# A Quantum-Behaved Neurodynamic Approach for Nonconvex Optimization with Constraints
## 1 Introduction
In the PSO algorithm, a simple velocity and displacement model to adjust the position according to the group optimal value and its own optimal value. It has strong global search ability and robustness. However, the PSO algorithm has at least three shortcomings. Firstly, it has been proved by [PERSON] in [25] that, when the number of iterations tends to infinity, the traditional PSO algorithm can not converge to the global optimal solution with probability 100%. Secondly, in the PSO algorithm, the velocity of particles has an upper limit to ensure that particles can aggregate and avoid divergence, which limits the search space of particles and thus makes the algorithm ineffective to jump out of local optimal solutions. Thirdly, as the evolution equation of the PSO algorithm is based on a set of simple state equations of velocity and position, which makes the randomness and swarm intelligence relatively low. Due to its simple calculation, easy implementation, and few control parameters, many improved PSO algorithms have been to overcome the drawbacks (e.g., see [26; 27; 28]). [PERSON] et al. [6] presented a collective neurodynamic optimization approach by combining the traditional PSO algorithm and a projection neural network to solve nonconvex optimization problems with box constraints. Later, [PERSON] et al. [29] apply an adaptive PSO algorithm together with a one-layer recurrent neural network to solve nonconvex general constrained optimization problems. Among the improved PSO algorithms, the quantum-behaved particle swarm optimization (QPSO) algorithm introduced in [30] is also one promising alternate, which describes particle behavior with probability. Compared with the traditional PSO algorithm, the QPSO algorithm has certain advantages in global optimization and speed of convergence. The convergence proof of QPSO is also given in [27]. Applying the QPSO algorithm for global search intermittently by incorporating a novel feedback neural network for local search in a hybrid mode can be a good choice. In this paper, we propose a quantum-behaved neurodynamic swarm optimization (QNSO) approach combined with an efficient neural network and the QPSO algorithm to solve the nonconvex optimization problems with constraints. We improve an efficient feedback neural network proposed in [14] and prove the global convergence of the neural network for a class of nonconvex optimization problems. Inspired by [6; 14; 29], we employ the feedback neural network and present a quantum-behaved neurodynamic swarm approach for solving the nonconvex optimization problems with inequality constraints efficiently. The proposed QNSO approach combining the QPSO algorithm [30] and the feedback neural network is applied to deal with constrained optimization problems with multiple global minima. Compared with the methods in [6; 29], it shows a better convergence performance. Both numerical examples and practical applications are provided to demonstrate the effectiveness of the QNSO approach. This paper is organized as follows. In Section 2, a constrained optimization problem is described and some theoretical analysis of an improved neural network is given. In Section 3, combined with the QPSO algorithm and the proposed neural network, the QNSO approach is developed for solving the constrained optimization problems. In Section 4, we perform experiments on two multimodal functions to demonstrate the performance of the QNSO algorithm. In Section 5, we apply the QNSO method to the optimization problems of the hollow transmission shaft, heat exchangers and crank-rocker mechanism. Finally, Section 6 concludes the paper.