# An Overview of Variants and Advancements of PSO Algorithm
## 4 Advances on Particle Swarm Optimization (PSO) Algorithm
### Theoretical Analysis on PSO  
By realizing the importance of the appropriate values of the control parameters in the PSO algorithm, such as the inertia weight, acceleration coefficients, number of particles, etc., the researchers diverted their attention towards the theoretical aspects of the PSO algorithm and derived many theories related to parametric selection to obtain an optimal solution in a lower number of iterations and with a higher rate of convergence. Their theoretical analysis includes parametric setting, convergence analysis, etc. [PERSON] and [PERSON] [14] had proposed a new PSO parameter, called inertia weight '\(\omega\)' into the original particle swarm optimizer to improve its convergence performance. The authors explained the impact of this new parameter through examples. [PERSON] and [PERSON] [44] investigated the convergence performance of the particle swarm optimization (PSO) algorithm with a linearly decreasing inertia weight, and tested it on four non-linear functions. On the basis of the results, the authors concluded that some of the disadvantages of PSO could be overcome by adjusting the inertia weight. By obtaining motivation from the Gaussian and Cauchy probability distribution in the PSO algorithm, the authors [PERSON] and [PERSON] [45] proposed a modified variant of the PSO, namely PSO-E, that made use of the exponential probability distribution to upgrade the convergence performance. The numerical results on well-known classical functions indicated the competence of PSO-E in finding the solution in a lower number of iterations. [PERSON] et al. [46] proposed a new adaptive inertia weight approach in the particle swarm optimization (PSP) algorithm. The performance of this newly proposed variant of PSO was checked on three classical benchmark functions and the results were also analyzed to prove its higher speed of convergence and more capability in locating the optimal solution in comparison to other optimization techniques. In order to balance between the local exploitation and the global exploration abilities for the PSO variant, [PERSON] et al. [47] introduced a modified variant of PSO, called the adaptive inertia weight PSO algorithm (AIW-PSO). The authors proved the efficiency of the proposed algorithm (AIW-PSO) by comparing its results with other modified variants of the PSO, namely fuzzy adaptive inertia weight PSO, random number inertia weight PSO and linearly decreasing inertia weight PSO. The results showed that this new algorithm, AIW-PSO, outperformed the other algorithms. Keeping in mind the importance of the parameter inertia weight in the PSO, [PERSON] et al. [48] proposed a dynamic inertia weight particle swarm optimization algorithm. The authors tested their improved PSO on six classical problems and concluded that the proposed variant improved the search performance of the standard particle swarm optimization (SPSO) algorithm. Inertia weight is considered to be the most important parameter in the PSO algorithm. Usually, \(\omega\) is chosen to be of small value in the range [0, 1] but [PERSON] et al. [49] considered varying the values of \(\omega\), and proposed two variants of PSO, namely the globally adaptive inertia weight (GAIW) PSO and the locally adaptive inertia weight (LAIW) PSO. By testing these variants on various benchmark problems, the authors concluded that the proposed variant outperformed the basic PSO in terms of accuracy, efficiency and convergence speed. To improvise the convergence speed and to overcome the stagnation behavior of the basic PSO algorithm, [PERSON] et al. [50] proposed a new variant of the basic PSO called fine grained inertia weight PSO (FGIWPSO). By testing their proposed variant of PSO on ten benchmark functions, the authors showed its effectiveness over the basic PSO, in terms of the quality of the solution obtained and the rate of convergence.