# A Cyclical Non-Linear Inertia-Weighted Teaching-Learning-Based Optimization Algorithm
## 4 Benchmark Tests
### CNIWTLBO vs. the Variants of ABC, DE  
The experiment in this section is aimed at identifying the performance of the CNIWTLBO algorithm to achieve the global optimum value by comparing with the variants of ABC and DE on 7 benchmark functions shown in Table 1. The variants of ABC, such as the gbest-guided artificial bee colony GABC algorithm [36] and the improved artificial bee colony IABC algorithm [37], and the variants of DE, such as SaDE and JADE, are used in this experiment. To be fair in the comparison, the parameters of the algorithms are same as in the literature [27], where the population size is 20 and dimension is 30. Alike other algorithms, TLBO and CNIWTLBO are tested with the same function evaluations listed in Table 6. The comparative results, in the form of mean value and average standard deviation, are listed in Table 6. The results of GABC, IABC, SaDE and JADE are taken from the literature [27] directly. The results of TLBO and CNIWTLBO are obtained after 30 independent runs on each benchmark function in the same way.  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**No.** & **Function** & & **PSO-w** & **PSO-cf** & **CPSO-H** & **CLPSO** & **TLBO** & **CNIWTLBO** \\ \hline _f1_ & Sphere & Mean & 7.96E-51 & 9.84E-105 & 4.98E-45 & 5.15E-29 & 0.00E+00 & **0.00E+00** \\  & & SD & 3.56E-50 & 4.21E-104 & 1.00E-44 & 2.16E-28 & 0.00E+00 & 0.00E+00 \\ \hline _f8_ & Rosenbrock & Mean & 3.08E+00 & **6.98E-01** & 1.53E+00 & 2.46E+00 & 4.21E+00 & 6.68E+00 \\  & & SD & 7.69E-01 & 1.46E+00 & 1.70E+00 & 1.70E+00 & 6.83E-01 & 6.32E-01 \\ \hline _f15_ & Ackley & Mean & 1.58E-14 & 9.18E-01 & 1.49E-14 & 4.32E-10 & 3.55E-15 & **8.44E-16** \\  & & SD & 1.60E-14 & 1.01E+00 & 6.97E-15 & 2.55E-14 & 8.32E-31 & 5.24E-17 \\ \hline _f16_ & Schwefel2.26 & Mean & 3.20E+02 & 9.87E+02 & 2.13E+02 & 0.00E+00 & **-4.01E+03** & -4.07E+03 \\  & & SD & 1.85E+02 & 2.76E+02 & 1.41E+02 & 0.00E+00 & 1.85E+02 & 1.14E+02 \\ \hline _f18_ & Rastrigin & Mean & 5.82E+00 & 1.25E+01 & 2.12E+00 & 0.00E+00 & 6.77E-08 & **0.00E+00** \\  & & SD & 2.96E+00 & 5.17E+00 & 1.33E+00 & 0.00E+00 & 3.68E-07 & 0.00E+00 \\ \hline _f19_ & Griewank & Mean & 9.69E-02 & 1.19E-01 & 4.07E-02 & 4.56E-03 & 0.00E+00 & **0.00E+00** \\  & & SD & 5.01E-02 & 7.11E-02 & 2.80E-02 & 4.81E-03 & 0.00E+00 & 0.00E+00 \\ \hline _f20_ & NCRastrigin & Mean & 4.05E+00 & 1.20E+01 & 2.00E-01 & 0.00E+00 & 2.65E-08 & **0.00E+00** \\  & & SD & 2.58E+00 & 4.99E+00 & 4.10E-01 & 0.00E+00 & 1.23E-07 & 0.00E+00 \\ \hline _f21_ & Weierstrass & Mean & 2.28E-03 & 6.69E-01 & 1.07E-15 & 0.00E+00 & 2.42E-05 & **0.00E+00** \\  & & SD & 7.04E-03 & 7.17E-01 & 1.67E-15 & 0.00E+00 & 1.38E-20 & 0.00E+00 \\ \hline \hline \end{tabular}  
* Population Size: **10**, D: **10**, Max. Eval.: **30,000** FEs. Source: Results of algorithms except CNIWTLBO are taken from References [28; 35].  
\end{table}
Table 5: Comparative results of CNIWTLBO and other variants of PSO.  
It can be observed from the results that the performance of CNIWTLBO performs much better than GABC, SaDE and JADE on all the benchmark functions in Table 6, and outperforms the IABC algorithm for _fl_(Sphere), _f4_(Schwefel 1.2), _f5_(Schwefel 2.22), _f6_(Schwefel 2.21) and _f15_(Ackley). Furthermore, CNIWTLBO is better than TLBO for _f15_(Ackley) and _f18_(Rastrigin). So, it is indicated that the CNIWTLBO algorithm has a good performance.