# Population Symmetrization in Genetic Algorithms
## 3 Experiments
### Numerical Test Results  
Figure 4 shows the graphs of the empirical cumulative distribution functions of the measured running time for the discussed four algorithms, averaged over the entire set of 24 BBOB benchmark test functions [42], successively for the dimensions \(D=2,\ 3,\ 5\). The detailed statistical data concerning performed numerical experiments are given in Appendix A (Table A1). Detailed analysis of cumulative distribution charts for the dimension \(D=5\) was performed in Figure 4.
* Having a computing budget, e.g., \(5\times 10^{5}\) calculations of the objective function, the GA algorithm solves approx. A total of 58% of problems (finds the global minimum for 58% of test functions with the assumed accuracy \(\epsilon=10^{-8}\)). With the same computing budget, the GASO algorithm solves 90% of problems. So we can conclude that with the above-mentioned calculation budget, GASO is 32 percentage points more effective than GA. Illustration on the chart: green dashed lines. Figure 4: Empirical cumulative distribution functions of the measured runtime (the number of calculations of the objective functions divided by dimension of search space) for four algorithms averaged for the entire set of 24 test functions of the BBOB benchmark successively for dimensions 2 (**a**), 3 (**b**) and 5 (**c**). The considered algorithms are: GA—reference generic algorithm with standard crossover, GASO—reference genetic algorithm with segment crossover, GASO—genetic algorithm with symmetrization operator and standard crossover, GASOSC—genetic algorithm with symmetrization operator and segment crossover. * Solving 60% of the problems needs \(5\times 10^{3.7}\) calculations of the objective function for GASO while using GA it requires \(5\times 10^{5.3}\) function calculations. GASO requires almost 40 times less computations of the objective function than the reference GA to solve 60% of the optimization problems considered. GASO is therefore 40 times faster than GA in solving 60% of problems. Illustration on the chart: blue dotted lines. Thus, the graphs show a qualitatively greater efficiency of the algorithm with the symmetrization operation in relation to the standard genetic algorithm. Due to the logarithmic scale in the graphs, a more accurate measurement of the difference in \"optimization time\" by GASO and GA seems pointless. The same phenomenon occurs in higher dimensions (\(D=10,\ 20,\ 40\)). Additionally, a synthetic measure of the effectiveness \(m_{e}\) of the analyzed algorithm \(A\) was introduced as the arithmetic mean of the values of the four cumulative values at the indicated points. In low dimensions (\(D=2,\ 3\)), most of the tested functions are optimized by algorithms with the symmetrization operation in the interval \(\langle 3;\ 6\rangle\) (interval on the axis with a logarithmic scale), while in higher dimensions, the number of calculations of the objective function value increases significantly (most of the tested functions are optimized in the interval \(\langle 4;\ 7\rangle\)). Therefore, we adopt the following definition of the algorithm efficiency measure:  
\[m_{e}(\,\mathrm{A})=\left\{ \begin{array}{ll}\frac{F(3)+F(4)+F(5)+F(6)}{4} \cdot 100\%,&D=2,3\\ \frac{F(4)+F(5)+F(6)+F(7)}{4}\cdot 100\%,&D\geq 5\end{array} \right. \tag{16}\]  
The method of calculating the efficiency measure so defined is illustrated in Figure 5.  
The calculated measures of effectiveness for the considered algorithms for all test functions can be found in Figure 6.  
Figure 5: The method of converting the values read from the graph of the empirical cumulative distribution function of operating time measured with the number of objective function calculations for the GASOSC algorithm, averaged for the entire set of 24 test functions of the BBOB benchmark for dimension 5 into a synthetic measure of the algorithm effectiveness \(m_{e}\) according to the given formula. The considered algorithm is GASOSC—a genetic algorithm with a symmetrization operator and segment crossover.