论文标题:优化计算的神经网络模型及其稳定性分析
Neural Networks for Optimization and Their Stability Analysis
论文作者
论文导师 程立新,论文学位 博士,论文专业 基础数学
论文单位 厦门大学,点击次数 129,论文页数 77页File Size1172K
2006-05-01论文网 http://www.lw23.com/lunwen_785943737/
neural network; global stability; saddle point problem; ball-coverings
递归神经网络具有较强的优化计算能力,是目前神经计算应用最为广泛的一类神经网络模型.本文针对约束鞍点问题和球覆盖最小半径的计算问题,分别利用投影法、对目标函数加上很小“扰动函数”的逼近法、罚函数法和梯度法等建立神经网络模型进行求解,并基于LaSalle不变集原理和Lyapunov直接法等工具,对模型的动态特性进行研究,从而设计出避免陷入局部极小的优化计算神经网络模型.全文共分五章: 第一章简要回顾了优化计算神经网络研究的发展概况,以及利用神经网络求解鞍点问题和球覆盖最小半径问题的研究现状. 第二章通过投影法把Hilbert空间中的鞍点问题转化为某动态系统的平衡点问题,并利用抽象空间常微分方程理论证明了该动态系统解的存在唯一性.然后通过把LaSalle不变集原理推广到Hilbert空间,给出了平衡点的大范围渐近稳定性条件. 第三章把约束鞍点问题转化为等价的无约束问题,然后利用投影法构造了一个神经网络模型进行求解,并利用第二章结论证明了在适当条件下,模型大范围收敛于问题的精确解.本模型还可用于求解目标函数具有连续和离散变量的极小极大问题.该模型包含文献[1, 2]作为特例,推广并减弱了文献[2–5]中的稳定性和收敛性条件.仿真结果表明,该模型是有效的. 第四章通过对目标函数加上一个很小但性质较好的扰动函数,利用逼近法构造一个新的神经网络模型来求解约束鞍点问题,并证明:无须另外的凸性假设,模型均大范围指数收敛于问题的逼近解,从而能够快速求解文献[1, 2]不能求解的问题.仿真结果表明,该模型是有效的. 第五章针对Rn中球覆盖最小半径的计算问题,给出了新的计算公式,然后基于罚函数法建立一个神经网络模型进行求解,并利用Lyapunov直接法和LaSalle不变集原理证明了模型的平衡点集具有大范围吸引性且问题的(严格)极大值点等价于模型的(渐近)稳定平衡点.仿真结果表明,模型是有效的.对于球数为2n和n + 1,还分别严格计算出了最小半径值.
Recurrent neural networks have powerful computational capabilities, and are oneof the most important types in neurocomputing. By means of the projection method,adding a small disturbing function to the objective function, the penalty method, andthe gradient method, this dissertation proposes several neural networks for solvingsaddle point problems and minimum radius of ball-coverings problems, and investi-gates the stability of the proposed networks by the LaSalle invariance principle andthe Lyapunov method, so as to design neural networks which avoid getting stuck inthe local minimum. It is divided into five chapters. Chapter 1 presents a survey of the study of optimization computing by neural net-works, including the stability analysis for recurrent neural networks, and neural net-works for solving saddle point problems and minimum radius of ball-coverings prob-lems. Chapter 2 converts the solutions of saddle point problems in Hilbert spaces intothe equilibrium points of some dynamic system by means of the projection method,and gives some conditions of global asymptotic stability by extending the LaSalleinvariance principle to Hilbert spaces. The results in this chapter are the theoreticalkey links in the construction of the neural networks in the next two chapters. In chapter 3, it first proposes a neural network for constrained saddle point prob-lems by means of the projection method, then shows that the proposed network isglobally asymptotically stable under some mild conditions. The proposed networkalso can be applied to minimax problems involving discrete and continuous variables.The proposed network contains those in [1, 2] as its special cases, and the obtainedstability results extend and weaken those in [2–5]. Simulation results demonstrate theeffectiveness of the proposed network. Chapter 4 presents a neural network for saddle point problems by adding a smalldisturbing function to the objective function, and shows that the proposed network isglobally exponentially stable and the solution of the problem is approximated globallyand exponentially, without any additional convexity assumptions. Thus, it can expo-nentially solve saddle point problems, including those problems which the existing
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