# Differential Evolution with Linear Bias Reduction in Parameter Adaptation
## 2 Materials and Methods
### Related Studies on Differential Evolution  
The original proposal of the original DE algorithm [13] included the three main control parameters of the algorithm: the population size \(NP\), scaling factor \(F\), and crossover rate \(Cr\). Most of the further studies on DE were focused on either developing methods to control or adapt these parameters or proposing new mutation strategies and modifying the algorithm scheme. Some of the earliest theoretical works [17; 18] considered the convergence properties of the DE and their connection to parameter settings on a set of benchmark problems, resulting in a set of recommendations for parameter values mainly in connection with the current population variance [19; 20]. Although these findings were important for the field, the adaptive DE variants appeared to have more influence on the field. Starting with one of the earlier studies [21], which proposed the popular jDE approach, which proposed the random generation of \(F\) and \(Cr\) with remembering successful values, there was a set of other adaptive DE variants proposed, including the SDE [22], SaDE (Self-adaptive Differential Evolution) [23], and the already mentioned JADE algorithm [15]. All these methods are based on sampling parameter values with random values distributed either with a uniform, normal, or Cauchy distribution. The most efficient scheme used guided adaptation, where more efficient parameter values were determined based on fitness values, like in the JADE [15] or SHADE [8] approach. In [24], the scale factor local search approach was proposed, where unlike the jDE algorithm, where the \(F\) parameter is generated randomly, the local search algorithms are used, such as the golden ratio or hill-climb. The idea of such a local search is similar to the steepest descent method, where the step size is optimized to get the best possible improvement. In [25], the parameter control strategy was proposed, where the scaling factor was linearly decreased with the iteration number, allowing achieving better results compared to other methods. Several studies [26; 27] have proposed different approach to the parameter adaptation, where a pool of fixed parameter values is maintained, and some of them are randomly chosen during the search process. A number of papers described the adaptive selection of mutation strategies used in the DE algorithm, for example adaptive DE with four mutation strategies [28] or a multi-population ensemble of mutations strategies [29]. In [30], the multi-population DE with scaling factor inheritance was proposed, where efficient solutions from one population were transferred to another together with the scaling factor. Despite the demonstrated efficiency of these approaches, the annual competitions on bound-constrained single-objective optimization organized within the Congress on Evolutionary Computation demonstrate that the most promising approaches are DE with adaptive parameter values, such as SHADE [31] or jDE variants, or hybrid methods, such as LSHADE-SPACMA [32], where L-SHADE was hybridized with CMA-ES (Covariance-Matrix Adaptation Evolution Strategy) with dynamic resource redistribution, or [33], where the super-fit individual generated with CMA-ES was included in the population, although the multi-strategy methods also demonstrate highly competitive results [34]. Several surveys were published discussing the problem of parameter adaptation in DE and considering the existing methods' advantages and drawbacks; a detailed taxonomy was presented in [35]. According to the results of this study, one of the most efficient schemes, the success-history adaptation, was initially introduced in the SHADE algorithm, the winner of CEC 2014 competition, and it represents the parameter tuning techniques introduced in JADE. SHADE maintains \(H\) memory cells, each keeping a pair \((M_{F,h},M_{Cr,h})\), where \(h\) is the current memory index.