# American Institute of Mathematical Sciences

doi: 10.3934/dcdsb.2022035
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

## 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. Chen, Y. 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. Ciapessoni, D. Cirio, S. 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. Kumtepeli, Y. Zhao, M. Naumann, A. Tripathi, Y. Wang, A. 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. Liu, J. 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. Ren, S. Guo, Z. 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. Chen, Y. 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. Ciapessoni, D. Cirio, S. 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. Kumtepeli, Y. Zhao, M. Naumann, A. Tripathi, Y. Wang, A. 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. Liu, J. 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. Ren, S. Guo, Z. 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.
The state trajectory of the time-delay neural network corresponding to Example 1
The state trajectory of the time-delay neural network corresponding to Example 2
The state trajectory of the time-delay neural network corresponding to Example 3
 [1] Serge Nicaise, Cristina Pignotti, Julie Valein. Exponential stability of the wave equation with boundary time-varying delay. Discrete and Continuous Dynamical Systems - S, 2011, 4 (3) : 693-722. doi: 10.3934/dcdss.2011.4.693 [2] Quan Hai, Shutang Liu. Mean-square delay-distribution-dependent exponential synchronization of chaotic neural networks with mixed random time-varying delays and restricted disturbances. Discrete and Continuous Dynamical Systems - B, 2021, 26 (6) : 3097-3118. doi: 10.3934/dcdsb.2020221 [3] Baowei Feng, Carlos Alberto Raposo, Carlos Alberto Nonato, Abdelaziz Soufyane. Analysis of exponential stabilization for Rao-Nakra sandwich beam with time-varying weight and time-varying delay: Multiplier method versus observability. Mathematical Control and Related Fields, 2022  doi: 10.3934/mcrf.2022011 [4] Bingru Zhang, Chuanye Gu, Jueyou Li. Distributed convex optimization with coupling constraints over time-varying directed graphs†. Journal of Industrial and Management Optimization, 2021, 17 (4) : 2119-2138. doi: 10.3934/jimo.2020061 [5] Abdelfettah Hamzaoui, Nizar Hadj Taieb, Mohamed Ali Hammami. Practical partial stability of time-varying systems. Discrete and Continuous Dynamical Systems - B, 2022, 27 (7) : 3585-3603. doi: 10.3934/dcdsb.2021197 [6] Xin-Guang Yang, Jing Zhang, Shu Wang. Stability and dynamics of a weak viscoelastic system with memory and nonlinear time-varying delay. Discrete and Continuous Dynamical Systems, 2020, 40 (3) : 1493-1515. doi: 10.3934/dcds.2020084 [7] Mokhtar Kirane, Belkacem Said-Houari, Mohamed Naim Anwar. Stability result for the Timoshenko system with a time-varying delay term in the internal feedbacks. Communications on Pure and Applied Analysis, 2011, 10 (2) : 667-686. doi: 10.3934/cpaa.2011.10.667 [8] Aowen Kong, Carlos Nonato, Wenjun Liu, Manoel Jeremias dos Santos, Carlos Raposo. Equivalence between exponential stabilization and observability inequality for magnetic effected piezoelectric beams with time-varying delay and time-dependent weights. Discrete and Continuous Dynamical Systems - B, 2022, 27 (6) : 2959-2978. doi: 10.3934/dcdsb.2021168 [9] Wei Feng, Xin Lu. Global stability in a class of reaction-diffusion systems with time-varying delays. Conference Publications, 1998, 1998 (Special) : 253-261. doi: 10.3934/proc.1998.1998.253 [10] Carlos Nonato, Manoel Jeremias dos Santos, Carlos Raposo. Dynamics of Timoshenko system with time-varying weight and time-varying delay. Discrete and Continuous Dynamical Systems - B, 2022, 27 (1) : 523-553. doi: 10.3934/dcdsb.2021053 [11] Roberta Fabbri, Russell Johnson, Sylvia Novo, Carmen Núñez. On linear-quadratic dissipative control processes with time-varying coefficients. Discrete and Continuous Dynamical Systems, 2013, 33 (1) : 193-210. doi: 10.3934/dcds.2013.33.193 [12] Yangzi Hu, Fuke Wu. The improved results on the stochastic Kolmogorov system with time-varying delay. Discrete and Continuous Dynamical Systems - B, 2015, 20 (5) : 1481-1497. doi: 10.3934/dcdsb.2015.20.1481 [13] Serge Nicaise, Julie Valein, Emilia Fridman. Stability of the heat and of the wave equations with boundary time-varying delays. Discrete and Continuous Dynamical Systems - S, 2009, 2 (3) : 559-581. doi: 10.3934/dcdss.2009.2.559 [14] Ruoxia Li, Huaiqin Wu, Xiaowei Zhang, Rong Yao. Adaptive projective synchronization of memristive neural networks with time-varying delays and stochastic perturbation. Mathematical Control and Related Fields, 2015, 5 (4) : 827-844. doi: 10.3934/mcrf.2015.5.827 [15] Rui Hu, Yuan Yuan. Stability, bifurcation analysis in a neural network model with delay and diffusion. Conference Publications, 2009, 2009 (Special) : 367-376. doi: 10.3934/proc.2009.2009.367 [16] Xin-Guang Yang. An Erratum on "Stability and dynamics of a weak viscoelastic system with memory and nonlinear time-varying delay" (Discrete Continuous Dynamic Systems, 40(3), 2020, 1493-1515). Discrete and Continuous Dynamical Systems, 2022, 42 (3) : 1493-1494. doi: 10.3934/dcds.2021161 [17] Di Wu, Yanqin Bai, Fusheng Xie. Time-scaling transformation for optimal control problem with time-varying delay. Discrete and Continuous Dynamical Systems - S, 2020, 13 (6) : 1683-1695. doi: 10.3934/dcdss.2020098 [18] Paul B. Hermanns, Nguyen Van Thoai. Global optimization algorithm for solving bilevel programming problems with quadratic lower levels. Journal of Industrial and Management Optimization, 2010, 6 (1) : 177-196. doi: 10.3934/jimo.2010.6.177 [19] Mohammed Abdulrazaq Kahya, Suhaib Abduljabbar Altamir, Zakariya Yahya Algamal. Improving whale optimization algorithm for feature selection with a time-varying transfer function. Numerical Algebra, Control and Optimization, 2021, 11 (1) : 87-98. doi: 10.3934/naco.2020017 [20] Dinh Cong Huong, Mai Viet Thuan. State transformations of time-varying delay systems and their applications to state observer design. Discrete and Continuous Dynamical Systems - S, 2017, 10 (3) : 413-444. doi: 10.3934/dcdss.2017020

2021 Impact Factor: 1.497