doi: 10.3934/dcdsb.2022035
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Stability analysis of time-varying delay neural network for convex quadratic programming with equality constraints and inequality constraints

Mathematics and Statistics Department, Qingdao University, Qingdao, China

*Corresponding author: Xiaoqi Sun

Received  August 2021 Revised  January 2022 Early access March 2022

Fund Project: The corresponding author is supported by Natural Science Foundation of Shangdong Province grant No.ZR2019PA007

This paper presented a class of neural networks with time-varying delays to solve quadratic programming problems. Compared with previous papers, the neural networks proposed in this paper replaced the constant time delays $ \tau $ with variable time delays $ \tau(t) $ and had a more concise structure. There was an improvement of previous method in proving the existence and uniqueness of solutions of the neural networks in this paper. Further, this paper gave the conditions to be satisfied for the global exponential stability of the proposed neural networks. Through numerical examples, this paper verified that the proposed neural networks were accurate and efficient in solving the quadratic programming problems.

Citation: Ling Zhang, Xiaoqi Sun. Stability analysis of time-varying delay neural network for convex quadratic programming with equality constraints and inequality constraints. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2022035
References:
[1]

K. D. Atalay, E. Eraslan and M. O. Çinar, A hybrid algorithm based on fuzzy linear regression analysis by quadratic programming for time estimation: An experimental study in manufacturing industry, Journal of Manufacturing Systems, 36 (2015).

[2]

K. ChenY. Leung and K. S. Leung, A neural network for solving nonlinear programming problems, Neural Computing & Applications, 11 (2002), 103-111. 

[3]

Y. H. Chen and S. C. Fang, Neurocomputing with time delay analysis for solving convex quadratic programming problems, IEEE Transactions on Neural Networks, 11 (2000), 230-240. 

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E. CiapessoniD. CirioS. Massucco and A. Pitto, A probabilistic risk assessment and Control methodology for HVAC electrical grids connected to multiterminal HVDC networks, IFAC Proceedings Volumes, 44 (2011), 1727-1732.  doi: 10.3182/20110828-6-IT-1002.02739.

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Y. Huang, Lagrange-type neural networks for nonlinear programming problems with inequality constraints, IEEE Conference on Decision and Control, (2005), 4129–4133.

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R. B. Jafarabadi, J. Sadeh and A. Soheili, Global optimum economic designing of grid-connected photovoltaic systems with multiple inverters using binary linear programming, Solar Energy, 183 (2019).

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A. K. Jain and S. C. Srivastava, Strategic bidding and risk assessment using genetic algorithm in electricity markets, International Journal of Emerging Electric Power Systems, 10 (2011). doi: 10.2202/1553-779X.2161.

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M. P. Kennedy and L. O. Chua, Space-time matched filter design for interference suppression in coherent frequency diverse array, Neural Networks for Nonlinear Programming, (1988).

[10]

V. KumtepeliY. ZhaoM. NaumannA. TripathiY. WangA. Jossen and H. Hesse, Design and analysis of an aging-aware energy management system for islanded grids using mixed-integer quadratic programming, International Journal of Energy Research, 43 (2019), 4127-4147.  doi: 10.1002/er.4512.

[11]

Q. Liu and J. Cao, Globally projected dynamical system and its applications, Neural Inf. Process., 7 (2005), 1-9. 

[12]

Q. LiuJ. Cao and Y. Xia, A delayed neural network for solving linear projection equations and its analysis, IEEE transactions on neural networks, 16 (2005), 834-843. 

[13]

R. C. Lozano and C. Schulte, Survey on combinatorial register allocation and instruction scheduling, ACM Computing Surveys (CSUR), 52 (2019).

[14]

Z. Lv, Z. Qiu and Q. Li, An interval reduced basis approach and its integrated framework for acoustic response analysis of coupled structural-acoustic system, Journal of Computational Acoustics, 25 (2017), 1750009, 26 pp. doi: 10.1142/S0218396X17500096.

[15]

A. Nazemi, A neural network model for solving convex quadratic programming problems with some applications, Engineering Applications of Artificial Intelligence, 32 (2014).

[16]

A. Nazemi, A capable neural network framework for solving degenerate quadratic optimization problems with an application in image fusion, Neural Processing Letters, 47 (2018).

[17]

J. Niu and D. Liu, A new delayed projection neural network for solving quadratic programming problems subject to linear constraints, Applied Mathematics and Computation, 219 (2012), 3139-3146.  doi: 10.1016/j.amc.2012.09.047.

[18]

E. C. $\ddot{O}$zelkan, $\acute{A}$. Galambosi, E. F. Gaucherand and L. Duckstein, Linear quadratic dynamic programming for water reservoir management, Applied Mathematical Modelling, 21 (1997).

[19]

Z. RenS. GuoZ. Li and Z. Wu, Adjoint-based parameter and state estimation in 1-D magnetohydrodynamic (MHD) flow system, Journal of Industrial & Management Optimization, 14 (2018), 1579-1594.  doi: 10.3934/jimo.2018022.

[20]

C. Sha and H. Zhao, A novel neurodynamic reaction-diffusion model for solving linear variational inequality problems and its application, Applied Mathematics and Computation, 346 (2019), 57-75.  doi: 10.1016/j.amc.2018.10.023.

[21]

C. Sha, H. Zhao, T. Huang and W. Hu, A projection neural network with time delays for solving linear variational inequality problems and its applications, Circuits, Systems, and Signal Processing, 35 (2016).

[22]

C. Sha, H. Zhao and F. Ren, A new delayed projection neural network for solving quadratic programming problems with equality and inequality constraints, Engineering Applications of Artificial Intelligence, 168 (2015).

[23]

C.-S. Shieh, Robust Output-Feedback Control for Linear Continuous Uncertain State Delayed Systems with Unknown Time Delay, Circuits, Systems & Signal Processing, 21 (2002), 309-318.  doi: 10.1007/s00034-004-7046-9.

[24]

D. Tank and J. Hopfield, Simple'neural'optimization networks: An A/D converter, signal decision circuit, and a linear programming circuit, IEEE Transactions on Circuits and Systems, 33 (1986), 533-541. 

[25]

H. Wang, G. Liao, J. Xu and S. Zhu, Space-time matched filter design for interference suppression in coherent frequency diverse array, IET Signal Processing, 14 (2020).

[26]

X. Wen, S. Qin and J. Feng, A delayed neural network for solving a class of constrained pseudoconvex optimizations, International Conference on Information Science and Technology, (2019), 29–35.

[27]

Y. Xia, G. Feng and J. Wang, A recurrent neural network with exponential convergence for solving convex quadratic program and related linear piecewise equations, Neural Networks, 17 (2004).

[28]

J. Xiao, Y. Situ, W. Yuan, X. Wang and Y. Jiang, Parameter identification method based on mixed-integer quadratic programming and edge computing in power internet of things, Mathematical Problems in Engineering, (2020) (2020), Article ID 4053825. doi: 10.1155/2020/4053825.

[29]

Y. Yang and J. Cao, Solving quadratic programming problems by delayed projection neural network, IEEE Transactions on Neural Networks, 17 (2006).

[30]

Y. Yang and J. Cao, A feedback neural network for solving convex constraint optimization problems, Applied Mathematics and Computation, 201 (2008), 340-350.  doi: 10.1016/j.amc.2007.12.029.

[31]

S. Zhang and A. G. Constantinides, Lagrange programming neural networks, IEEE Transactions on Circuits and Systems II: Analog and Digital Signal Processing, 39 (1986), 441-452. 

show all references

References:
[1]

K. D. Atalay, E. Eraslan and M. O. Çinar, A hybrid algorithm based on fuzzy linear regression analysis by quadratic programming for time estimation: An experimental study in manufacturing industry, Journal of Manufacturing Systems, 36 (2015).

[2]

K. ChenY. Leung and K. S. Leung, A neural network for solving nonlinear programming problems, Neural Computing & Applications, 11 (2002), 103-111. 

[3]

Y. H. Chen and S. C. Fang, Neurocomputing with time delay analysis for solving convex quadratic programming problems, IEEE Transactions on Neural Networks, 11 (2000), 230-240. 

[4]

E. CiapessoniD. CirioS. Massucco and A. Pitto, A probabilistic risk assessment and Control methodology for HVAC electrical grids connected to multiterminal HVDC networks, IFAC Proceedings Volumes, 44 (2011), 1727-1732.  doi: 10.3182/20110828-6-IT-1002.02739.

[5]

J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-4342-7.

[6]

Y. Huang, Lagrange-type neural networks for nonlinear programming problems with inequality constraints, IEEE Conference on Decision and Control, (2005), 4129–4133.

[7]

R. B. Jafarabadi, J. Sadeh and A. Soheili, Global optimum economic designing of grid-connected photovoltaic systems with multiple inverters using binary linear programming, Solar Energy, 183 (2019).

[8]

A. K. Jain and S. C. Srivastava, Strategic bidding and risk assessment using genetic algorithm in electricity markets, International Journal of Emerging Electric Power Systems, 10 (2011). doi: 10.2202/1553-779X.2161.

[9]

M. P. Kennedy and L. O. Chua, Space-time matched filter design for interference suppression in coherent frequency diverse array, Neural Networks for Nonlinear Programming, (1988).

[10]

V. KumtepeliY. ZhaoM. NaumannA. TripathiY. WangA. Jossen and H. Hesse, Design and analysis of an aging-aware energy management system for islanded grids using mixed-integer quadratic programming, International Journal of Energy Research, 43 (2019), 4127-4147.  doi: 10.1002/er.4512.

[11]

Q. Liu and J. Cao, Globally projected dynamical system and its applications, Neural Inf. Process., 7 (2005), 1-9. 

[12]

Q. LiuJ. Cao and Y. Xia, A delayed neural network for solving linear projection equations and its analysis, IEEE transactions on neural networks, 16 (2005), 834-843. 

[13]

R. C. Lozano and C. Schulte, Survey on combinatorial register allocation and instruction scheduling, ACM Computing Surveys (CSUR), 52 (2019).

[14]

Z. Lv, Z. Qiu and Q. Li, An interval reduced basis approach and its integrated framework for acoustic response analysis of coupled structural-acoustic system, Journal of Computational Acoustics, 25 (2017), 1750009, 26 pp. doi: 10.1142/S0218396X17500096.

[15]

A. Nazemi, A neural network model for solving convex quadratic programming problems with some applications, Engineering Applications of Artificial Intelligence, 32 (2014).

[16]

A. Nazemi, A capable neural network framework for solving degenerate quadratic optimization problems with an application in image fusion, Neural Processing Letters, 47 (2018).

[17]

J. Niu and D. Liu, A new delayed projection neural network for solving quadratic programming problems subject to linear constraints, Applied Mathematics and Computation, 219 (2012), 3139-3146.  doi: 10.1016/j.amc.2012.09.047.

[18]

E. C. $\ddot{O}$zelkan, $\acute{A}$. Galambosi, E. F. Gaucherand and L. Duckstein, Linear quadratic dynamic programming for water reservoir management, Applied Mathematical Modelling, 21 (1997).

[19]

Z. RenS. GuoZ. Li and Z. Wu, Adjoint-based parameter and state estimation in 1-D magnetohydrodynamic (MHD) flow system, Journal of Industrial & Management Optimization, 14 (2018), 1579-1594.  doi: 10.3934/jimo.2018022.

[20]

C. Sha and H. Zhao, A novel neurodynamic reaction-diffusion model for solving linear variational inequality problems and its application, Applied Mathematics and Computation, 346 (2019), 57-75.  doi: 10.1016/j.amc.2018.10.023.

[21]

C. Sha, H. Zhao, T. Huang and W. Hu, A projection neural network with time delays for solving linear variational inequality problems and its applications, Circuits, Systems, and Signal Processing, 35 (2016).

[22]

C. Sha, H. Zhao and F. Ren, A new delayed projection neural network for solving quadratic programming problems with equality and inequality constraints, Engineering Applications of Artificial Intelligence, 168 (2015).

[23]

C.-S. Shieh, Robust Output-Feedback Control for Linear Continuous Uncertain State Delayed Systems with Unknown Time Delay, Circuits, Systems & Signal Processing, 21 (2002), 309-318.  doi: 10.1007/s00034-004-7046-9.

[24]

D. Tank and J. Hopfield, Simple'neural'optimization networks: An A/D converter, signal decision circuit, and a linear programming circuit, IEEE Transactions on Circuits and Systems, 33 (1986), 533-541. 

[25]

H. Wang, G. Liao, J. Xu and S. Zhu, Space-time matched filter design for interference suppression in coherent frequency diverse array, IET Signal Processing, 14 (2020).

[26]

X. Wen, S. Qin and J. Feng, A delayed neural network for solving a class of constrained pseudoconvex optimizations, International Conference on Information Science and Technology, (2019), 29–35.

[27]

Y. Xia, G. Feng and J. Wang, A recurrent neural network with exponential convergence for solving convex quadratic program and related linear piecewise equations, Neural Networks, 17 (2004).

[28]

J. Xiao, Y. Situ, W. Yuan, X. Wang and Y. Jiang, Parameter identification method based on mixed-integer quadratic programming and edge computing in power internet of things, Mathematical Problems in Engineering, (2020) (2020), Article ID 4053825. doi: 10.1155/2020/4053825.

[29]

Y. Yang and J. Cao, Solving quadratic programming problems by delayed projection neural network, IEEE Transactions on Neural Networks, 17 (2006).

[30]

Y. Yang and J. Cao, A feedback neural network for solving convex constraint optimization problems, Applied Mathematics and Computation, 201 (2008), 340-350.  doi: 10.1016/j.amc.2007.12.029.

[31]

S. Zhang and A. G. Constantinides, Lagrange programming neural networks, IEEE Transactions on Circuits and Systems II: Analog and Digital Signal Processing, 39 (1986), 441-452. 

Figure 1.  The state trajectory of the time-delay neural network corresponding to Example 1
Figure 2.  The state trajectory of the time-delay neural network corresponding to Example 2
Figure 3.  The state trajectory of the time-delay neural network corresponding to Example 3
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