## 2 Algorithms
### Dimension-Wise PSO  
The Dimension-wise PSO (DPSO) algorithm has been designed with the intention of handling relatively large dimensional spaces with a sparse population size. This is accomplished by determining the global and local attraction points separately for each dimension. DPSO's velocity is defined as  
\[\vec{v}_{t+1}=c_{0}\cdot\vec{v}_{t}+c_{1}\cdot\overrightarrow{rand}(-1,1) \cdot|\vec{v}_{t}|+c_{2}\cdot\vec{b}_{3}\cdot(\vec{p}_{\text{global}}-\vec{p }_{t})+c_{4}\cdot\vec{b}_{5}\cdot(\vec{p}_{\text{local}}-\vec{p}_{t}), \tag{7}\]  
and is also subject to dimension-wise limitations  
\[v_{t+1}= \begin{cases}v_{\text{lim}}&\text{if }v_{\text{lim}}<v_{t+1},\\ -v_{\text{lim}}&\text{if }-v_{\text{lim}}>v_{t+1},\\ v_{t+1}&\text{otherwise},\end{cases}  \tag{8}\]  
followed by Equation (5). As with PSO, the individual's position update relies on Equation (4) subject to Equation (6). Percent noise injection--regulated by \(c_{1}\)--is used as a precaution to improve variations in movement and increase the likelihood of escaping from local minima without causing a large change in course. Noise injection replaces the random values affecting attractor strengths and allows it to have an effect independent of a particle's distance from the optima. Visually, it can be likened to roughening a surface dependent on the speed of a ball rolling down its surface, causing the said ball to deviate from a trivial path. The trajectory and speed are often disrupted, but the general direction of travel is not necessarily changed as it has three other elements that are largely responsible for determining velocity, i.e., momentum, global best attraction, and local best preference. The velocity limit, \(v_{\text{lim}}\), partially opposes the effect of percent noise injection, as it attempts to restrict the overshooting caused by an excessive build up of momentum and attractive strength along a given dimension. Situations where a particle builds up too much momentum and is slingshot further away from the known optimal locations can help with increasing diversity in exploration, but scattering particles away from the optimal attractors can also impair the rate of convergence [16]. To prevent an excessive expansion in the required search parameters, \(v_{\text{lim}}\) is simplified to  
\[v_{\text{lim}}=c_{6}\cdot(p_{\text{max}}-p_{\text{min}}), \tag{9}\]  
where \(c_{6}\) is the percentage of a given dimension's span a particle can traverse in one iteration, and \(p_{\text{max}}\) and \(p_{\text{min}}\) are the position boundaries bracketing the valid range of exploration along the given dimension. The primary change that gives DPSO the ability to search on a per-dimension basis is that, in Equation (7), the uniform random attraction values are replaced with probabilistic binary values, \(b_{3}\) and \(b_{5}\), respectively. These values determine if the individual should move toward the respective best position of a given dimension. The activation of \(b_{3}\) and \(b_{5}\) can be described as  
\[b_{n}= \begin{cases}1&\text{if }rand(0,1)<c_{n},\\ 0&\text{otherwise},\end{cases}  \tag{10}\]  
where \(c_{3}\) and \(c_{5}\) denote the respective probability of activation. It must be stressed that \(b_{3}\) and \(b_{5}\) are evaluated on a per-dimension basis, as opposed to being a single value applied to each dimension. Although this discretized approach reduces the coverage of points between the two known best locations, it increases the exploration of points in  
Figure 1: Flow diagrams depicting the process of each algorithm. the surrounding neighborhood (see Figure 2). When momentum and noise injection are accounted for, the individual is less likely to become fixated on a single vector and can partially explore the area around each point. Without noise or momentum, the probability of moving to a given point is based on the products of \(b_{3}\), \(b_{5}\), \(1-b_{3}\), and/or \(1-b_{5}\) for each dimension.