≤ k μ}
2 while not happy
3 Q = {}
4 for k ε {1,... λ}
5 selected = select_mates(ρ, P)
6 xk = recombine(selected)
7 xk = mutate(xk)
8 Q = Q + (xk; f(xk))
9 P = P U Q
10 P = select_by_age(P)
11 P = select_best(μ, P) // by f-ranking

In which we initially have a set of µ parents. In each iteration of the algorithm, the offspring (λ) is created, for this ρ parents are randomly selected to recombine, the recombination product is mutated and the new individual is formed. After the set of offspring is formed, the best µ individuals are selected from the old population and the new offspring.

One of the distinctive features of Evolutionary Strategies within Evolutionary Algorithms is the mutation operator. Said operator is carried out through a multivariate normal distribution:

• A non-dimensional X random vector, distributes normal multivariant with positive defined covariance parameter and matrix C if its density function is: fx(x)=1(2π π )n/2⋅ ⋅ det(C)1/2⋅ ⋅ Exp (− − 12(x− − x! ! )TC− − 1(x− − x! ! )){displaystyle fx(x)={frac {1}{(2pi)^{n/2cdot det(C)^{1/2}}}}cdot exp(-{frac {1}{2}}}(x-{bar {x}}})^{TC^{-1}(x-{bar {x})})}}}
• In short notation:

X▪ ▪ N(x! ! ,C){displaystyle Xbacksim N({bar {x}},C)}The most widely used distributions in Evolutionary Strategies are: N(0,1),N(0,diag(δ δ 2)),N(0,C){displaystyle N(0.1),N(0,diag(delta ^{2})),N(0,C}

There are other variants of Evolutionary Strategies:

• (1+1)-ES (Only a parent generates a mutating offspring, then selects the best of both. It needs other parameters that are self-adjusted)
• (μ, λ)-MSC-ES
• DR1, DR2, DR3
• CMA-ES (It is one of the most used in practice, maintains a matrix of parameters that is self-adjusted)