近年来，国内外学者发表了许多关于线性代数问题实时求解的方法，其中包括了矩阵求逆和线性方程组的并行求解方法。在研究了基于梯度法的递归神经网络用于Sylvester矩阵方程的实时求解后，通过使用Kronecker乘积和矩阵向量化等技术进行了MATLAB仿真从而验证了相关理论分析。计算机仿真的结果证实了这类神经网络方法在解决Sylvester矩阵方程中的有效性和高效率（特别是在使用幂S型激励函数的情况下）。

In view of parallel-processing nature and circuit-implementation convenience, recurrent neural networks are often employed to solve optimization problems. Recently, a primal-dual neural network based on linear variational inequalities (LVI) was developed by Zhang et al. for the online solution of linear-programming (LP) and quadratic-programming (QP) problems simultaneously subject to equality, inequality and bound constraints. For the final purpose of field programmable gate array (FPGA) and application-specific integrated circuit (ASIC) realization, we investigate in this paper the MATLAB Simulink modeling and simulative verification of such an LVI-based primal-dual neural network (LVI-PDNN). By using click-and-drag mouse operations in MATLAB Simulink environment, we could quickly model and simulate complicated dynamic systems. Modeling and simulative results substantiate the theoretical analysis and efficacy of the LVI-PDNN for solving online the linear and quadratic programs.

One of state-of-the-art recurrent neural networks (RNN) is dual neural network (DNN). It can solve quadratic programs (QP) in real time. The dual neural network is of simple piecewise-linear dynamics and has global (exponential) convergence to optimal solutions. In this chapter, we firstly introduce the QP problem formulation and its online solution based on recurrent neural networks. Some related concepts and definitions are also given. Secondly, we present the dual neural network and its design method. In addition to the general design method, for non-diagonal, non-analytical and/or time-varying cases, a matrix-inverse neural network could be combined into such a design procedure of dual neural network for online computation of its matrixinverse related term. Thirdly, we show the analysis results of dual neural networks. In addition to the general analysis results, we investigate the proof complexity of the exponential convergence condition of dual neural networks. Fourthly, we present the numerical simulation and illustrative example of using the dual neural network to solve static QP problems. Finally, we exploit the dual neural network to online solve motion planning problems of redundant robot manipulators, which is illustrated as engineering-application examples.

A recurrent neural network called the dual neural network is proposed in this Letter for solving the strictly convex quadratic programming problems. Compared to other recurrent neural networks, the proposed dual network with fewer neurons can solve quadratic programming problems subject to equality, inequality, and bound constraints. The dual neural network is shown to be globally exponentially convergent to optimal solutions of quadratic programming problems. In addition, compared to neural networks containing high-order nonlinear terms, the dynamic equation of the proposed dual neural network is piecewise linear, and the network architecture is thus much simpler. The global convergence behavior of the dual neural network is demonstrated by an illustrative numerical example. Ù 2002 Elsevier Science B.V. All rights reserved.

为解决在速度层上无穷范数最小化模型中可能出现的不连续点问题，提出一种基于双判据方法的二次型优化模型。冗余机器人运动规划与控制模型可以统一各种关节物理极限，如关节变量极限与关节速度极限。同时该模型又可以最终转化为一个标准的二次规划问题。为了实时求解该二次规划问题，提出一种基于线性变分不等式（LVI）的原对偶神经网络。该神经网络作为实时求解器具有简单的分段线性结构和较高的计算效率。计算机对PUMA560机器手臂的模拟仿真表明，该方案具有灵活性和有效性。