# Differential Evolution with Linear Bias Reduction in Parameter Adaptation
## 2 Materials and Methods  
### Differential Evolution Algorithm  
The original differential evolution algorithm was proposed by [PERSON] and [PERSON] [13], and its main feature was the usage of differences between the vectors representing solutions, hence the name of the algorithm. It has received much attention from researchers mainly because it is easy to implement and it has high efficiency. The DE starts with a population of randomly initialized solutions \(x_{i,j}=rand(x_{max,j}-x_{min,j})\), where \(i=1,\dots,NP\), \(j=1,\dots,D\), \(x_{min,j}\) and \(x_{max,j}\) are the lower and upper bounds of the search area for variable \(j\), NP is the initial population size, and \(D\) is the problem dimension. After initialization, the algorithm proceeds with the mutation, crossover, and selection operations. The theoretical considerations of all these steps were studied in [14]. There are several mutation schemes known for DE, and the basic strategy is called rand/1 and implemented as follows:  
\[v_{i,j}=x_{r1,j}+F(x_{r2,j}-x_{r3,j}), \tag{1}\]  
where \(F\) is the scaling factor parameter and indexes \(r1\), \(r2\) and \(r3\) are mutually different and randomly generated. There were several more advanced strategies proposed, including the current-to-pbest/1, introduced in JADE [15] and also used in the SHADE framework [8]:  
\[v_{i,j}=x_{i,j}+F(x_{pbest,j}-x_{i,j})+F(x_{r1,j}-x_{r2,j}) \tag{2}\]  
where _pbest_ is the index of one of the \(pb*100\%\) best in terms of fitness individuals and _pbest_\(\
eq r1\
eq r2\
eq i\). The mutation strategies significantly influence the algorithm performance and require specific parameter tuning techniques; most of them randomly sample \(F\) and \(Cr\) values. There were several modifications to the _current-to-pbest/1_ strategy, including the _current-to-pbest-w/1_ strategy introduced in jSO [11], which significantly improved the performance:  
\[v_{i,j}=x_{i,j}+F_{w}(x_{pb,j}-x_{i,j})+F(x_{r1,j}-x_{r2,j}), \tag{3}\]  
where \(F_{w}\) is defined as follows:  
\[F_{w}= \begin{cases}0.7F,&\text{if }NFE<0.2 NFE_{max},\\ 0.8F,&\text{if }0.2 NFE_{max}\leq NFE<0.4 NFE_{max},\\ 1.2F,&\text{otherwise}.\end{cases}  \tag{4}\]where \(NFE\) is the number of function evaluations and \(NFE_{max}\) is the total computational resource. In L-SHADE-RSP (L-SHADE with Rank-based Selective Pressure) [16], the mutation strategy was modified with selective pressure resulting in _current-to-pbest-w/r_, which allowed faster convergence. In _current-to-pbest-w/r_, the last index \(r2\) is selected considering the fitness values of the individuals in the population, i.e., the rank values \(rank_{i}=i\), \(i=1\ldots NP\) are set, with the largest rank assigned to the best individual (smallest goal function value in the case of minimization). The probabilities of individuals being selected are calculated as follows:  
\[p_{i}^{sel}=\frac{rank_{i}}{\sum_{j=1}^{NP}rank_{j}}. \tag{5}\]  
After the mutation step, the crossover is performed, where the newly generated donor vector \(v_{i}\) is combined with the current vector \(x_{i}\). There are two types of crossover operations known for DE, namely the binomial and exponential crossover. The exponential crossover is mostly used when there are known connections between adjacent variables, while the binomial crossover is more popular and implemented as follows:  
\[u_{i,j}= \begin{cases}v_{i,j},&\text{if }rand(0,1)<Cr\text{ or }j=jrand\\ x_{i,j},&\text{otherwise}\end{cases}  \tag{6}\]  
The \(jrand\) index is randomly generated in \([1,D]\) and is needed to make sure that at least one component is taken from the donor vector, and \(Cr\) is the crossover rate parameter. The \(\beta\)O algorithm additionally changes the \(Cr\) generation, so that large values of the crossover rate are not allowed at the beginning of the search:  
\[Cr= \begin{cases}max(Cr,0.7),&\text{if }NFE<0.25 NFE_{max},\\ max(Cr,0.6),&\text{if }0.25 NFE_{max}\leq NFE\\ &\text{and }NFE<0.5 NFE_{max}.\end{cases}