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New five-step DTZD algorithm for future nonlinear minimization with quartic steady-state error pattern

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Abstract

In this paper, a new five-step discrete-time zeroing dynamics (DTZD) algorithm, discretized from a continuous-time zeroing dynamics (CTZD) model, is proposed and investigated for online future nonlinear minimization (OFNM), i.e., online discrete-time dynamic nonlinear minimization. For approximating more accurately the first-order derivative and discretizing more effectively the CTZD model, a six-node g-cube discretization (6Ng CD) formula with higher precision is presented to obtain the new five-step DTZD algorithm. Besides, the corresponding theoretical result shows that the proposed five-step DTZD algorithm is with a quartic steady-state error pattern, i.e., O(g4) pattern, with g denoting the sampling gap. Moreover, a general DTZD algorithm is constructed by applying the general linear multistep method, and a specific DTZD algorithm based on the 4th-order Adams-Bashforth method (termed DTZD-AB algorithm for short) is further developed for OFNM. Several numerical experiments are conducted to substantiate the efficacy, accuracy, and superiority of the proposed five-step DTZD algorithm (as well as the DTZD-AB algorithm) for solving the OFNM problem, as compared with the one-step and three-step DTZD algorithms developed and investigated in previous works.

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References

  1. Nocedal, J., Wright, S.J.: Numerical Optimization, 2nd edn. Springer, New York (2006)

    Google Scholar 

  2. Boyd, S., Vandenberghe, L.: Convex Optimization. Cambridge University, New York (2004)

    Book  MATH  Google Scholar 

  3. Cheng, L., Hou, Z.-G., Lin, Y., Tan, M., Zhang, W.C., Wu, F.-X.: Recurrent neural network for non-smooth convex optimization problems with application to the identification of genetic regulatory networks. IEEE Trans. Neural Netw. 22, 714–726 (2011)

    Article  Google Scholar 

  4. Miao, P., Shen, Y., Li, Y., Bao, L.: Finite-time recurrent neural networks for solving nonlinear optimization problems and their application. Neurocomputing 177, 120–129 (2016)

    Article  Google Scholar 

  5. Jin, L., Li, S., La, H.M., Luo, X.: Manipulability optimization of redundant manipulators using dynamic neural networks. IEEE Trans. Ind. Electron. 64, 4710–4720 (2017)

    Article  Google Scholar 

  6. Wang, F.-S., Jian, J.-B., Wang, C.-L.: A model-hybrid approach for unconstrained optimization problems. Numer. Algorithms 66, 741–759 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  7. Yue, L., Yang, Y.: A new integral filter algorithm for unconstrained global optimization. Numer. Algorithms 63, 419–430 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  8. Ahookhosh, M., Amini, K.: A nonmonotone trust region method with adaptive radius for unconstrained optimization problems. Comput. Math. Appl. 60, 411–422 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  9. Huang, S., Wan, Z., Chen, X.: A new nonmonotone line search technique for unconstrained optimization. Numer. Algorithms 68, 671–689 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  10. Wang, J., Zhu, D.: Conjugate gradient path method without line search technique for derivative-free unconstrained optimization. Numer. Algorithms 73, 957–983 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  11. Tan, M.: Exponential convergence and stability of delayed fuzzy cellular neural networks with time-varying coefficients. J. Control Theory Appl. 9, 500–504 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  12. Zhang, Y., Xiao, L., Ruan, G., Li, Z.: Continuous and discrete time Zhang dynamics for time-varying 4th root finding. Numer. Algorithms 57, 35–51 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  13. Jin, L., Zhang, Y., Qiu, B.: Neural network-based discrete-time Z-type model of high accuracy in noisy environments for solving dynamic system of linear equations. Neural Comput. Appl. 29, 1217–1232 (2018)

    Article  Google Scholar 

  14. Zhang, Y., Yi, C., Guo, D., Zheng, J.: Comparison on Zhang neural dynamics and gradient-based neural dynamics for online solution of nonlinear time-varying equation. Neural Comput. Appl. 20, 1–7 (2011)

    Article  Google Scholar 

  15. Xiao, L.: A nonlinearly-activated neurodynamic model and its finite-time solution to equality-constrained quadratic optimization with nonstationary coefficients. Appl. Soft Comput. 40, 252–259 (2016)

    Article  Google Scholar 

  16. Stanimirović, P.S., živković, I.S., Wei, Y.: Recurrent neural network for computing the Drazin inverse. IEEE Trans. Neural Netw. Learn. Syst. 26, 2830–2843 (2015)

    Article  MathSciNet  Google Scholar 

  17. Xiao, L., Liao, B.: A convergence-accelerated Zhang neural network and its solution application to Lyapunov equation. Neurocomputing 193, 213–218 (2016)

    Article  Google Scholar 

  18. Chen, K., Yi, C.: Robustness analysis of a hybrid of recursive neural dynamics for online matrix inversion. Appl. Math. Comput. 273, 969–975 (2016)

    MathSciNet  MATH  Google Scholar 

  19. Wang, X.-Z., Ma, H., Stanimirović, P.S.: Recurrent neural network for computing the W-weighted Drazin inverse. Appl. Math. Comput. 300, 1–20 (2017)

    MathSciNet  MATH  Google Scholar 

  20. Stanimirović, P.S., Petković, M.D.: Gradient neural dynamics for solving matrix equations and their applications. Neurocomputing 306, 200–212 (2018)

    Article  Google Scholar 

  21. Liu, Y.-J., Tong, S., Li, D.-J., Gao, Y.: Fuzzy adaptive control with state observer for a class of nonlinear discrete-time systems with input constraint. IEEE Trans. Fuzzy Syst. 24, 1147–1158 (2016)

    Article  Google Scholar 

  22. Wang, H., Liu, X., Liu, K.: Robust adaptive neural tracking control for a class of stochastic nonlinear interconnected systems. IEEE Trans. Neural Netw. Learn. Syst. 27, 510–523 (2016)

    Article  MathSciNet  Google Scholar 

  23. Na, J., Chen, Q., Ren, X., Guo, Y.: Adaptive prescribed performance motion control of servo mechanisms with friction compensation. IEEE Trans. Ind. Electron. 61, 486–494 (2014)

    Article  Google Scholar 

  24. Liu, Q., Cao, J.: Global exponential stability of discrete-time recurrent neural network for solving quadratic programming problems subject to linear constraints. Neurocomputing 74, 3494–3501 (2011)

    Article  Google Scholar 

  25. Li, S., Li, Y.: Nonlinearly activated neural network for solving time-varying complex Sylvester equation. IEEE Trans. Cybern. 44, 1397–1407 (2014)

    Article  Google Scholar 

  26. Miao, P., Shen, Y., Huang, Y., Wang, Y.-W.: Solving time-varying quadratic programs based on finite-time Zhang neural networks and their application to robot tracking. Neural Comput. Appl. 26, 693–703 (2015)

    Article  Google Scholar 

  27. Sun, J., Wang, S., Wang, K.: Zhang neural networks for a set of linear matrix inequalities with time-varying coefficient matrix. Inf. Process. Lett. 116, 603–610 (2016)

    Article  MATH  Google Scholar 

  28. Zhang, Y., Li, Z., Guo, D., Ke, Z., Chen, P.: Discrete-time ZD, GD and NI for solving nonlinear time-varying equations. Numer. Algorithms 64, 721–740 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  29. Zhang, Y., Xiao, L., Xiao, Z., Mao, M.: Zeroing Dynamics, Gradient Dynamics, and Newton Iterations. CRC Press, Florida (2015)

    Book  MATH  Google Scholar 

  30. Petković, M.D., Stanimirović, P.S., Katsikis, V.N.: Modified discrete iterations for computing the inverse and pseudoinverse of the time-varying matrix. Neurocomputing 289, 155–165 (2018)

    Article  Google Scholar 

  31. Stanimirović, P.S., Srivastava, S., Gupta, D.K.: From Zhang neural network to scaled hyperpower iterations. J. Comput. Appl. Math. 331, 133–155 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  32. Wang, X. -Z., Stanimirović, P.S., Wei, Y.: Complex ZFs for computing time-varying complex outer inverses. Neurocomputing 275, 983–1001 (2018)

    Article  Google Scholar 

  33. Guo, D., Zhang, Y.: Neural dynamics and Newton-Raphson iteration for nonlinear optimization. ASME J. Comput. Nonlinear Dyn. 9, 021016 (2014)

    Article  Google Scholar 

  34. Jin, L., Zhang, Y.: Continuous and discrete Zhang dynamics for real-time varying nonlinear optimization. Numer. Algorithms 73, 115–140 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  35. Guo, D., Lin, X., Su, Z., Sun, S., Huang, Z.: Design and analysis of two discrete-time ZD algorithms for time-varying nonlinear minimization. Numer. Algorithms 77, 23–36 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  36. Jin, L., Zhang, Y.: Discrete-time Zhang neural network of o(τ 3) pattern for time-varying matrix pseudoinversion with application to manipulator motion generation. Neurocomputing 142, 165–173 (2014)

    Article  Google Scholar 

  37. Zhang, Y., Jin, L., Guo, D., Yin, Y., Chou, Y.: Taylor-type 1-step-ahead numerical differentiation rule for first-order derivative approximation and ZNN discretization. J. Comput. Appl. Math. 273, 29–40 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  38. Liao, B., Zhang, Y., Jin, L.: Taylor o(h 3) discretization of ZNN models for dynamic equality-constrained quadratic programming with application to manipulators. IEEE Trans. Neural Netw. Learn. Syst. 27, 225–237 (2016)

    Article  MathSciNet  Google Scholar 

  39. Zhao, Y., Swamy, M.N.S.: A novel technique for tracking time-varying minimum and its applications. In: Proceedings of the 11th Canadian Conference on Electrical and Computer Engineering, pp. 910–913 (1998)

  40. Zhao, Y., Feng, C.B.: Time-dependent optimization for information processing and its applications (i) basic concept and system identification. J. Southeast Univ. 29, 92–97 (1999)

    MathSciNet  MATH  Google Scholar 

  41. Guo, D., Nie, Z., Yan, L.: Theoretical analysis, numerical verification and geometrical representation of new three-step DTZD algorithm for time-varying nonlinear equations solving. Neurocomputing 214, 516–526 (2016)

    Article  Google Scholar 

  42. Mathews, J.H., Fink, K.D.: Numerical Methods Using MATLAB, 4th edn. Prentice Hall, New Jersey (2004)

    Google Scholar 

  43. Suli, E., Mayers, D.F.: An Introduction to Numerical Analysis. Cambridge University Press, Oxford (2003)

    Book  MATH  Google Scholar 

  44. Griffiths, D.F., Higham, D.J.: Numerical Methods for Ordinary Differential Equations: Initial Value Problems. Springer, England (2010)

    Book  MATH  Google Scholar 

  45. Butcher, J.C.: Numerical Methods for Ordinary Differential Equations, 2nd edn. Wiley, New York (2008)

    Book  Google Scholar 

Download references

Acknowledgements

The authors thank the editors and anonymous reviewers for their valuable suggestions and constructive comments, which helped to improve the presentation and quality of the paper.

Funding

This work is supported by the National Natural Science Foundation of China (with number 61473323), by the Foundation of Key Laboratory of Autonomous Systems and Networked Control, Ministry of Education, China (with number 2013A07), and also by the Laboratory Open Fund of Sun Yat-sen University (with number 20160209).

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Authors and Affiliations

  1. School of Data and Computer Science, Sun Yat-sen University (SYSU), Guangzhou, 510006, China

    Binbin Qiu, Yunong Zhang, Jinjin Guo, Zhi Yang & Xiaodong Li

  2. Key Laboratory of Autonomous Systems and Networked Control, Ministry of Education, Guangzhou, 510640, China

    Binbin Qiu, Yunong Zhang, Jinjin Guo, Zhi Yang & Xiaodong Li

  3. SYSU-CMU Shunde International Joint Research Institute, Foshan, 528300, China

    Binbin Qiu, Yunong Zhang, Jinjin Guo, Zhi Yang & Xiaodong Li

  4. Key Laboratory of Machine Intelligence and Advanced Computing, Ministry of Education, Guangzhou, 510006, China

    Binbin Qiu, Yunong Zhang, Jinjin Guo, Zhi Yang & Xiaodong Li

Authors
  1. Binbin Qiu
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  2. Yunong Zhang
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  3. Jinjin Guo
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  4. Zhi Yang
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  5. Xiaodong Li
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Correspondence to Binbin Qiu.

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Qiu, B., Zhang, Y., Guo, J. et al. New five-step DTZD algorithm for future nonlinear minimization with quartic steady-state error pattern. Numer Algor 81, 1043–1065 (2019). https://doi.org/10.1007/s11075-018-0581-4

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  • DOI: https://doi.org/10.1007/s11075-018-0581-4

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