# Multi-Guide Set-Based Particle Swarm Optimization for Multi-Objective Portfolio Optimization
## 6 Results
### Sgp 100  
Table 11 shows that, for the S&P 100 benchmark problem, NSGA-II, SPEA2, MGPSO, and MGSBPSO obtained similar averages for return and risk. SBPSO, on the other hand, obtained solutions with more return and risk. NSGA-II and SPEA2 were the most risk-averse algorithms with lower return values. The low return and risk values of NSGA-II and \begin{table}
\begin{tabular}{l l c c c c} \hline \hline  & & **GD** & **IGD** & **HV** & **Overall** \\ \hline \multirow{4}{*}{SBPSO} & Wins & 4 & 2 & 0 & 6 \\  & Losses & 0 & 1 & 4 & 5 \\  & Draws & 0 & 1 & 0 & 1 \\  & Difference & 4 & 1 & \(-\)4 & 1 \\  & Rank & 1 & 2 & 5 & 2 \\ \hline \multirow{4}{*}{MGPSO} & Wins & 1 & 4 & 4 & 9 \\  & Losses & 3 & 0 & 0 & 3 \\  & Draws & 0 & 0 & 0 & 0 \\  & Difference & \(-\)2 & 4 & 4 & 6 \\  & Rank & 4 & 1 & 1 & 1 \\ \hline \multirow{4}{*}{NSGA\(-\)II} & Wins & 3 & 0 & 1 & 4 \\  & Losses & 1 & 4 & 3 & 8 \\  & Draws & 0 & 0 & 0 & 0 \\  & Difference & 2 & \(-\)4 & \(-\)2 & \(-\)4 \\  & Rank & 2 & 4 & 4 & 5 \\ \hline \multirow{4}{*}{SPEA2} & Wins & 2 & 1 & 2 & 5 \\  & Losses & 2 & 3 & 2 & 7 \\  & Draws & 0 & 0 & 0 & 0 \\  & Difference & 0 & \(-\)2 & 0 & \(-\)2 \\  & Rank & 3 & 3 & 3 & 4 \\ \hline \multirow{4}{*}{MGSBPSO} & Wins & 0 & 2 & 3 & 5 \\  & Losses & 4 & 1 & 1 & 6 \\ \cline{1-1}  & Draws & 0 & 1 & 0 & 1 \\ \cline{1-1}  & Difference & \(-\)4 & 1 & 2 & \(-\)1 \\ \cline{1-1}  & Rank & 5 & 2 & 2 & 3 \\ \hline \hline \end{tabular}
\end{table} 
Table 10: FTSE 100 rankings for each performance measure. Figure 5: Pareto-optimal fronts obtained for FTSE 100. SPEA2 make sense, as Figure 6 shows that these algorithms were (only) able to approximate the lower part of the true POP. SBPSO ranked first, as shown in Table 12, but with the worst HV ranking. MGPSO and MGSBPSO ranked second and third, respectively, while the multi-objective GAs, SPEA2, and NSGA-II ranked fourth and last, respectively. Lastly, Figure 6c shows that MGSBPSO was able to approximate the upper part of the true POP the best out of the multi-objective algorithms. MGSBPSO reached the stopping condition in 110 s, and MGPSO reached it in 160 s. SPEA2 and NSGA-II, on the other hand, were faster and reached the stopping condition in approximately 80 s. Note that the difference in time could be due to the different programming libraries used to implement the algorithms (see Section 5.1). \begin{table}
\begin{tabular}{l l c c c c c} \hline \hline  & & \(R\) & \(\tilde{\mathbf{\sigma}}\) & **GD** & **IGD** & **HV** \\ \hline \multirow{2}{*}{SBPSO} & \(\tilde{\mathbf{\sigma}}\) & 0.007716 & 0.001248 & 0.000275 & 0.000247 & 0.875057 \\  & \(\sigma\) & 0.000006 & 0.000001 & 0.000044 & 0.000003 & 0.044175 \\ \hline \multirow{2}{*}{MGPSO} & \(\tilde{\mathbf{\sigma}}\) & 0.004993 & 0.000579 & 0.001344 & 0.000276 & 1.195282 \\  & \(\sigma\) & 0.000359 & 0.000150 & 0.000235 & 0.000037 & 0.001887 \\ \hline \multirow{2}{*}{NSGA-II} & \(\tilde{\mathbf{\sigma}}\) & 0.004521 & 0.000254 & 0.000626 & 0.000594 & 1.188502 \\  & \(\sigma\) & 0.000130 & 0.000015 & 0.000036 & 0.000033 & 0.001069 \\ \hline \multirow{2}{*}{SPEA2} & \(\tilde{\mathbf{\sigma}}\) & 0.004352 & 0.000243 & 0.000580 & 0.000541 & 1.187435 \\  & \(\sigma\) & 0.000170 & 0.000019 & 0.000045 & 0.000052 & 0.001274 \\ \hline \multirow{2}{*}{MGSBPSO} & \(\tilde{\mathbf{\sigma}}\) & 0.004819 & 0.000388 & 0.001263 & 0.000330 & 1.190151 \\  & \(\sigma\) & 0.000195 & 0.000053 & 0.000099 & 0.000038 & 0.001808 \\ \hline \hline \end{tabular}
\end{table} 
Table 11: S&P 100 results for each performance measure.