Skip to main content
SpringerLink
Log in

Continuous and discrete Zhang dynamics for real-time varying nonlinear optimization

  • Original Paper
  • Published:
Numerical Algorithms Aims and scope Submit manuscript

Abstract

Online solution of time-varying nonlinear optimization problems is considered an important issue in the fields of scientific and engineering research. In this study, the continuous-time derivative (CTD) model and two gradient dynamics (GD) models are developed for real-time varying nonlinear optimization (RTVNO). A continuous-time Zhang dynamics (CTZD) model is then generalized and investigated for RTVNO to remedy the weaknesses of CTD and GD models. For possible digital hardware realization, a discrete-time Zhang dynamics (DTZD) model, which can be further reduced to Newton-Raphson iteration (NRI), is also proposed and developed. Theoretical analyses indicate that the residual error of the CTZD model has an exponential convergence, and that the maximum steady-state residual error (MSSRE) of the DTZD model has an O(τ 2) pattern with τ denoting the sampling gap. Simulation and numerical results further illustrate the efficacy and advantages of the proposed CTZD and DTZD models for RTVNO.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Courvoisier, Y., Gander, M.J.: Optimization of Schwarz waveform relaxation over short time windows. Numer. Algor 64(2), 221–243 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  2. Amini, K., Ahookhosh, M., Nosratipour, H.: An inexact line search approach using modified nonmonotone strategy for unconstrained optimization. Numer. Algor 66(1), 49–78 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  3. Andrei, N.: An accelerated subspace minimization three-term conjugate gradient algorithm for unconstrained optimization. Numer. Algor 65(4), 859–874 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  4. Martínez, J.M., Prudente, L.F.: Handling infeasibility in a large-scale nonlinear optimization algorithm. Numer. Algor 60(2), 263–277 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  5. Zhang, Y., Ge, S.S., Lee, T.H.: A unified quadratic programming based dynamical system approach to joint torque optimization of physically constrained redundant manipulators. IEEE Trans. Syst., Man, Cybern. B, Cybern 34(5), 2126–2132 (2004)

    Article  Google Scholar 

  6. Ahookhosh, M., Amini, K.: An efficient nonmonotone trust-region method for unconstrained optimization. Numer. Algor 59(4), 523–540 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  7. Wang, F., Jian, J., Wang, C.: A model-hybrid approach for unconstrained optimization problems. Numer. Algor 66(4), 741–759 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  8. Gaviano, M., Lera, D.: Properties and numerical testing of a parallel global optimization algorithm. Numer. Algor 60(4), 613–629 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  9. Cai, X., Wang, G., Zhang, Z.: Complexity analysis and numerical implementation of primal-dual interior-point methods for convex quadratic optimization based on a finite barrier. Numer. Algor 62(2), 289–306 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  10. Birgin, E.G., Martínez, J.M.: A spectral conjugate gradient method for unconstrained optimization. Appl. Math. Opt 43(2), 117–128 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  11. Dai, Y., Liao, L.: New conjugacy conditions and related nonlinear conjugate gradient methods. Appl. Math. Opt 43(1), 87–101 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  12. Narushima, Y., Yabe, H.: Conjugate gradient methods based on secant conditions that generate descent search directions for unconstrained optimization. J. Appl. Math. Comput. 236(17), 4303–4317 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  13. Xia, Y., Sun, C.: A novel neural dynamical approach to convex quadratic program and its efficient applications. Neural Netw. 22(1), 1463–1470 (2009)

    Article  Google Scholar 

  14. Liao, B., Zhang, Y.: Different complex ZFs leading to different complex ZNN models for time-varying complex generalized inverse matrices. IEEE Trans. Neural Netw. Learn. Syst. 25(9), 1621–1631 (2014)

    Article  Google Scholar 

  15. Xiao, L., Lu, R.: Finite-time solution to nonlinear equation using recurrent neural dynamics with a specially-constructed activation function. Neurocomputing 151(1), 246–251 (2015)

    Article  MathSciNet  Google Scholar 

  16. Barbarosou, M.P., Maratos, N.G.: A nonfeasible gradient projection recurrent neural network for equality-constrained optimization problems. IEEE Trans. Neur. Netw. 19(10), 1665–1677 (2008)

    Article  Google Scholar 

  17. Cao, J., Mao, X., Luo, Q.: Neurodynamic system theory and applications. Abstr. Appl. Anal. 2013(1), 1–1 (2013)

    Google Scholar 

  18. Liao, L., Qi, H., Qi, L.: Neurodynamical optimization. J. Global Optim. 28(2), 175–195 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  19. Mathews, J.H., Fink, K.D.: Numerical Methods Using MATLAB. Prentice-Hall, Englewood Cliffs, NJ (2005)

    Google Scholar 

  20. Bhaya, A., Kaszkurewicz, E.: Control Perspectives on Numerical Algorithms and Matrix Problems. SIAM, Philadelphia, PA (2006)

    Book  MATH  Google Scholar 

  21. Bhaya, A., Kaszkurewicz, E.: A control-theoretic approach to the design of zero finding numerical methods. IEEE Trans. Autom. Control 52(6), 1014–1026 (2007)

    Article  MathSciNet  Google Scholar 

  22. Bhaya, A., Kaszkurewicz, E.: Iterative methods as dynamical systems with feedback control. In: Proceedings of IEEE Conference on Decision and Control, pp 2374–2380 (2003)

  23. Zhang, Y., Yi, C.: Zhang Neural Networks and Neural-Dynamic Method. Nova, New York (2011)

    Google Scholar 

  24. Zhang, Y., Li, Z., Guo, D., Ke, Z., Chen, P.: Discrete-time ZD, GD and NI for solving nonlinear time-varying equations. Numer. Algor. 64(4), 721–740 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  25. Zhang, Y., Xiao, L., Ruan, G., Li, Z.: Continuous and discrete time Zhang dynamics for time-varying 4th root finding. Numer. Algor. 57(1), 35–51 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  26. Piepmeier, J.A., McMurray, G.V., Lipkin, H.: A dynamic quasi-Newton method for uncalibrated visual servoing. In: Proceedings of IEEE Conference on Robotics and Automation, pp 1595–1600 (1999)

  27. Zhang, Y., Ge, S.S.: Design and analysis of a general recurrent neural network model for time-varying matrix inversion. IEEE Trans. Neur. Netw. 16(6), 1477–1490 (2005)

    Article  Google Scholar 

  28. Xiao, L., Zhang, Y.: Zhang neural network versus gradient neural network for solving time-varying linear inequalities. IEEE Trans. Neur. Netw. 22(10), 1676–1684 (2011)

    Article  MathSciNet  Google Scholar 

  29. Zhang, Y., Chen, K., Tan, H.: Performance analysis of gradient neural network exploited for online time-varying matrix inversion. IEEE Trans. Autom. Control 54 (8), 1940–1945 (2009)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

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

    Long Jin & Yunong Zhang

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

    Long Jin & Yunong Zhang

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

    Long Jin & Yunong Zhang

Authors
  1. Long Jin
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yunong Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yunong Zhang.

Additional information

This work is supported by the National Natural Science Foundation of China (with numbers 61473323), by the Foundation of Key Laboratory of Autonomous Systems and Networked Control, Ministry of Education, China (with number 2013A07), and also by the Science and Technology Program of Guangzhou, China (with number 2014J4100057). Besides, kindly note that both authors of the paper are jointly of the first authorship.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Jin, L., Zhang, Y. Continuous and discrete Zhang dynamics for real-time varying nonlinear optimization. Numer Algor 73, 115–140 (2016). https://doi.org/10.1007/s11075-015-0088-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11075-015-0088-1

Keywords

Search

Navigation