# Multi-Guide Set-Based Particle Swarm Optimization for Multi-Objective Portfolio Optimization
## 7 Conclusions  
This paper proposed the multi-guide set-based particle swarm optimization (MGS-BPSO) algorithm, a multi-objective adaptation of the set-based particle swarm optimization (SBPSO) algorithm that incorporates elements from multi-guide particle swarm optimization.  
\begin{table}
\begin{tabular}{l l 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 & 0 & 0 & 1 & 1 \\  & Losses & 4 & 4 & 1 & 9 \\  & Draws & 0 & 0 & 2 & 2 \\  & Difference & \(-\)4 & \(-\)4 & 0 & \(-\)8 \\  & Rank & 4 & 4 & 3 & 4 \\ \hline \multirow{4}{*}{NSGA\(-\)II} & Wins & 2 & 2 & 1 & 5 \\  & Losses & 1 & 1 & 2 & 4 \\  & Draws & 1 & 1 & \(-\)1 & 3 \\  & Difference & 1 & 1 & \(-\)1 & 1 \\  & Rank & 2 & 2 & 4 & 2 \\ \hline \multirow{4}{*}{SPEA2} & Wins & 2 & 4 & 4 & 10 \\  & Losses & 1 & 0 & 0 & 1 \\  & Draws & 1 & 0 & 0 & 1 \\  & Difference & 1 & 4 & 4 & 9 \\  & Rank & 2 & 1 & 1 & 1 \\ \hline \multirow{4}{*}{MGSBPSO} & Wins & 1 & 1 & 2 & 4 \\  & Losses & 3 & 3 & 1 & 7 \\ \cline{1-1}  & Draws & 0 & 0 & 1 & 1 \\ \cline{1-1}  & Difference & \(-\)2 & \(-\)2 & 1 & \(-\)3 \\ \cline{1-1}  & Rank & 3 & 3 & 2 & 3 \\ \hline \hline \end{tabular}
\end{table}
Table 14: Nikkei 225 rankings for each performance measure.  
Figure 7: Pareto-optimal obtained fronts for Nikkei 225.  
tion (MGPSO). MGSBPSO uses two set-based swarms, where the first selects assets that minimize risk, and the second swarm selects assets that maximize return. For the purpose of optimizing the asset weights, MGPSO, which samples' control parameter values that satisfy theoretically derived stability criteria, was used. The performance of MGSBPSO was compared with that of MGPSO, NSGA-II, SPEA2, and the single-objective SBPSO across five portfolio optimization benchmark problems.  
The results showed that all algorithms, in general, were able to approximate the true Pareto-optimal front (POF). NSGA-II and SPEA2 generally ranked quite high in comparison with the other algorithms. However, visual analysis of the POF obtained by NSGA-II and SPEA2 shows that these algorithms were only able to approximate part of the true POF. SBPSO, MGPSO, and MGSBPSO were able to approximate the full true POF for each benchmark problem, with the exception of MGPSO, for the last (and largest) benchmark problem. MGPSO, as well as MGSBPSO and SBPSO, did not scale to the largest portfolio problem, which redefined portfolio optimization as a set-based optimization problem. SBPSO optimized the mean-variance portfolio model of Equation (3) for a given risk-return tradeoff value. By optimizing for multiple tradeoff values, SBPSO obtained a variety of solutions with differing risk and return characteristics. Thus, an advantage of MGSBPSO over its single-objective counterpart is that MGSBPSO does not require multiple runs to obtain a diverse set of optimal solutions, as risk (minimized) and return (maximized) are optimized independently of each other. It should also be noted that MGPSO without control parameter tuning performed similar to or better than NSGA-II and SPEA2 with tuned parameters.  
Overall, the results show that the benefits of redefining the portfolio optimization problem as a set-based problem are also applicable to multi-objective portfolio optimization.