# A Quantum-Behaved Neurodynamic Approach for Nonconvex Optimization with Constraints
## 1 Introduction  
Constrained optimization problems arise in many scientific and engineering applications including robot control [1], regression analysis [2], economic forecasting [3], filter design [4] and so on. In many real-time applications, the optimization problems are often subject to complex and time-varying nature, which makes it difficult to compute global optimal solutions in real time using traditional numerical optimization techniques [4, 5], such as Lagrange methods, descent methods and penalty function methods. A promising approach for handling these optimization problems is to employ neurodynamic optimization which is available for hardware implementation and possesses parallel and distributed computing ability. However, as pointed out in [6], the dynamic behaviors of a neural network could change drastically and become unpredictable, when applying neurodynamic optimization to deal with general nonconvex optimization problems. To compute global optimal solutions in such optimization problems, one solution is resorting to strategies in the meta-heuristics research field. Recently, neurodynamic optimization based on recurrent neural networks has attracted much focus in solving various linear and nonlinear optimization problems. The essence of such neurodynamic optimization approaches is fundamentally different from those of iterative numerical algorithms such as the sequential programming method and interior point algorithms. The gradient method has been widely used in optimization problems [7, 8]. Recurrent neural networks are also based on the gradient method and many other methods have been used to improve recurrent neural networks such as dual neural network [9, 10], projection neural network [11], delayed neural network [12], weight-constrained neural networks [3, 13] and so on. Some researchers focused on improving the performance of neural networks. [PERSON] et al. [14] addressed a high-performance feedback neural network for solving convex nonlinear programming problems. [PERSON] [15] exploreda high performance neural network model to solve chance constrained optimization problem. Some researchers give different forms of neural networks for specific problems. [PERSON] et al. presented a kind of recurrent neural network to solve quadratic programming problems and nonlinear programming problems with fuzzy parameters in [16; 17]. [PERSON] and [PERSON] presented a cooperative projection neural network to solve the least absolute deviation problems with general linear constraints [18]. [PERSON] and [PERSON] presented a two-timescale duplex neurodynamic system for constrained biconvex optimization in [19]. For some special nonconvex and nonsmooth problems, some achievements have been made in neurodynamic optimization. Based on projection method and differential inclusions, [PERSON] et al. [20] proposed a generalized neural network. In [21], a one-layer recurrent projection neural network was utilized for solving pseudoconvex optimization problems with general convex constraints. [PERSON] et al. [22] provided a one-layer recurrent neural network based on an exact penalty function method for solving nonconvex optimization problems subject to general inequality constraints. [PERSON] et al. [23] proposed a one-layer recurrent neural network for solving a class of nonsmooth, pseudoconvex optimization problems with general convex constraints. Despite great progresses of neurodynamic optimization approaches in solving optimization problems with convex or some pseudoconvex functions with constraints, it is difficult to find global optimal solutions for the optimization problems with more nonconvex functions. In addition, many neural networks seems inadequate when dealing with constrained optimization problems containing multiple local minima. As one of population-based evolutionary computation approaches, the well-known particle swarm optimization (PSO) introduced by by [PERSON] and [PERSON] [24] is a popular meta-heuristic method for global optimization with multimodal objective functions.