# A Brain Storm and Chaotic Accelerated Particle Swarm Optimization Hybridization
## 2 Background
### The BSO Algorithm
In general, when an algorithm is applied to an open or complex problem, it is preferable to choose boundary conditions that cooperate sufficiently with the problem's characteristics and constraints. A flowchart diagram of the BSO-CAPSO hybrid algorithm is presented in \[x_{new}^{d}=x_{chosen}^{d}+\xi\mathcal{N}(0,1),\]2

## 4 Considerations and Setup for the Numerical Experimentation

The BSO-CAPSO hybrid and stand-alone BSO/CAPSO algorithms were tested using the same benchmark functions to showcase the advantages of the hybrid approach. It is noted that in this work, the proposed hybrid was compared to its parental algorithms; this is a fairly common practice, as can be seen in, e.g., [26; 29; 33], or similarly when a developed variant is examined against the main algorithm and/or closely-related variants [17; 39; 40; 45]. \[x_{new}^{d}=x_{chosen}^{d}+\xi\mathcal{N}(0,1),\]3 Flowchart diagram of the BSO–CAPSO hybrid algorithm. BSO and PSO are very popular algorithms and their advantages and disadvantages are well documented. Moreover, CAPSO shares the same family tree with its main parental algorithm, PSO. Additionally, for CAPSO it has been demonstrated that it serves as an improvement to APSO and that it outperforms chaotic PSO (CPSO) [17]. Based on these considerations, the BSO-CAPSO hybrid is specifically examined as an improved alternative to the use of stand-alone BSO or CAPSO. ### Benchmark Functions

Benchmark functions are typically used for testing optimization algorithms; [PERSON]'s proposed test functions are widely utilized [46], while a recently organized collection is presented in [47]. In this work, we selected the functions \(f_{k}\) presented in \[x_{new}^{d}=x_{chosen}^{d}+\xi\mathcal{N}(0,1),\]4 Specifically, functions \(f_{1},f_{2},f_{3}\) are unimodal, while functions \(f_{4},f_{5},f_{6},f_{7}\) are multimodal. In related works, the benchmarking experimentation process of an optimization algorithm has been examined in various manners. The original BSO [10] was initially only tested on two functions. In the hybrids presented in [27; 29; 32; 33], the proposed algorithms were tested on two or three complex problems or designs in order to be evaluated. In this work, the above collection of seven popular functions served as an efficient and reliable experimental basis, with our approach being similar to [17; 45], where six popular benchmarking functions were selected and examined. It is also important to note that multimodal functions are usually significantly more complex to optimize than unimodal ones. For this reason, more multimodal than unimodal functions were selected. ### Experimental Parameters and Conditions

The stand-alone BSO, CAPSO and BSO-CAPSO hybrid algorithms were developed with Python3, with all experiments executed in the same computational platform. The employed \(k\)-means clustering algorithm was the one included in the scikit-learn library [48], which by default uses the \(k\)-means++ variant. For each test, each algorithm was run 25 distinct times, for a maximum number of iterations \(T=2000\), and the size of the population, \(N\), was set as \(4D+1\) for \(D=10,20,30\), yielding \(N=41,81,121\), respectively. The numbers of clusters were set as one fifth or less of the population sizes, as is recommended in Ref. [14], thus \(m=8,16,24\), respectively.