# A Brain Storm and Chaotic Accelerated Particle Swarm Optimization Hybridization
## 2 Background
### The BSO Algorithm
## 6 Discussion and Conclusions

### Observations and Remarks

The results demonstrated that the BSO-CAPSO Hybrid showcased improved behaviors compared to the stand-alone BSO and CAPSO algorithms. The hybrid provided better solutions regarding all unimodal functions, compared to both BSO and CAPSO. The only exception to this was the Schwefel 2.21 function, \(f_{3}\), for \(D=10\) dimensions, where CAPSO offered the best results. However, for larger numbers of dimensions the hybrid severely outperformed CAPSO for the said benchmark. For multimodal functions, the results were less absolute, but they were in favor of the hybrid approach. The hybrid outperformed CAPSO for all multimodal functions--with the exception of the Ackley function, \(f_{5}\), only for \(D=30\). In some cases, this was very impactful, such as for the Griewank function, \(f_{6}\). Compared to BSO it showcased both more and less advantageous results. Nevertheless, it decreased BSO's computational time significantly-- up to a third or less--while providing acceptable and comparable results. Additionally, the BSO-CAPSO hybrid accelerated convergence at \(t_{switch}\). This means that the BSO initialization could, indeed, be beneficial for CAPSO's local exploitation. Additionally, acceleration at \(t_{switch}\) often showcases the ability to discover a high-quality local search area (a high-quality attraction basin) very early during the iterative process, which means access to high-quality solutions with minimal computational costs; see, e.g., Figures 3 and 6. Moreover, it was observed that in some functions (e.g., \(f_{3}\), \(f_{4}\), \(f_{6}\)) discovering a good value for \(\gamma\) of \(\alpha(t)\) (cf. Equation (9)) was challenging and demanded some intuition and a trial-and-error approach. This possibly implies that a different \(\alpha(t)\) function could be more beneficial for these cases, since this function is also problem-dependent to some degree. In this work, however, guidelines regarding the tuning of the hybrid's parameters were provided; thus, a solid baseline regarding parameter setting was solidified. Good results were obtained, and they supported the usage of the hybrid approach. Hence, it is also implied that the BSO-CAPSO's performance on multimodal functions could be further improved; this was also discussed in Sections 2.2 and 2.3, above. Future investigation into

Figure 10: Convergence diagram of the stand-alone BSO, BSO–CAPSO hybrid and stand-alone CAPSO algorithms for the Alpine 1 (\(f_{7}\)) function with \(D=30\). Convergence acceleration occurred after \(t_{switch}=200\). \(\alpha(t)\) could benefit not only the BSO-CAPSO hybrid but also CAPSO's use of hybridization in general. Additionally, although the sinusoidal chaotic map was showcased as the top performer in previous research regarding CAPSO [17], it is possible that different chaotic maps could be beneficial for some problems and should be considered in the BSO-CAPSO hybridization. ### Further Discussion

Since BSO-CAPSO outperformed BSO significantly when applied to unimodal functions, we can assume that this hybrid approach could be efficiently utilized for local search in combination with global search optimization algorithms when applied to complex problems. If BSO-CAPSO is provided with a single attraction basin, or a small number of attraction basins, the results of this paper imply that it could outperform stand-alone BSO and provide a better result faster. This could extend to several multimodal functions, since the hybrid provided optimal results in some cases. Regarding CAPSO, the hybrid approach demonstrated significant improvements compared to the stand-alone version, so it is advantageous to propose replacing CAPSO with the BSO-CAPSO hybrid when it is applied to a complex problem. An important point of discussion is that several optimization problems remain open, very complex and computationally heavy (e.g., engineering optimization problems such as antenna design).