2.4. Differential evolution

Differential Evolution (DE) is a stochastic population-based metaheuristic, proposed by Storn and Price ^[35], to solve single-objective optimization problems over continuous spaces. Essentially, the evolutionary strategy follows three fundamental steps: mutation, crossover and selection. The initial population containing NP individuals is randomly created, covering the entire search space. The population in a given generation G is composed of d-dimensional individuals denoted by xj(G), for j=1,…,NP. During G _max generations, the three genetic operators are applied sequentially, so that one hopes the population evolves towards the optimizer of the problem.

In the first step, mutant vectors are created by adding the balanced difference between two individuals to a third individual, by means of vj(G+1)=xκ1(G)+F(xκ2(G)−xκ3(G)). Individuals are mutually different and selected at random, and F represents the scale factor, such that F ∈ [0, 2]. In the crossover procedure, the second step, new candidate solutions are created by combining the attributes of the original population with the mutant vectors. Thus, trial vectors are created by ujk(G+1)=vjk(G+1) if randb(k) ≤ CR or k=rnbr(j). Otherwise, ujk(G+1)=xjk(G), where k=1,…,d and randb(k) ∈ [0, 1] is an uniformly distributed random number. The crossover probability CR ∈ [0, 1] is a predefined constant parameter. In turn, rnbr(j) ∈ [1, d] is a randomly chosen index. After generating the trial vectors, the best individuals are selected according to a greedy strategy, during the third step. Further details on the algorithm can be found in Price et al. ^[36].