基于时滞神经网络的二次规划的全局最优性条件
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摘要
全局优化问题是数学规划理论中重要而又困难的一个研究领域，它研究的是非线性函数在某个区域上全局最优点的特征和计算方法。由于目标函数通常不是凸函数，特定区域也有可能不是凸集，所以全局优化也可以称为非凸优化。全局最优的必要性条件是刻画一个解不是全局最优解的基本工具，而充分性条件是刻画一个解是全局最优解的基本工具。本文系统的讨论了时滞神经网络模型对于二次规划的全局最优性的建立、解的存在性、稳定性分析以及最优解的求法。
首先本文讨论了时滞神经网络模型的来源，详尽的分析了全局优化的研究现状，给出了全局优化的确定性算法。
随后研究了带有等式约束和不等式约束的二次规划的全局最优性，得到了系统的全局指数收敛，求得了收敛点为二次规划的最优解。
证明了当M是半正定矩阵时，时滞神经网络模型全局收敛于退化的二次优化的最优解，同时也验证了稳定点一定为最优解。
最后本文针对M是广义正定矩阵的一类新的带有不等式约束的不定二次规划问题，借助时滞神经网络模型，得到了系统解的存在性和唯一性。应用定性分析方法，构造恰当的李雅普诺夫函数，得到了该系统全局收敛于二次优化的最优解的充分条件，并在M是广义正定矩阵条件下，求得了系统的最优解。
Mathematical programming problem of global optimization theory is an importantand difficult area of research. It is a nonlinear function of a region on the global optimalcharacteristics and methods of calculation. Since the objective function is usually notconvex function, specific regions are probably not convex; Therefore, global optimizationcan also be referred to as non-convex optimization. The need for global optimum solutionis not a condition that depicts a basic tool for the global optimal solution and sufficientcondition is portrayed a global optimum solution is a basic tool. This article discusses thesystem delay neural network model for quadratic programming, the establishment ofglobal optimality、existence of solutions、stability analysis and find the optimal solutionmethod.
The first article discusses the sources of delay neural network model, detailedanalysis of the global optimization of research, gives the deterministic global optimizationalgorithm.
Then studied with equality constraints and inequality constraints of the globaloptimum of quadratic programming, get the global exponential convergence of the system,Obtained a convergence point for the optimal solution of quadratic programming.
Proved that when M is positive semi-definite matrix, delay neural network model onthe degradation of the global convergence of the optimal solution of quadraticoptimization, also verify that a stable point for the optimal solution.
Finally, M is a generalized positive definite matrix for a new class of inequalityconstraints with indefinite quadratic programming problem; neural network model withdelay, system has been the existence of solutions and uniqueness. Application ofqualitative methods, construct an appropriate Lyapunov function, global convergence ofthe system has been optimized in the second sufficient condition for the optimal solution,and the matrix M is positive definite under the conditions of generalized, obtain theoptimal solution of the system.
首先本文讨论了时滞神经网络模型的来源，详尽的分析了全局优化的研究现状，给出了全局优化的确定性算法。
随后研究了带有等式约束和不等式约束的二次规划的全局最优性，得到了系统的全局指数收敛，求得了收敛点为二次规划的最优解。
证明了当M是半正定矩阵时，时滞神经网络模型全局收敛于退化的二次优化的最优解，同时也验证了稳定点一定为最优解。
最后本文针对M是广义正定矩阵的一类新的带有不等式约束的不定二次规划问题，借助时滞神经网络模型，得到了系统解的存在性和唯一性。应用定性分析方法，构造恰当的李雅普诺夫函数，得到了该系统全局收敛于二次优化的最优解的充分条件，并在M是广义正定矩阵条件下，求得了系统的最优解。
Mathematical programming problem of global optimization theory is an importantand difficult area of research. It is a nonlinear function of a region on the global optimalcharacteristics and methods of calculation. Since the objective function is usually notconvex function, specific regions are probably not convex; Therefore, global optimizationcan also be referred to as non-convex optimization. The need for global optimum solutionis not a condition that depicts a basic tool for the global optimal solution and sufficientcondition is portrayed a global optimum solution is a basic tool. This article discusses thesystem delay neural network model for quadratic programming, the establishment ofglobal optimality、existence of solutions、stability analysis and find the optimal solutionmethod.
The first article discusses the sources of delay neural network model, detailedanalysis of the global optimization of research, gives the deterministic global optimizationalgorithm.
Then studied with equality constraints and inequality constraints of the globaloptimum of quadratic programming, get the global exponential convergence of the system,Obtained a convergence point for the optimal solution of quadratic programming.
Proved that when M is positive semi-definite matrix, delay neural network model onthe degradation of the global convergence of the optimal solution of quadraticoptimization, also verify that a stable point for the optimal solution.
Finally, M is a generalized positive definite matrix for a new class of inequalityconstraints with indefinite quadratic programming problem; neural network model withdelay, system has been the existence of solutions and uniqueness. Application ofqualitative methods, construct an appropriate Lyapunov function, global convergence ofthe system has been optimized in the second sufficient condition for the optimal solution,and the matrix M is positive definite under the conditions of generalized, obtain theoptimal solution of the system.
引文
[1] Bazraa M S, Sherali H D and Shetty C M. Nonlinear Programming Theory and Algorithms[J].1993
[2] Hiriart J B. Conditions for Global Optimality, in Handbook, Global Optimization, KluwerAcademic Publishers, Dordrecht, The Netherlands[J].1995,1-26
[3] Hiriart J B. Conditions for Global optimality of Global Optimization[J].1998,13,349-367
[4] Hiriart J B. Global Optimality Conditions in Maximizing a Convex Quadratic Function underConvex Quadratic Constraints[J].2001,445-455
[5] Horst R. On The Global Optimization Concave Functions Introduction and Survey OperationsResearch[J].Spektrum6,1984,195-205
[6] Horst R. Global Optimization[M]. Berlin Germany,1994
[7] Michel Y H, Wang A N. Global Stability and Local Stability of Hopfield Neural Networks withDelays[J]. Physical Review E,1994,50:4206-4213
[8]王占山,张化光,王智良.一类混沌神经网络的全局同步[J].物理学报,2006,55(6):2687-2693
[9] Wang Z S, Zhang H G. Exponential Stability Analysis of Neural with Multiply time VaryingDelays[J].Chinese Journal of Electronics,2006,15(4):649-653
[10] Cao Y, Wu Q. A note on Stability Analysis of Neural l with Delays[J].IEEE Transactions onNeural Networks,1996,7(6):1533-1535
[11] Arik S. On the Global Asymptotic Stability of Delayed Cellar Neural Network[J].IEEETransactions on Circuits and Systems,2000,47(4):571-574
[12] Cao J D. A General Framework for Global Asymptotic Stability Analysis of Delayed NeuralNetworks based on LMI approach chaos Solutions and Fractals[J].2005,24:1317-1329
[13] Zhang H G, Wang Z S. Stability Analysis of BAM Neural Networks with time varyingDelays[J]. Progress in Natural Science,2007,17(2):206-211
[14] Hiriart J B. Conditions for Global Optimality, Handbook for Global Optimization[J]. KluwerAcademic Publishers, Dordrecht, Holland,1999,1-26
[15] Hiriart J B. Conditions for Global Optimality [J]. Journal of Global Optimization,1998,13:349-367
[16] Barhen J. A Deterministic Arithm for Global Optimization[J].Science1997,276:1094-1097
[17] Horst R. Introduction to Global Optimization[J].The Netherlands,1995
[18] Xia Y S, Wang J A. General Projection Neural Network for Solving Monotone VariationInequalities and Related Optimization Problems[J]. IEEE Trans Neural Networks,2004,15:318–328
[19] Cao J and Wang L. Exponential Stability And Periodic Oscillatory Solution in BAM Networkwith Delays[J]. IEEE Trans. Neural Netw, Mar2002, vol.13:457-463
[20] Sun C Y, Zhang K J. On Exponential Stability of Delayed Neural Networks with a General classof Activation Functions[J].Phys. Lett.2002, vol.298:122-132
[21] Avrel M. Nonlinear Programming Analysis and Methods[J].Prentic Hall Inc. En-Cli. NJ,1976.
[22] Peng J M and Yuan Y X. Optimality Continuous for The Minimization D, A Quadratic withTwo Quadratic Constraints. Siam Joptim,1997,3:579-594
[23] Xia Y S, Feng G. A Recurrent Neural Network with Exponential Convergence for SolvingConvex Quadratic Program and Related Linear Piecewise Equation[J]. Neural Networks,2004,17:1003–1015
[24] Hiriart J B. Global Optimality Conditions in Maximizing a Convex Quadratic Function underConvex Quadratic Constraints of Global Optimization[J]. Neural Networks,2001,21:445-455
[25] Xia Y S. A Recurrent Neural Network for Solving Linear and Quadratic ProgrammingProblems[J].IEEE Trans. Neural Networks,1996,7:1544–1547
[26] Xia Y S, Feng G. A Recurrent Neural Network with Exponential Convergence for SolvingConvex Quadratic Program and Related Linear Piecewise Equation[J]. Neural Networks,2004,17:1003–1015
[27] Xia Y S, Wang G. An Improved Network for Convex Quadratic Optimization with Application toReal-time Beam forming[J].Neural Computing,2005,64:359–374
[28] Liao X F, Wong K W. Novel Robust Stability Criteria for Interval Delayed Hopfield NeuralNetworks[J]. IEEE Trans,2001,48:1355–1359
[29] Wang Z D. Stability Analysis for Stochastic Cohen Grasberg Neural Networks with Mixed TimeDelays[J]. IEEE Trans. Neural Networks,2006,17:814–820
[30] Wang Z D. Robust Stability Analysis of Generalized Neural Networks with Discrete andDistributed Time Delays, Chaos, Solutions and Fractals[J].2006,30:886–896
[31] Bazraa M S, Sherali H D. Nonlinear Programming Theory and Algorithms.1993
[32] Hiriart J B. Conditions for Global Optimality, in Handbook, Global Optimization.1995,1-26
[33] Hiriart J B. Conditions for global optimality of Global Optimization.1998,349-367
[34] Hiriart J B. From Convex to No convex Optimization necessary and Sufficient Conditions forGlobal Optimization[J].New York,1989,219-239
[35] Hansen E R. Global Optimization Using Interval Analysis the One dimensional Case[J].Journalof Optimization Theory and Application,29,331-344,1979
[36] Hansen E R. Global Optimization Using Interval Analysis-the Multidimensional Case numericMathematic,1980,34:247-270
[37] Horst R. Introduction to Global itemization, Kluwer Academic Publishers[J].The Netherlands,1995.
[38] Horst R. A General Class of Branch and Bound Methods in Global Optimization with Some NewApproaches for Concave Minimization[J]. Journal of Optimization Theory and Applications,1986,51,271-291
[39] Horst R. Deterministic Global Optimization with Partition Sets Whose Feasibility Is Not Known,Application to Concave Minimization[J]. De Programming, Reverse Convex Constraints andLipschitization Optimization Journal of Optimization Theory and Applications,1985,58, ll-37
[40] Horst R. On Consistency of Bounding Operations in Deterministic Global Optimization Journalof Optimization Theory and Applications,1989,61,143-146
[41] Horst R.Convergence and Restart in Branch and Bound Algorithms for Global OptimizationApplication to Concave Minimization and Optimization Problems, Mathematical Programming,1989,41,161-183
[42] Horst R. Deterministic Methods in Constrained Global Optimization, Some Recent Advancesand New Fields of Application, Naval Research Logistics,1990,37,433-471
[43] Zheng Q. Integral Global Optimization, Lecture Notes in Economics and Mathematics Systems,No.198. Springer-VEerlag,1988.
[44] Zheng Q and Zhuan D M. Integral Global Minimization, Algorithms, implementations andNumerical of Global Optimization.1995,7(4),421-454
[45]邬冬华,田蔚文,张连生,一个求总极值的实现算法及其收敛性[J].运筹学学报,3(2),1999,82-89
[46]邬冬华,田蔚文,张连生,黄伟,一种修正的求总极值的积分水平集方法的实现算法收敛性[J].应用数学学报,2001,24(1),100-110
[47] Ge R P. A Filled Function Method or Finding A Global Minimize of A Functional SeveralVariables[J].Paper Presented at the Dundee Biennial Conference on Numerical Analysis,1983
[48] Liu X. Finding Global Minima with a Compute Filled Function. Journal of Global Optimization,2001,19,151-161
[49] Liu X. A Glass of Generalities Filled Functions with Improved Computability[J].Journal ofComputational and Applied Mathematics,2001,137,62-69
[50] Ng C K. High Performa Continuous Discrete Global Optimization Methods[M].The ChineseUniversity of Hong Kong, Ph. D. Thesis,2003
[51] Rockafellar R T. Convex Analysis[M]. Princeton University Press, Princet NJ,1970
[52] Bazaraa M S, Herali H D S. Nonlinear programming theory and algorithms [M].2nd ed. NewYork: Wiley,1993
[53] Xia Y S, WANG J. A General Projection Neural Network for Solving monotone VariationInequalities and Related Optimization Problems[J]. IEEE Trans Neural Networks,2001,124,251-260. Network,2006,17:1630–1634
[54] Feng G, Wang J. A Recurrent Neural Network with Exponential Convergence for SolvingConvex Quadratic Program and Related Linear Piecewise Equation[J]. Neural Networks,2004,17:1003-1015
[55] Xia Y S,Wang J. A General Projection Neural Network for Solving Monoton Variation Inequalityties and Related Optimization Problems [J].IEEE Trans Neural Networks,2004,15:318–328
[56] Baiocchi C. Variation and Inequalities Applications to free Boundary Problems [M]. New York:Wiley,1984
[57] LaSalle J P. The Stability of Dynamical Systems [M]. Philadelphia: SIMA,1976
[58] Kennedy M P. Neural Networks for Nonlinear Programming[J].IEEE Trans. Circuits Syst.35,1998,554-562
[59] Xia Y S, Wang J. A Recurrent Neural Network with Exponential Convergence for SolvingConvex Quadratic Program and Related Linear Piecewise Equation[J].Neural Networks,17,2004,1003-1015
[60] Yang Y, Cao J. Solving Quadratic Programming Problems by Delayed projection NeuralNetwork[J].IEEE Trans. Neural Network172006,1630-1634
[61] Yang Y, Cao J. A Delayed Network Method for Solving Convex Optimization Problems[J]Neural Syst16,2006,295-303
[62] Gao X, Liao L. A Delayed Novel Neural Network for Solving Linear Projection Equations andits Analysis[J]. IEEE Trans, Neural Network16,2005,835-843
[63] Cao J and Wang J, Exponential Stability and Periodic Oscillatory solution in BAM Network withDelays[J]. IEEE Trans, Neural Networks, vol.13, no.2, Mar2002,457-463
[64] Xia Y S, Wang J. A General Projection Neural Network for Solving Monotone VariationInequalities and Related Optimization Problems[J]. IEEE Trans Neural Networks,2004,15:318–328
[65] Xia Y S, Wang J. A Recurrent Neural Network for Solving Linear Projection Equations[J].Neural Netw, vol.13,2000,337-350
[66] Kinder D. An Introduction to Variation Inequalities and Applications[M].New York: Academic,1980
[67] Hale J, Theory of Functional Differential Equation[M]. New York: SpringerVer-lag,1977.
[68] Lasalle J P. The Stability of Dynamical Systems[M].Philadelphia. PA: SILAM,1976
[2] Hiriart J B. Conditions for Global Optimality, in Handbook, Global Optimization, KluwerAcademic Publishers, Dordrecht, The Netherlands[J].1995,1-26
[3] Hiriart J B. Conditions for Global optimality of Global Optimization[J].1998,13,349-367
[4] Hiriart J B. Global Optimality Conditions in Maximizing a Convex Quadratic Function underConvex Quadratic Constraints[J].2001,445-455
[5] Horst R. On The Global Optimization Concave Functions Introduction and Survey OperationsResearch[J].Spektrum6,1984,195-205
[6] Horst R. Global Optimization[M]. Berlin Germany,1994
[7] Michel Y H, Wang A N. Global Stability and Local Stability of Hopfield Neural Networks withDelays[J]. Physical Review E,1994,50:4206-4213
[8]王占山,张化光,王智良.一类混沌神经网络的全局同步[J].物理学报,2006,55(6):2687-2693
[9] Wang Z S, Zhang H G. Exponential Stability Analysis of Neural with Multiply time VaryingDelays[J].Chinese Journal of Electronics,2006,15(4):649-653
[10] Cao Y, Wu Q. A note on Stability Analysis of Neural l with Delays[J].IEEE Transactions onNeural Networks,1996,7(6):1533-1535
[11] Arik S. On the Global Asymptotic Stability of Delayed Cellar Neural Network[J].IEEETransactions on Circuits and Systems,2000,47(4):571-574
[12] Cao J D. A General Framework for Global Asymptotic Stability Analysis of Delayed NeuralNetworks based on LMI approach chaos Solutions and Fractals[J].2005,24:1317-1329
[13] Zhang H G, Wang Z S. Stability Analysis of BAM Neural Networks with time varyingDelays[J]. Progress in Natural Science,2007,17(2):206-211
[14] Hiriart J B. Conditions for Global Optimality, Handbook for Global Optimization[J]. KluwerAcademic Publishers, Dordrecht, Holland,1999,1-26
[15] Hiriart J B. Conditions for Global Optimality [J]. Journal of Global Optimization,1998,13:349-367
[16] Barhen J. A Deterministic Arithm for Global Optimization[J].Science1997,276:1094-1097
[17] Horst R. Introduction to Global Optimization[J].The Netherlands,1995
[18] Xia Y S, Wang J A. General Projection Neural Network for Solving Monotone VariationInequalities and Related Optimization Problems[J]. IEEE Trans Neural Networks,2004,15:318–328
[19] Cao J and Wang L. Exponential Stability And Periodic Oscillatory Solution in BAM Networkwith Delays[J]. IEEE Trans. Neural Netw, Mar2002, vol.13:457-463
[20] Sun C Y, Zhang K J. On Exponential Stability of Delayed Neural Networks with a General classof Activation Functions[J].Phys. Lett.2002, vol.298:122-132
[21] Avrel M. Nonlinear Programming Analysis and Methods[J].Prentic Hall Inc. En-Cli. NJ,1976.
[22] Peng J M and Yuan Y X. Optimality Continuous for The Minimization D, A Quadratic withTwo Quadratic Constraints. Siam Joptim,1997,3:579-594
[23] Xia Y S, Feng G. A Recurrent Neural Network with Exponential Convergence for SolvingConvex Quadratic Program and Related Linear Piecewise Equation[J]. Neural Networks,2004,17:1003–1015
[24] Hiriart J B. Global Optimality Conditions in Maximizing a Convex Quadratic Function underConvex Quadratic Constraints of Global Optimization[J]. Neural Networks,2001,21:445-455
[25] Xia Y S. A Recurrent Neural Network for Solving Linear and Quadratic ProgrammingProblems[J].IEEE Trans. Neural Networks,1996,7:1544–1547
[26] Xia Y S, Feng G. A Recurrent Neural Network with Exponential Convergence for SolvingConvex Quadratic Program and Related Linear Piecewise Equation[J]. Neural Networks,2004,17:1003–1015
[27] Xia Y S, Wang G. An Improved Network for Convex Quadratic Optimization with Application toReal-time Beam forming[J].Neural Computing,2005,64:359–374
[28] Liao X F, Wong K W. Novel Robust Stability Criteria for Interval Delayed Hopfield NeuralNetworks[J]. IEEE Trans,2001,48:1355–1359
[29] Wang Z D. Stability Analysis for Stochastic Cohen Grasberg Neural Networks with Mixed TimeDelays[J]. IEEE Trans. Neural Networks,2006,17:814–820
[30] Wang Z D. Robust Stability Analysis of Generalized Neural Networks with Discrete andDistributed Time Delays, Chaos, Solutions and Fractals[J].2006,30:886–896
[31] Bazraa M S, Sherali H D. Nonlinear Programming Theory and Algorithms.1993
[32] Hiriart J B. Conditions for Global Optimality, in Handbook, Global Optimization.1995,1-26
[33] Hiriart J B. Conditions for global optimality of Global Optimization.1998,349-367
[34] Hiriart J B. From Convex to No convex Optimization necessary and Sufficient Conditions forGlobal Optimization[J].New York,1989,219-239
[35] Hansen E R. Global Optimization Using Interval Analysis the One dimensional Case[J].Journalof Optimization Theory and Application,29,331-344,1979
[36] Hansen E R. Global Optimization Using Interval Analysis-the Multidimensional Case numericMathematic,1980,34:247-270
[37] Horst R. Introduction to Global itemization, Kluwer Academic Publishers[J].The Netherlands,1995.
[38] Horst R. A General Class of Branch and Bound Methods in Global Optimization with Some NewApproaches for Concave Minimization[J]. Journal of Optimization Theory and Applications,1986,51,271-291
[39] Horst R. Deterministic Global Optimization with Partition Sets Whose Feasibility Is Not Known,Application to Concave Minimization[J]. De Programming, Reverse Convex Constraints andLipschitization Optimization Journal of Optimization Theory and Applications,1985,58, ll-37
[40] Horst R. On Consistency of Bounding Operations in Deterministic Global Optimization Journalof Optimization Theory and Applications,1989,61,143-146
[41] Horst R.Convergence and Restart in Branch and Bound Algorithms for Global OptimizationApplication to Concave Minimization and Optimization Problems, Mathematical Programming,1989,41,161-183
[42] Horst R. Deterministic Methods in Constrained Global Optimization, Some Recent Advancesand New Fields of Application, Naval Research Logistics,1990,37,433-471
[43] Zheng Q. Integral Global Optimization, Lecture Notes in Economics and Mathematics Systems,No.198. Springer-VEerlag,1988.
[44] Zheng Q and Zhuan D M. Integral Global Minimization, Algorithms, implementations andNumerical of Global Optimization.1995,7(4),421-454
[45]邬冬华,田蔚文,张连生,一个求总极值的实现算法及其收敛性[J].运筹学学报,3(2),1999,82-89
[46]邬冬华,田蔚文,张连生,黄伟,一种修正的求总极值的积分水平集方法的实现算法收敛性[J].应用数学学报,2001,24(1),100-110
[47] Ge R P. A Filled Function Method or Finding A Global Minimize of A Functional SeveralVariables[J].Paper Presented at the Dundee Biennial Conference on Numerical Analysis,1983
[48] Liu X. Finding Global Minima with a Compute Filled Function. Journal of Global Optimization,2001,19,151-161
[49] Liu X. A Glass of Generalities Filled Functions with Improved Computability[J].Journal ofComputational and Applied Mathematics,2001,137,62-69
[50] Ng C K. High Performa Continuous Discrete Global Optimization Methods[M].The ChineseUniversity of Hong Kong, Ph. D. Thesis,2003
[51] Rockafellar R T. Convex Analysis[M]. Princeton University Press, Princet NJ,1970
[52] Bazaraa M S, Herali H D S. Nonlinear programming theory and algorithms [M].2nd ed. NewYork: Wiley,1993
[53] Xia Y S, WANG J. A General Projection Neural Network for Solving monotone VariationInequalities and Related Optimization Problems[J]. IEEE Trans Neural Networks,2001,124,251-260. Network,2006,17:1630–1634
[54] Feng G, Wang J. A Recurrent Neural Network with Exponential Convergence for SolvingConvex Quadratic Program and Related Linear Piecewise Equation[J]. Neural Networks,2004,17:1003-1015
[55] Xia Y S,Wang J. A General Projection Neural Network for Solving Monoton Variation Inequalityties and Related Optimization Problems [J].IEEE Trans Neural Networks,2004,15:318–328
[56] Baiocchi C. Variation and Inequalities Applications to free Boundary Problems [M]. New York:Wiley,1984
[57] LaSalle J P. The Stability of Dynamical Systems [M]. Philadelphia: SIMA,1976
[58] Kennedy M P. Neural Networks for Nonlinear Programming[J].IEEE Trans. Circuits Syst.35,1998,554-562
[59] Xia Y S, Wang J. A Recurrent Neural Network with Exponential Convergence for SolvingConvex Quadratic Program and Related Linear Piecewise Equation[J].Neural Networks,17,2004,1003-1015
[60] Yang Y, Cao J. Solving Quadratic Programming Problems by Delayed projection NeuralNetwork[J].IEEE Trans. Neural Network172006,1630-1634
[61] Yang Y, Cao J. A Delayed Network Method for Solving Convex Optimization Problems[J]Neural Syst16,2006,295-303
[62] Gao X, Liao L. A Delayed Novel Neural Network for Solving Linear Projection Equations andits Analysis[J]. IEEE Trans, Neural Network16,2005,835-843
[63] Cao J and Wang J, Exponential Stability and Periodic Oscillatory solution in BAM Network withDelays[J]. IEEE Trans, Neural Networks, vol.13, no.2, Mar2002,457-463
[64] Xia Y S, Wang J. A General Projection Neural Network for Solving Monotone VariationInequalities and Related Optimization Problems[J]. IEEE Trans Neural Networks,2004,15:318–328
[65] Xia Y S, Wang J. A Recurrent Neural Network for Solving Linear Projection Equations[J].Neural Netw, vol.13,2000,337-350
[66] Kinder D. An Introduction to Variation Inequalities and Applications[M].New York: Academic,1980
[67] Hale J, Theory of Functional Differential Equation[M]. New York: SpringerVer-lag,1977.
[68] Lasalle J P. The Stability of Dynamical Systems[M].Philadelphia. PA: SILAM,1976