# Effects of Random Values for Particle Swarm Optimization Algorithm
## 4 Experiments and Analysis
### Experimental Results and Comparisons  
For some benchmark functions, the comparisons of standard PSO algorithm with different types of random values are shown in Table 2. The bold numbers indicate the best solutions for each test function in the certain space dimension. Obviously, the performances of deterministic PSO algorithm (\(r=0.5\)) are the worst for all the benchmark functions. This is because the random values are deterministic which decreases the diversity of particles. For the random values generated by \(U[0,1]\),the standard PSO algorithm can only obtain the optimal solutions of Sphere1, Sphere2, Alpine and Moved axis parallel hyper-ellipsoid when the space dimension is 10, but this algorithm is useless for other test functions or higher space dimensions. However, for the random values generated by \(U[-1,1]\) or \(G(0,1)\), the standard PSO algorithm can obtain the optimal solutions of all test functions in every space dimension except Rosenbrock. In the low space dimension (10), the best solution of Rosenbrock is obtained by the standard PSO algorithm with random values distributed in \(U[0,1]\). However, in the high space dimension (30 or 100), its best solution is obtained by the standard PSO algorithm with random values generated by \(U[-1,1]\) or \(G(0,1)\). In addition, the best solutions of Levy and Montalvo 2 (30 dimensions) and Sinusoidal (100 dimensions) are also obtained by the standard PSO algorithm with random values distributed in \(U[0,1]\). Obviously, the random values make an important effect on the performance of standard PSO algorithm, and its performance is highly improved when the range of random value is expanded. This implies that the standard PSO algorithm with large-scale random values can avoid falling into local optima and obtain the global optima. The comparisons of LDIW-PSO algorithm with different types of random values for benchmark functions are shown in Table 3. In addition, the bold numbers indicate the best solutions for each  
Figure 4: Random values with Gauss distribution. test function in the certain space dimension. For all the benchmark functions, the performances of deterministic LDIW-PSO algorithm (\(r=0.5\)) are also the worst. For the \(U[-1,1]\) and \(G(0,1)\), the performance of LDIW-PSO algorithm is similar with that of standard PSO algorithm. However, for the random values distributed in \(U[0,1]\), the performance of LDIW-PSO algorithm is improved compared to the standard PSO algorithm, which is also reported in some references [5; 20; 45; 46]. It should be noted that the LDIW-PSO algorithm with random values generated by \(U[0,1]\) can obtain the optimal solutions of all test function in low space dimension (10 and 30) except Sphere2, Rotated Expanded Scaffer and Schwefel. In addition, for Levy and Montalvo 2, Sinusoidal and Alpine in 30 dimensions, the performance of LDIW-PSO algorithm with random values generated by \(U[0,1]\) is better than that with random values generated by \(U[-1,1]\) or \(G(0,1)\). However, in the high space dimension (100), the LDIW-PSO algorithm with random values generated by \(U[0,1]\) cannot obtain the optimal solutions of these test functions, which implies that this algorithm is useless for the problems in high dimension space. Therefore, the performance of improved PSO (LDIW-PSO) algorithm is also influenced by the random values, especially for solving the problems in high dimension space. Furthermore, the LDIW-PSO algorithm with a wide range of random values is more beneficial to escape from local optima and obtain the global optima. Figure 5 shows the mean best fitness of the standard PSO algorithm with different types of random values for benchmark functions. For the same type of random values, the performance of the standard PSO algorithm improves with decreasing the space dimension, and the convergence velocity also improves with decreasing the space dimension. For the same space dimension of each test function, the performances of deterministic PSO algorithm (\(r=0.5\)) are the worst.