## 3 Results
### Algorithm Performance
As expected, the AIS Gbest and Hybrid algorithms converged the fastest to the minimum cost. The Eggholder and Holder  
Figure 2: Convergence traces of various BPSO algorithms while optimizing (**a**) Sphere, (**b**) Griewank, (**c**) Holder Table 2, and (**d**) Eggholder functions showing global best output over 200 iterations, averaged over 500 runs. Table 2 functions, however, are significantly more complicated multi-modal cost functions. The Eggholder function is especially difficult due to the presence of several local minima of comparable magnitude to the global minimum. These functions are therefore better optimized by Lbest algorithms. Results in Figure 2c,d confirm this. AIS Lbest and DH-BPSO outperformed AIS Gbest and AIS Hybrid. It is also with these functions that the superiority of the DH-BPSO algorithm can be seen. The DH-BPSO algorithm consistently reached the lowest minimum of any of the other algorithms. It also discovered the lowest minimum quicker than the AIS Lbest algorithm, shown by the steeper slope of the DH-BPSO convergence trace. These results also indicate that dynamic hybrid algorithms such as DH-BPSO promise better performance on complex, difficult-to-optimize cost functions than static hybrid algorithms such as AIS Hybrid. Table 2 provides column plots of performance statistics calculated during the optimizations. Three metrics are provided. First, the average minimum cost is provided. This is calculated by taking the average over all trials of the minimum cost achieved by each optimization. Secondly, the average iterations to reach the minimum cost is given. This is calculated as an average over the 500 trials of the iterations required to first reach that trial's final minimum cost. Lastly, the average time to reach the minimum cost is given. For each of the 500 trials, the time required to first reach the ultimate minimum cost is calculated and averaged over the 500 trials. These benchmarks were performed on a computer equipped with an Intel i7-6560u 2.2 GHz quad-core processor and 8 GB DDR3 RAM. The raw values of the performance metrics are given in Appendix A in Table 1.  
The qualitative conclusions gleaned from Figure 2 can now be quantified. For the simple Sphere function, the difference in average minimum cost achieved is negligible, varying from 0.014 for AIS Hybrid to 0.03 for AIS Lbest. The average iterations to the minimum cost show a similar pattern, with AIS Hybrid leading with 115.028 iterations on average to converge to a minimum value. Although it did not outperform AIS Gbest or Hybrid, DH-BPSO did outperform AIS Lbest in both average minimum cost and average iterations required to minimize. For the Sphere function, DH-BPSO required 115.348 \(\pm\) 26.48 average iterations to find an average minimum of 0.022 \(\pm\) 0.147, compared to AIS Lbest's average minimum of 0.03 \(\pm\) 0.171 achieved in 123.88 \(\pm\) 24.822 average iterations. Note that the DH-BPSO average minimum cost has a smaller standard deviation, which is an indication of consistency. The performance of these two algorithms on the Griewank function exhibits a similar pattern, namely that DH-BPSO achieves a lower cost in less iterations on average. These results once again show that dynamic hybrid algorithms can exceed in performance compared to their static hybrid or non-hybrid counterparts. The performance of DH-BPSO and AIS Lbest excel when optimizing the more complex Holder Table 2 and Eggholder functions. In these trials, DH-BPSO outperformed the other algorithms in minimum cost achieved. For Holder Table 2, DH-BPSO achieved a minimum cost of \(-9.336\)\(\pm\) 0.597 compared to the next best minimum cost of \(-9.278\)\(\pm\) 0.811 achieved by AIS Lbest. The true global minimum of \(-9.45799\) lies within one standard deviation of both results, however, DH-BPSO consistently achieved a lower minimum cost, evidenced by the lower average minimum and smaller standard deviation. A similar conclusion can be made comparing AIS Lbest and DH-BPSO performance optimizing the Eggholder function.