# Cyber Firefly Algorithm Based on Adaptive Memory Programming for Global Optimization
## 4 Experimental Results and Discussions
### Analysis of CFA Strategies
#### 4.2.5 Worst-Case Analysis  
As the proposed CFA is a stochastic optimization algorithm, every single execution of the program may produce a different solution. It thus becomes very important to measure the solution variation of the worst case obtained by the CFA. We conduct the worst-case analysis on the three most challenging functions, namely [PERSON] (30), [PERSON] (30), and [PERSON] (30) as follows. One thousand independent runs of the CFA program are performed. We record the worst function value (fitness) that could be obtained by allowing different number of repetitive runs in the experiment. It is seen from Figure 2 that the [PERSON] (30) fitness value of less than 5 can be obtained with 99.7% confidence because three out of the one thousand runs report a fitness value greater than 5. It also indicates the worst fitness value we could obtain is no more than 5 if we can run the CFA program at least three times. Actually, there are two major components with the fitness distribution. One is close to 4 (a local optimum) with about 20% of the distribution, the other is near zero (the global optimum) with about 79% confidence. The worst-case analysis with Rastrigin (30) is illustrated in Figure 3. We observe a Gaussian-like distribution and the major component falls in the fitness value between 0 and 12. The Gaussian-like distribution provides a reliable guarantee that the mean performance value can be obtained with high confidence. Figure 4 shows the worst-case analysis for Griewank (30). The distribution has a long tail, and the major component concentrates at the high quality part. This situation manifests good properties that there is only one major outcome which is near the global optimum, and that a near-optimal solution can be obtained with a few runs in the worst case. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Functions**} & \multicolumn{2}{c}{**No Multi-Start Rebuilding**} & \multicolumn{2}{c}{**With Multi-Start Rebuilding**} \\ \cline{2-5}  & **Mean** & **Std** & **Mean** & **Std** \\ \hline [PERSON] (2) & \(-1.0\) & \(0.0\) & \(-1.0\) & \(0.0\) \\ [PERSON] (2) & \(-186.03\) & \(0.708244\) & \(-186.723\) & \(0.016331\) \\ [PERSON] (2) & \(0.0\) & \(0.0\) & \(0.0\) & \(0.0\) \\ [PERSON] (2) & \(0.0\) & \(0.0\) & \(0.0\) & \(0.0\) \\ [PERSON] (3) & \(0.0\) & \(0.0\) & \(0.0\) & \(0.0\) \\ [PERSON] (4, 5) & \(-5.56783\) & \(3.517801\) & \(-8.09613\) & \(3.003044\) \\ [PERSON] (4, 7) & \(-9.12732\) & \(2.851160\) & \(-10.2611\) & \(0.440954\) \\ [PERSON] (4, 10) & \(-7.65202\) & \(3.724529\) & \(-10.3554\) & \(0.966638\) \\ [PERSON] (10) & \(0.0\) & \(0.0\) & \(0.0\) & \(0.0\) \\ [PERSON] (10) & \(0.000006\) & \(0.00005\) & \(0.002511\) & \(0.009392\) \\ [PERSON] (10) & \(1.790926\) & \(1.294722\) & \(0.646338\) & \(0.800856\) \\ [PERSON] (10) & \(0.018083\) & \(0.015917\) & \(0.01645\) & \(0.009914\) \\ [PERSON] (10) & \(0.0\) & \(0.0\) & \(0.0\) & \(0.0\) \\ [PERSON] (20) & \(0.0\) & \(0.0\) & \(0.0\) & \(0.0\) \\ [PERSON] (20) & \(0.931281\) & \(1.685579\) & \(0.267050\) & \(0.994099\) \\ [PERSON] (20) & \(3.349695\) & \(1.807129\) & \(2.466102\) & \(1.765681\) \\ [PERSON](20) & \(0.008904\) & \(0.011122\) & \(0.007213\) & \(0.012479\) \\ [PERSON] (20) & \(0.0\) & \(0.0\) & \(0.0\) & \(0.0\) \\ [PERSON] (30) & \(0.0\) & \(0.0\) & \(0.0\) & \(0.0\) \\ [PERSON] (30) & \(1.205757\) & \(1.838271\) & \(0.801094\) & \(1.598004\) \\ [PERSON] (30) & \(4.974795\) & \(1.973266\) & \(4.680690\) & \(2.632701\) \\ [PERSON] (30) & \(0.005735\) & \(0.011983\) & \(0.003364\) & \(0.007581\) \\ [PERSON] (30) & \(0.000001\) & \(0.000004\) & \(0.0\) & \(0.0\) \\ \hline \hline \end{tabular}
\end{table} 
Table 5: The mean best function value and the standard deviation obtained by CFA with or without multi-start strategy.