## 2 Accelerated Particle Swarm Optimization and Its Improved Version  
Particle swarm optimization is now a very famous metaheuristic optimization algorithm that has shown promising results when implemented on many types of complex, non-linear, and multi-modal optimization problems. The canonical version of PSO is given in Equation (1) as:  
\[\begin{split} v_{i}^{n+1}&=\theta v_{i}^{n}+\alpha \epsilon_{1}\odot[g^{*}-x_{i}^{n}]+\beta\epsilon_{2}\odot[x_{i}^{*}-x_{i}^{n}] \\ x_{i}^{n+1}&=x_{i}^{n+1}+v_{i}^{n+1}\end{split} \tag{1}\]In Equation (1), \(v_{i}^{n+1}\) is the velocity update of particle \(i\), at iteration \(n+1\). This velocity is added to the particle position \(x_{i}^{n}\) of particle \(i\), at iteration \(n\), to get the updated position \(x_{i}^{n+1}\) of particle \(i\). \(x_{i}^{*}\) is the local best position of particle \(i\), in iteration history, and \(g^{*}\) is the global best among all the particles, at iteration \(n\). \(\epsilon_{1}\) and \(\epsilon_{2}\) are random number ranging from 0-1.  
There are almost more than two dozen variants of this canonical version of the PSO algorithm, as discussed in [29; 30]. Accelerated particle swarm optimization is its variant, presented in [1; 2], which is very elegant, i.e., easy to understand, the easiest to program, and proven itself to give good approximates to global optimum solutions. Its single step particle update equation is given as Equation (2):  
\[x_{i}^{n+1}=(1-\beta)x_{i}^{n}+\beta g^{*}+\alpha(\epsilon-0.5) \tag{2}\]  
As can be seen, this single step update equation has ignored the use of the velocity update and did not use the local best position in the updating of particle \(i\). The typical values of \(\alpha\) and \(\beta\) are 0.2 and 0.5, respectively. \(\epsilon\) is a random variable, and its value is between zero and one, as given in [1; 2]. This variant has been proven to find the global optimum of highly complex multi-modal functions like the Michaelwics2D function, as presented in [1; 2]. There can be many possible variants of Equation (2), as discussed in [1; 2]. The variant discussed in this paper is related to the dynamic search space squeezing method in which, at every iteration, the search space of the particles, in all dimensions, is readjusted by taking the influence of the global best particle, i.e., of \(g^{*}\), each iteration. This is how the chance of each particle to oscillate in the search space is decreased, and there is a lesser chance to get stuck to local optima. By reducing the search space, the range of particles at each dimension is reduced, i.e., the constraints are readjusted, and new constraints are bounded by the original constraints of the optimization problem, by using the dynamic search space squeezing Equations (3)-(7), as presented in [25]. The work in [25] has applied the concept of the dynamic search squeezing technique on the canonical form of PSO, i.e., on Equation (1), on the velocity update equation. These equations help to squeeze the search space dynamically from the given constraints to newer constraints from one iteration to the next iteration. This paper presents this concept of dynamic search space squeezing on the APSO algorithm, i.e., Equation (2), to further enhance its performance. Therefore, rather than squeezing the search space of velocity terms, the search space of particles themselves will be directly squeezed. The dynamic search space squeezing phenomenon is shown in Figure 1.  
\[\Delta_{low,i}^{n}=\frac{g_{i}^{*n}-x_{i,min}^{n}}{x_{i,max}^{n}-x_{i,min}^{n}} \tag{3}\]  
\[\Delta_{high,i}^{n}=\frac{x_{i,max}^{n}-g_{i}^{*n}}{x_{i,max}^{n}-x_{i,min}^{n}} \tag{4}\]  
\[\Delta_{low,i}^{n}+\Delta_{high,i}^{n}=1 \tag{5}\]  
\[x_{i,min}^{n+1}=x_{i,min}^{n}+(g_{i}^{*n}-x_{i,min}^{n})(\Delta_{low,i}^{n}) \tag{6}\]  
\[x_{i,max}^{n+1}=x_{i,max}^{n}+(x_{i,max}^{n}-g_{i}^{*n})(\Delta_{high,i}^{n}) \tag{7}\]  
Figure 1: Dynamic search space squeezing under the influence of \(g^{*}\) at iteration \(n+1\). The arrows indicate the squeezing of the search space from iteration \(n\)–\(n+1\).