# A Quantum-Behaved Neurodynamic Approach for Nonconvex Optimization with Constraints
## 3 Quantum-Behaved Neurodynamic Swarm Approach  
To solve the nonconvex programming problems with box constraints, a collective neurodynamic approach is proposed in [6], which can be seen as a combination of neurodynamic optimization and particle swarm optimization. In order to improve the convergence speed of the optimization algorithm, we explore a quantum-behaved neurodynamic approach combining the QPSO algorithm with the feedback neural network model (7) for nonconvex programming problems with general constraints. In the QPSO algorithm, each particle represents a position, which represents a potential optimal solution of optimization problem. Imitating the swarm intelligence behavior of animals, the renewal of particles is related to their optimal position and global optimal position. Unlike the basic PSO algorithm, the updating equation of particle position follows the quantum-behaved model. The QPSO algorithm has the aggregation of particles with the iteration, increases the intelligence of particle behavior, and makes the possible search range of particles wider. Therefore, the ability of global optimization increases. Given \(q\) particles, define the position of the \(i\)th particle as \(X_{i}=(X_{i1},X_{i2},\cdots,X_{in})\), the individual best position of the \(i\)th particle as \(P_{i}=(P_{i1},P_{i2},\cdots,P_{in})\) and the best position of the swarm as \(P_{g}=(G_{1},G_{2},\cdots,G_{n})\). The particle renewal equation of the QPSO algorithm is given by  
\[X_{ij}(k+1)=p_{ij}(k)\pm\beta|C_{j}(k)-X_{ij}(k)|\ \ln\frac{1}{\eta_{ij}(k)}, \tag{8}\]  
where  
\[p_{ij}(k) =\alpha_{j}(k)P_{ij}(k)+[1-\alpha_{j}(k)]G_{j}(k)\] \[C(k) =(C_{1}(k),C_{2}(k),\cdots,C_{n}(k))=\frac{1}{q}\sum_{i=1}^{q}P_{ i}(k)\] \[=\frac{1}{q}\big{(}\sum_{i=1}^{q}P_{i1}(k),\sum_{i=1}^{q}P_{i2}(k),\cdots,\sum_{i=1}^{q}P_{in}(k)\big{)}\]  
and \(\alpha_{j}(k),\eta_{ij}(k)\) are random numbers between \((0,1)\), \(j\in\{1,2,\cdots,n\}\), \(i\in\{1,2,\cdots,q\}\), \(\beta\) is an adjustable parameter, \(p_{ij}(k)\) is the center of attraction potential field of \(X_{ij}(k+1)\). \(C(k)\) is the average optimal position of particle. The iteration stops when the maximum number of iterations is reached or the given conditions is achieved. Note that, despite of its excellent global optimization ability, the QPSO algorithm lacks the ability for constraint processing and deep local searching. Inspired by neurodynamic optimization, during the local search process, we apply the feedback neural network to optimize each particle and improvethe search efficiency. Next, we explore the QNSO approach to deal with constrained optimization problems, where the QPSO algorithm is applied to adjust the initial conditions of the feedback neural network at each local search phrase. The steps of the QNSO algorithm are summarized as follows:  
1. Initialize the position of particles. 2. Initialize the individual best position \(P_{i}\) of the \(i\)th particle, \(i=1,2,\cdots,q\) and the global best position \(P_{g}\) of the particle swarm. 3. Each particle reaches a local optimal solution based on the feedback neural network (7). 4. Update the individual best position \(P_{i}\) of the \(i\)th particle, \(i=1,2,\cdots,q\) and the global best position \(P_{g}\) of the particle swarm. 5. Update the position of each particles based on (8). 6. Repeat _Step 3_ to _Step 5_ before reaching the given conditions. Following the description in Section 2, we simply consider the energy function in (3) as the fitness value of the QPSO algorithm. Then, the individual best position \(P_{i}\) of the \(i\)th particle is updated according to  
\[P_{i}(k)=\left\{ \begin{array}{ll}X_{i}(k),&\text{when }E(X_{i}(k))<E(P_{i}(k-1)),\\ P_{i}(k-1),&\text{when }E(X_{i}(k))\geq E(P_{i}(k-1)),\end{array} \right. \tag{9}\]  
where \(k-1\) and \(k\) represent the \(k-1\)th iteration and \(k\)th iteration, respectively. The global best position \(P_{g}\) of the particle swarm is updated by  
\[P_{g}(k)=\min_{1\leq i\leq q}\{E(P_{i}(k))\}. \tag{10}\]  
**Remark 4**.