# An Enhanced Particle Swarm Optimization (PSO) Algorithm Employing Quasi-Random Numbers
## 1 Introduction
The approaches discussed in this paper [9], including the inertia weight technique, mutation operator, opposition-based learning, hybridization with other metaheuristics, and dynamic parameter tuning, can be used individually or in combination to address the limitations of PSO. [PERSON] and a team [10] of researchers conducted a comprehensive survey on PSO. The survey discusses techniques such as varying the inertia weight and hybridizations, which are discussed above. The survey also states that the ability of PSO to be hybridized with other optimization algorithms has contributed to its popularity. Another technique described in the survey is velocity clamping [10], a technique introduced by [PERSON] and [PERSON]. Velocity clamping involves setting bounds for the values of the velocities of the particles in all the dimensions. Another approach to improving the efficiency of the PSO algorithm discussed in this paper [10] is varying the controlling parameters, such as using the varying inertia weight technique in which inertia weight changes throughout the optimization process, or acceleration coefficient techniques in which the two constant controlling parameters for PSO other than the inertia, are chosen in different ways to yield optimal solutions while evading premature convergence. Many other approaches have been discussed both in the survey and elsewhere. The choice of approach depends on the problem characteristics and the available computational resources. However, most of these approaches can provide problem-dependent solution methods. In this paper, we proposed a new approach to replace the random numbers used for this method with quasi-random numbers [11; 12; 13], like Halton and Sobol, by maintaining the k-dimensional uniformity of these quasi-random numbers. This not only provides a generalized approach to any optimization problem, but this method can be used in conjunction with the earlier enhancement techniques like the inertia weight technique, mutation operator, opposition-based learning, hybridization with other metaheuristics, and dynamic parameter tuning. In this research, two enhanced versions of PSO (one using Sobol random numbers and the other using Halton random numbers) were proposed with the intention of speeding up the convergence of the standard PSO algorithm. To test the efficiency improvement of the two proposed enhancements of the standard PSO algorithm, the number of iterations taken to achieve the optimum of the well-known cigar, ellipsoid, and paraboloid functions [14], along with the number of iterations taken to obtain an optimal path for the famous Travelling Salesman Problem (TSP) [15; 16; 17; 18], were noted. Following this, improvement in terms of the optimum of the objective function and the number of iterations needed to reach the global optimum were calculated for both the PSO enhanced with Sobol random number samplings and the PSO enhanced with Halton random number samplings, with respect to the standard PSO, which uses Monte Carlo random number samplings. All the results for each benchmark function and TSP unanimously show efficiency improvement due to the use of the Sobol and Halton sequences. Additionally, we noted that the more decision variables an optimization problem has, the improvement due to Sobol and Halton sequences increases. In conclusion, both the enhancements of the standard PSO presented in this research, one utilizing Sobol random numbers and the other utilizing Halton random numbers, consistently show efficiency improvement and a better optimum, meaning that they successfully have increased the speed of convergence of the standard PSO algorithm. In addition, we have also compared our enhanced PSO with the SALP meta-heuristic variant [19]. This is a recent algorithmic approach developed to improve the convergence rate. Our enhancement is compared with the SALP to see whether the approach of avoiding clusters in random number generation can make the enhanced PSO algorithm perform better than the SALP algorithm.