## 4 Hybrid Particle Swarm Optimization  
PSO, proposed by [PERSON] and [PERSON] [36], was inspired by motions of birds or swimming fishes. PSO is a heuristic random search method with a simple principle. It easily gets into a local optimum [37] due to smaller step size, or misses the global optimum due to larger step size. For a global optimum with high efficiency, each particle should balance the exploration ability and the exploitation ability. The exploration ability contributes to global search at initial stage, while the exploitation ability focuses on local search at final stage. In order to balance the two abilities, various particle updating mechanisms are proposed such as the dynamic adaptation, the switch mechanism and the exponential form. In this paper, HPSO with self-adaptive ability is proposed. The velocity and the position of each particle in HPSO is updated as followed:  
\[\left\{ \begin{array}{l}v_{i,k}=w_{i,k}v_{i,k}+c_{1i,k}r_{1}\left(x_{b_{i,k}} -x_{i,k}\right)++c_{2i,k}r_{2}\left(x_{g_{k}}-x_{i,k}\right)\\ x_{i,k}=x_{i,k}+v_{i,k}\end{array} \right. \tag{29}\]  
In Equation (29) \(i=1,\ldots,m\) and \(k=1,\ldots,N\). \(m\) and \(N\) respectively depict the population size and the maximum iterations. The symbols \(v_{i,k}\) and \(x_{i,k}\) represent the velocity and the position of \(i\)th particle at \(k\)th iteration, respectively. In conjunction with the Section 3.3, the position \(x_{i,k}\) represents a group assignment of \((P_{1,5},\ldots,P_{n,5})\). The constants \(r_{1}\) and \(r_{2}\) are uniformly distributed in the region of \([0,1]\). \(x_{b_{i,k}}\) represents the best position of \(i\)th particle so far, while \(x_{g_{k}}\) represents the best position of the population so far. \(w_{i,k}\), \(c_{1i,k}\) and \(c_{2i,k}\) are three control parameters with self-adaptive abilities. Inspired by the increasing abilities of the exponential function, \(w_{i,k}\), \(c_{1i,k}\) and \(c_{2i,k}\) are formulated as:  
\[w_{i,k}=\left(w_{0}-w_{f}\right)\cdot e^{\left(-\frac{w_{0}-w_{f}}{g_{x_{b}}} \cdot\frac{i}{N}\right)}+w_{f} \tag{30}\]  
\[c_{1i,k}=\left(c_{10}-c_{1f}\right)\cdot e^{\left(-\frac{c_{10}-c_{1f}}{g_{x_{ b}}}\cdot\frac{i}{N}\right)}+c_{1f} \tag{31}\]  
\[c_{2i,k}=\left(c_{20}-c_{2f}\right)\cdot e^{\left(-\frac{c_{20}-c_{2f}}{g_{x_{ b}}}\cdot\frac{i}{N}\right)}+c_{2f} \tag{32}\]  
where \(w_{0}\), \(c_{10}\) and \(c_{20}\) are initial values, \(w_{f}\), \(c_{1f}\) and \(c_{2f}\) are final values. The control parameters are defined in an exponential way, in order to balance the exploration ability and the exploitation ability. Furthermore, an self-adaptive strategy is introduced to adjust each control parameter. \(g_{x_{g}}=\text{d}\left(x_{g_{k}},\,x_{b_{i,k}}\right)=\|x_{g_{k}}-x_{b_{i,k}}\|\) represents the distance between the best position of the \(i\)th particle and the global best position. As the iterations increase, \(c_{1i,k}\) and \(c_{2i,k}\) decrease while \(w_{i,k}\) increases. Therefore, each particle has high exploration ability at early stage and high exploitation ability at final stage. Furthermore, \(g_{x_{b}}\) influences the variation velocities of control parameters shown in Figure 3. Large \(g_{x_{b}}\) makes a slow variation of \(\omega\), \(c_{1}\) and \(c_{2}\), contributing to superior exploration ability of particles. However, small \(g_{x_{b}}\) contributes to fast variation of control parameters and makes particles exhibit superior exploitation ability. Actually, if the distance between a particle position and the global best position is large, searching through the whole space is required. Then, the particle moves towards the global best position as fast as possible. If the distance is small, searching through the local space and around the global best position is required. Utilizing the dynamic system theory, a stable system meets the sufficient condition that the magnitude of solution is less than 1. Namely, eigenvalues of coefficient matrix in Equation (29) is less smaller than 1.