# A Hybrid Adaptive Particle Swarm Optimization Algorithm for Enhanced Performance
## 3 Improved PSO Algorithm
### Mutation Strategies for Ordinary Particles  
In particle swarm optimization algorithms, ordinary particles often become trapped in local optima and rely heavily on the current optimal individual within the population, thereby neglecting the exploration of new regions of the solution space. This phenomenon can adversely affect the efficiency of the global search. It is important to note that the global optimal individual may simply be a local optimum rather than the true optimal solution for the entire population. Consequently, this dependence not only diminishes the convergence speed of the algorithm but also reduces the efficiency of the particles in conducting a global search. Due to the inherent differences among particles, various updating methods can be employed, which enhances the diversity of the population. For the updating method of standard particles, iterative updates can be performed using the differential evolution algorithm. The DE/best/1 strategy, a variant of the differential evolution algorithm, is notable for its approach of guiding the search based on the data from the best individual in the population, along with the characteristics of other individuals, thereby improving the efficiency and effectiveness of the search [30]. In this algorithm, two distinct individuals from the population are randomly selected to generate a difference vector by calculating the difference between them. This difference vector, in conjunction with the best individual in the population, serves as a mutation factor applied to the target individual, resulting in the creation of a mutated new individual. The formula for the update rate of DE/best/1 can be defined as shown in Equation (23):  
\[v_{ij}^{t+1}=gbt_{gj}^{t}+rand([0,2])(x_{r1}^{t}-x_{r2}^{t}) \tag{23}\]  
where \(gbt_{gj}^{t}\) denotes the global optimal position of the current population in dimension \(j\), \(rand([0,2])\) denotes the differential scaling factor taking values in the interval [0, 2], \(x_{r1}^{t}\) and \(x_{r2}^{t}\) denote the random unequal values in the population that satisfy \(r1\
eq r2\
eq i\), and \(t\) denotes the current iteration number. The DE/rand/1 strategy in the differential evolution algorithm involves a mutation stage where three distinct individuals from the population are randomly selected. The algorithm computes a difference vector from these individuals and linearly combines it with one of the randomly selected individuals to generate new mutated individuals, thereby enhancing the diversity of the population. Unlike the stochastic mutation approach employed in composite chaos theory, the DE/rand/1 strategy is more systematic and directed during the evolutionary process, facilitating efficient search and optimization of the particles [31]. The velocity update formula for DE/rand/1 is as follows:  
\[v_{ij}^{t+1}=x_{r1}^{t}+rand([0,2])(x_{r2}^{t}-x_{r3}^{t}) \tag{24}\]  
In this context, let \(x_{r1}^{t},x_{r2}^{t}\), and \(x_{r3}^{t}\) represent three distinct individual particles that have been randomly selected from the current population and meet the criterion \(r1\
eq r2\
eq i\). The differential evolution algorithm, particularly the DE/best/1 and DE/rand/1 variants, is employed in the velocity update formula for standard particles. During the iterative process, if the absolute value of the ratio of the current fitness to the previous fitness is equal to or exceeds 0.9, the individual particles may face the challenge of local optima, categorizing the particle in question as a mutated particle. This mutated particle is assigned a predetermined crossover mutation probability, denoted as CR. The algorithm then determines which gene loci will be directly inherited from the parent individual and which loci will undergo mutation operations, thereby enhancing the diversity of the population. The definition of crossover mutation is provided below:  
\[nx_{ij}^{t+1}= \begin{cases}v_{ij}^{t+1},\text{ if }(rand([0,\,1])<CR)\text{ or }d=rand(0,D-1)\\ x_{ij}^{t},\text{ otherwise}\end{cases}