# A Brain Storm and Chaotic Accelerated Particle Swarm Optimization Hybridization
## 2 Background
### The BSO Algorithm
On the other hand, the cluster center is defined after the clusters are obtained as the solution with the fittest objective function evaluation. ### Chaotic Accelerated Particle Swarm Optimization (CAPSO)

The original PSO algorithm [15; 16] is inspired by the swarm intelligence observed in the behavior of swarms of birds in search for food. Swarms of birds (and other species, e.g., schools of fish) have the ability to cooperate instead of compete in order to find food sources. The members of the swarm have the ability to maintain cohesion, alignment and separation, while they also exhibit cognitive behaviors. They have memory for their own past success during the exploration process, while they can also communicate useful information to each other regarding food allocation. Usually, in PSO algorithms, we do not refer to animals, but particles, meaning points (vectors) in the \(D\)-dimensional solution space. Each particle, \(i\), represents a member of the swarm with position \(\mathbf{x}_{i}\) and velocity \(\mathbf{v}_{i}\), both being vectors of \(\mathbb{R}^{D}\). In the original PSO algorithm, each particle takes into account both the global best position, \(\mathbf{g}\), and its individual (local) best position, \(\mathbf{x}_{i}^{*}\), when it updates its velocity, \(\mathbf{v}_{i}\). The updated velocity is then used to update the position \(\mathbf{x}_{i}\). So, the update scheme of PSO is performed in two steps. The accelerated PSO (APSO) algorithm was proposed by [PERSON] in 2008 [21; 22]. Since the individual best of PSO is mainly used for necessary diversity, in the APSO algorithm this is instead simulated by randomness. Additionally, in APSO, each particle's position updates in a single step (contrasting the two steps of PSO) as follows:

\[x_{new,i}^{d}=(1-\beta)x_{i}^{d}+\beta g^{d}+\alpha\mathcal{N}(0,1), \tag{8}\]

where \(\beta\), commonly taken in \([0.2,0.7]\), is referred to as the attraction parameter for the global best and \(\alpha\), multiplied by a probability distribution, offers useful randomness to the updates. Here, a Gaussian random distribution is chosen. Moreover, it has been shown that a decreasing \(\alpha\) is beneficial for the algorithm since, in this manner, it controls the exploration-exploitation trade-off more adequately during the iterative process. To this end, a strictly-decreasing function, \(\alpha(t)\), was chosen, with a commonly used one being

\[\alpha(t)=\gamma^{t}, \tag{9}\]

where \(\gamma\in(0,1)\) is a control parameter and \(t\) refers to time (the current iteration). It is important to fine-tune \(\alpha(t)\) to the nature of the optimization problem and search area [17; 21; 22]. The chaotic APSO (CAPSO) [17] is a variant of the APSO algorithm. In the CAPSO algorithm, the same single-step routine with APSO is utilized (i.e., Equation (8)), but a varying \(\beta\) is deemed beneficial for improved performance. This global attraction parameter is updated through a chaotic map. In [17], many chaotic maps were tested, and the most advantageous results stemmed from the sinusoidal and singer maps. Here, a simplified sinusoidal map was chosen for \(\beta\), particularly

\[\beta_{k+1}=\sin(\pi\beta_{k}),\quad\beta_{0}=0.7. \tag{10}\]

Algorithm 3 contains simple pseudocode for CAPSO. ```
Set parameters;  Initialize the population with \(N\) randomly generated particles;  Initialize iteration number, \(t\); whileT is not reacheddo  Evaluate the particles of the population;  Find the global best, \(g^{*}\), for the current iteration;  Update \(\alpha\) through a decreasing function;  Update \(\beta\) through a chaotic map; for each particle in \(N\)do  Update particle's position;  end while  Update \(t\);  end while
```

**Algorithm 3**CAPSO Algorithm

## 3 The BSO-CAPSO Hybrid Concept

The BSO-CAPSO hybrid approach is fairly simple. Since BSO has better initial exploration compared to PSO [33], we can safely assume that this could be similar for CAPSO.