# A Cyclical Non-Linear Inertia-Weighted Teaching-Learning-Based Optimization Algorithm
## 4 Benchmark Tests
### CNIWTLBO vs. the Variants of TLBO in Different Dimensions  
In order to identify the performance of the CNIWTLBO, the experiments are carried out to compare the CNIWTLBO algorithm with some other variants of TLBO in different dimensions. The variants of the TLBO algorithm including WTLBO [25], ITLBO [26], I-TLBO [28], and NIWTIBO [29] are adopted. In the experiments, 9 unimodal and multimodal benchmark functions listed in Table 1 are used to evaluate the performance of the algorithms. In order to make fair comparisons, the CNIWTIBO and all adopted TLBO variants use the same parameters. In this work, evolutionary generation is used to evaluate the performance of CNIWTLBO and the TLBO variants. The population size is set to 30 and the number of evolutionary generations is set to 2000. In I-TLBO algorithm, the number of teachers is 4. The learning cycle \(T\) in CNIWTLBO is 500. The CNIWTIBO and the TLBO variants are tested on 9 benchmark functions with 20, 50 and 100 dimensions, respectively. To eliminate the randomness impact on the results, 30 independent runs for the algorithms are conducted. The experimental results are reported in Table 7, which is in form of the mean solution.  
From the Table 7, it could be observed that the TLBO and its different variants perform identically for functions _fl_, _f5_, _f8_, _f15_, _f16_ and _f18_-_f21_. The performance of the CNIWTLBO is better than rest of the algorithms for _f5_ (Schwefel2.22), _fl5_(Ackley), _f16_(Schwefel2.26) function in 20, 50 and 100 dimensions. Moreover, it outperforms other algorithms for _f20_(NCRastrigin) in 100 dimensions and _f21_(Weierstrass) in 50 and 100 dimensions. For the _fl_(Sphere), _fl8_(Rastrigin) and _fl9_(Griewank), the performance of CNIWTLBO and NIWTLBO is similar. But the CNIWTLBO performs worse than other algorithms for the _f8_(Rosenbrock) function.  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**No.** & **Function** & & **GABC** & **IABC** & **SaDE** & **JADE** & **TLBO** & **CNIWTLBO** \\ \hline _fl_ & Sphere & Mean & 3.6E-63 & 5.34E-178 & 4.5E-20 & 1.8E-60 & 0.00E+00 & **0.00E+00** \\  & FEs:\(1.5\times 10^{5}\) & SD & 5.7E-63 & 0 & 1.9E-14 & 8.4E-60 & 0.00E+00 & 0.00E+00 \\ \hline _f4_ & Schwefel 1.2 & Mean & 4.3E+02 & 1.78E-65 & 9.0E-37 & 5.7E-61 & 0.00E+00 & **0.00E+00** \\  & FEs:\(5.0\times 10^{5}\) & SD & 8.0E+02 & 2.21E-65 & 5.4E-36 & 2.7E-60 & 0.00E+00 & 0.00E+00 \\ \hline _f5_ & Schwefel 2.22 & Mean & 4.8E-45 & 8.82E-127 & 1.9E-14 & 1.8E-25 & 0.00E+00 & **0.00E+00** \\  & FEs:\(2.0\times 10^{5}\) & SD & 1.4E-45 & 3.49E-126 & 1.1E-14 & 8.8E-25 & 0.00E+00 & 0.00E+00 \\ \hline _f6_ & Schwefel 2.21 & Mean & 3.6E-06 & 4.98E-38 & 7.4E-11 & 8.2E-24 & 0.00E+00 & **0.00E+00** \\  & FEs:\(5.0\times 10^{5}\) & SD & 7.6E-07 & 8.59E-38 & 1.82E-10 & 4.0E-23 & 0.00E+00 & 0.00E+00 \\ \hline _f15_ & Ackley & Mean & 1.8E-09 & 3.87E-14 & 2.7E-03 & 8.2E-10 & 4.44E-15 & **8.46E-16** \\  & FEs:\(5.0\times 10^{4}\) & SD & 7.7E-10 & 8.52E-15 & 5.1E-04 & 6.9E-10 & 2.58E-30 & 2.15E-31 \\ \hline _f18_ & Rastrigin & Mean & 1.5E-10 & 0.00E+00 & 1.2E-03 & 1.0E-04 & 1.88E+01 & **0.00E+00** \\  & FEs:\(1.0\times 10^{5}\) & SD & 2.7E-10 & 0.00E+00 & 6.5E-04 & 6.0E-05 & 4.65E+00 & 0.00E+00 \\ \hline _f19_ & Griewank & Mean & 6.0E-13 & 0.00E+00 & 7.8E-04 & 9.9E-08 & 0.00E+00 & **0.00E+00** \\  & FEs:\(5.0\times 10^{5}\) & SD & 7.7E-13 & 0.00E+00 & 1.2E-03 & 6.0E-07 & 0.00E+00 & 0.00E+00 \\ \hline \hline \end{tabular}  
* Population Size: **20**, D: **30**, Source: Results of algorithms except TLBO and CNIWTLBO are taken from Reference [27].  
\end{table}
Table 6: Comparative results of CNIWTLBO and the variants of ABC, DE.