## 5 Conclusions and Future Work  
This article examines the recently introduced dimension-wise particle swarm optimization (DPSO) algorithm and compares its performance with other commonly used metaheuristic optimization systems. Specifically, it compares the statistical mean of the global fitness values for DPSO, PSO, GA, and DE in a two-step process: in-sample tuning and out-of-sample evaluation. The optimal parameters for each algorithm were selected by applying self-tuning while evaluating 30 independent runs of a generic in-sample problem \begin{table}
\begin{tabular}{c c c c c c c c} \hline  & **DPSO** & **PSO** & **GA** & & **DE** \\ \hline
**Dims** & **[bytes]** & **[s]** & **[bytes]** & **[s]** & **[bytes]** & **[s]** & **[bytes]** & **[s]** \\ \hline
7 & 6872 & 1.849 & 6552 & 1.218 & 5528 & 1.849 & 4944 & 1.849 \\
14 & 7904 & 1.879 & 7616 & 1.060 & 6280 & 1.856 & 5448 & 1.851 \\
21 & 8968 & 1.866 & 8680 & 1.474 & 7096 & 1.860 & 5952 & 1.857 \\ \hline \end{tabular}
\end{table} 
Table 5: The Computational costs in terms of average memory and time for each dimension size on the elliptical problem. that approximated the set of all functions used for evaluation. To evaluate the performance, each algorithm was tested on 24 separate problems, and the mean and standard deviation were obtained from 30 separate runs of each for 7, 14, and 21 dimensions. The obtained results show that DPSO performs better than DE, PSO, and GA when the population is sparse w.r.t the number of dimensions to be explored. In future work, it may be worth incorporating other methods such as Adaptive PSO to reduce the number of parameters that must be tuned. It may also be interesting to investigate other options for rules on warping particles and setting the population size as one of the adjustable parameters. To better grasp the benefits of DPSO, it would also be worth comparing this algorithm with more state-of-the-art variations in PSO such as Dual Gbest PSO without regard for the computational costs. Given that there is no conflict with momentum degradation methods such as the one found in continuous PSO w.r.t. the changes made in DPSO, integrating them into DPSO may allow for further improvements in fitness with minimal addition to computational costs [9]. Conceptualization, P.M.; methodology, J.S. ; software, J.S. ; investigation, J.S. ; resources, P.M.; writing--original draft preparation, J.S. ; writing--review and editing, P.M.; supervision, P.M.; project administration, P.M.; and funding acquisition, P.M. Both authors have read and agreed to the published version of the manuscript. This research was funded by the Natural Sciences and Engineering Research Council (NSERC) of Canada grant number RGPIN-2017-05866 and by the Government of Alberta under the Major Innovation Fund, project RCP-19-001-MIF. The authors declare no conflict of interest. The following symbols are used in this manuscript: \begin{tabular}{l l l l l l l} \hline \hline \multirow{2}{*}{DPSO} & \(c_{0}\) & \(c_{1}\) & \(c_{2}\) & \(c_{3}\) & \(c_{4}\) & \(c_{5}\) & \(c_{6}\) \\  & 0.862056 & 0.859008 & 0.620299 & 0.526893 & 0.103819 & 0.986512 & 0.801248 \\ \hline \multirow{2}{*}{PSO} & \(c_{0}\) & \(c_{1}\) & \(c_{2}\) & & & & \\  & 0.107710 & 0.741607 & 0.492210 & & & & \\ \hline \multirow{2}{*}{GA} & \(c_{0}\) & \(c_{1}\) & & & & & \\  & 0.974085 & 0.558012 & & & & & \\ \hline \multirow{2}{*}{DE} & \(c_{0}\) & \(c_{1}\) & & & & & \\  & 0.251732 & 0.000000 1 & & & & & \\ \hline \hline \end{tabular}   
* DE is forced to choose one parameter to crossover.