**4. AKQPSO**

The AKQPSO algorithm is a mixed metaheuristic algorithm, which combines the advantages of annealing krill herd (AKH) and QPSO [79]. AKH solves the disadvantage that KH cannot escape from the local optimal solution. At the same time, the efficiency of KH algorithm is improved by the simulated annealing strategy. Quantum behaved PSO (QPSO) is a new algorithm proposed by Sun et al. [79] in 2004. By introducing the quantum behavior and combining it with the idea of simulated annealing, the search ability of AKQPSO is greatly improved, and the new algorithm has better performance. The principle of QPSO is shown below.

Quantum computing is a new computing mode, which follows the laws of quantum mechanics and regulates the quantum information unit. In quantum space, when the aggregation state property is satisfied, particles can search in the whole feasible solution space, thus greatly improving the exploration search ability of QPSO. According to the analysis theory of particle convergence trajectory, if every particle can converge to its local attraction point *Pi* = (*pi*1, *pi*2 ... , *pid*), then the algorithm has the possibility of convergence. The particle position update expression of the standard QPSO algorithm is as follows:

$$\mu\_{ij}^{t+1} = p\_{ij}^t \pm \alpha \cdot \left| \mathbf{C}\_{ij}^t - \mathbf{x}\_{ij}^t \right| \cdot \ln[1/u\_{ij}^t], \\ u\_{ij}^t \sim \mathcal{U}(0, 1) \tag{10}$$

where α is the only parameter that needs to be adjusted in the algorithm, called the compression-expansion factor, which is used to control the convergence rate of particles. During the iterative process, the calculation method of individual and global optimal position is the same as that of the PSO algorithm, and the biggest difference is that the QPSO algorithm removes the speed information.

First of all, we initialize the whole population, and all the individuals in the population are randomly generated. After initialization, we divide the population into two subpopulations according to the ratio of 3:7, which is discussed in Section 5.2. The population with the proportion of 3/10 is optimized by the improved KH algorithm, which is called subpopulation-AKH, and the population with the proportion of 7/10 is optimized by the QPSO algorithm, which is called subpopulation-QPSO. The two subpopulations will be re-integrated into a population after iterative optimization. If the optimization result of the new population meets the output conditions, then the result can be output. However, if it does not meet the output conditions, then it can be re-divided according to the proportion of 3:7, and repeat the above process until the results meet the termination conditions. In the process of decentralized and re-fusion of this population, the location information of individuals is shared, so it will greatly improve the efficiency of fusion search and ge<sup>t</sup> the best results faster. In the process of the AKQPSO algorithm, we also optimized the KH algorithm by the idea of simulated annealing, named annealing KH (AKH), so we used QPSO and simulated annealing strategy to guarantee the algorithm to escape from the local optimal value. The framework of AKQPSO is shown in Figure 1 and the process of the AKQPSO algorithm (Algorithm 3) is as follows.

Determine whether the best solution meets the termination conditions, if yes, stop. Otherwise, return to step **Evaluation**.

**Figure 1.** Framework of annealing krill quantum particle swarm optimization (AKQPSO).
