光滑神经网络解决非李普西茨优化问题的研究

J4 ›› 2015, Vol. 37 ›› Issue (12): 2262-2269.

光滑神经网络解决非李普西茨优化问题的研究

喻昕,谢缅,李晨宇

（广西大学计算机与电子信息学院，广西南宁 530004）

收稿日期:2014-10-28 修回日期:2015-03-23 出版日期:2015-12-25 发布日期:2015-12-25
基金资助:
国家自然科学基金资助项目（61462006）；广西区自然科学基金资助项目（2014GXNSFAA118391）

Solving non-Lipschitz optimization problems by
smoothing neural networks

YU Xin,XIE Mian,LI Chenyu

(School of Computer & Electronics and Information，Guangxi University,Nanning 530004,China）

Received:2014-10-28 Revised:2015-03-23 Online:2015-12-25 Published:2015-12-25

摘要/Abstract

摘要：

为寻求满足约束条件的优化问题的最优解，针对目标函数是非李普西茨函数，可行域由线性不等式或非线性不等式约束函数组成的区域的优化问题，构造了一种光滑神经网络模型。此模型通过引进光滑逼近技术将目标函数由非光滑函数转换成相应的光滑函数以及结合惩罚函数方法所构造而成。通过详细的理论分析证明了不论初始点在可行域内还是在可行域外，光滑神经网络的解都具有一致有界性和全局性，以及光滑神经网络的任意聚点都是原始优化问题的稳定点等结论。最后通过几个简单的仿真实验证明了理论的正确性。

关键词: 非李普西茨函数, 光滑逼近技术, 稳定点, 聚点

Abstract:

In order to seek to optimal solution satisfying the necessary conditions of optimality, aiming at the optimization problems that objective functions are nonLipschitz and the feasible region consists of linear inequality or nonlinear inequality, we design a new smooth neural network by the penalty function method and the smoothing approximate techniques which convert nonsmoothing objective functions into smoothing functions. Detailed theoretical analysis proves the uniform boundedness and globality of the solutions to smooth neural networks, regardless of the initial points inside or outside of the feasible domain. Moreover, any accumulation point of the solutions of to smooth neural networks is a stationary point of the optimization promble. Numerical examples also demonstrate the effectiveness of the method.

Key words: non-Lipschitz , function;smoothing approximate techniques;stationary point;accumulation point

喻昕,谢缅,李晨宇. 光滑神经网络解决非李普西茨优化问题的研究[J]. J4, 2015, 37(12): 2262-2269.

YU Xin,XIE Mian,LI Chenyu. Solving non-Lipschitz optimization problems by
smoothing neural networks [J]. J4, 2015, 37(12): 2262-2269.

参考文献

［1］Gao Yun.The neural network for soving optimation problems［D］.Wuxi:Jiangnan University，2011．(in Chinese)
［2］Tank D W,Hopfield J J.Simple neural optimization networks:An a/dconverter,signal decision circuit,and a linear programming circuit［J］.IEEE Transactions on Circuits System，1986，33（5）:533541．
［3］Kennedy M,Chua L.Neural networks for nonlinear programming［J］.IEEE Transactions on Circuits System,1988，35（5）:554562.
［4］Huang Yuancan.Lagrangetype neural networks for nonlinear programming problems with inequality constraints ［C］∥Proc of the 44th IEEE Conference on Decision and Control，and the European Control Conference,2005:41294133．
［5］Huang Yuancan.A new type of Lagrange nonlinear programming neural network［J］.ACTA Electronica Sinica，2002，30（1）:2729．(in Chinese)
［6］Li Guocheng,Song Shiji,Wu Cheng.Generalized gradient projection neural networks for nonsmooth optimization problems［J］.Science China Information Sciences,2010,53（5）:9901005．
［7］Liu Qingshan,Wang Jun.A onelayer recurrent neural network for constrained nonsmooth optimization［J］.IEEE Transactions on Systems,Man,and Cybernetics—PART B:Cybernetics，2011，41（5）:13231333．
［8］Liu Qingshan，Tang Chuangying，Huang Tingwen.A onelayer recurrent neural network for realtime portfolio optimization with probability criterion［J］.IEEE Transactions on Cybernetics，2013，43（1）:1423．
［9］Chen Xiaojun.Smoothing methods for nonsmooth nonconvex minimization［J］.Mathematical Programming，2012，134（1）:7199．
［10］Bian Wei,Chen Xiaojun.Smoothing neural network for constrained nonLipschitz optimization with application［J］.IEEE Transactions on Neural Network and Learning Systems,2012,23（3）:399411．
［11］Forti M,Nistri P,Quincampoix M.Generalized neural network for nonsmooth nonlinear programming problems［J］.IEEE Transactions on Circuits and Systems—I，2004，41（5）:17411754．
［12］Han Zhengzhi,Cai Xiushan,Huang Jun.The theory for control systems described by differential conclusions［M］.Shanghai:Shanghai Jiao Tong University Press，2013．(in Chinese)
［13］Filippov A F.Differential equations with discontinuous righthand side［M］∥Mathematics and Its Applications （Soviet Series）.Boston,MA:Kluwer,1988．
［14］Betounes D.Differential equations:Theory and applications［M］.New York:SpringerVerlag,2009．
［15］Aubin J P,Cellina A.Differential inclusion:Setvalued maps and viability theory［M］.New York:SpringerVerlag,1984．
［16］Ham F M,Kostanic I.Principles of neurocomputing for science & engineering［M］.Ye Shiwei，Wang Haijuan Translated.Beijing:China Machine Press，2007．(in Chinese)
附中文参考文献
［1］高云.求解优化问题的神经网络方法［D］.无锡:江南大学，2011．
［4］黄远灿.一种新型的Lagrange非线性规划神经网络［J］.电子学报，2002，30（1）:2729．
［12］韩正之，蔡秀珊，黄俊.微分包含控制系统理论［M］.上海:上海交通大学出版社，2013．
［16］Ham F M,Kostanic I.神经计算原理［M］.叶世伟,王海娟,译.北京:机械工业出版社,2007．