General four-step discrete-time zeroing and derivative dynamics applied to time-varying nonlinear optimization

Zhang, Yunong; He, Ling; Hu, Chaowei; Guo, Jinjin; Li, Jian; Shi, Yang

doi:10.1016/j.cam.2018.08.017

Cited by 63 publications

(54 citation statements)

References 29 publications

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“…. , 7) satisfy (25). Similarly, substituting the bilinear transform = (1 + /2)/(1 − /2) into the characteristic equation yields a similar equation as (B.14) but with Since 5 = −94/15 < 0, according to eorem 2.6, the general six-step h-order ZeaD formula (24) is divergent.…”

Section: Appendixmentioning

confidence: 91%

“…en, according to eorem 2.6, we have and other elements do not exist. en, by the Routh-Hurwitz stability criterion and De nitions 1-3 in [25], general six-step fourth-order ZeaD formula (21) is convergent if and only if 1 and 2 satisfy the six inequalities listed in (23). Now, we are going to prove any six-step h-order ZeaD formula is divergent.…”

Section: Appendixmentioning

confidence: 97%

“…. , 7) satisfy the system (25). Substituting (25) into (B.23), we get general six-step fourth-order ZeaD formula (24).…”

Section: Appendixmentioning

confidence: 99%

“…, 7) satisfy the system (25). Substituting (25) into (B.23), we get general six-step fourth-order ZeaD formula (24). Now let us analyze the convergence of general six-step h-order ZeaD formula (24), whose characteristic polynomial is the same as that of (21), and ( = 1, 2, .…”

Section: Appendixmentioning

confidence: 99%

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General Six-Step Discrete-Time Zhang Neural Network for Time-Varying Tensor Absolute Value Equations

Sun

Liu

2019

Discrete Dynamics in Nature and Society

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is article presents a general six-step discrete-time Zhang neural network (ZNN) for time-varying tensor absolute value equations. Firstly, based on the Taylor expansion theory, we derive a general Zhang et al. discretization (ZeaD) formula, i.e., a general Taylor-type 1-step-ahead numerical di erentiation rule for the rst-order derivative approximation, which contains two free parameters. Based on the bilinear transform and the Routh-Hurwitz stability criterion, the e ective domain of the two free parameters is analyzed, which can ensure the convergence of the general ZeaD formula. Secondly, based on the general ZeaD formula, we design a general six-step discrete-time ZNN (DTZNN) for time-varying tensor absolute value equations (TVTAVEs), whose steady-state residual error changes in a higher order manner than those presented in the literature. Meanwhile, the feasible region of its step size, which determines its convergence, is also studied. Finally, experiment results corroborate that the general six-step DTZNN model is quite e cient for TVTAVE solving.

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Section: Appendixmentioning

confidence: 91%

Section: Appendixmentioning

confidence: 97%

“…. , 7) satisfy the system (25). Substituting (25) into (B.23), we get general six-step fourth-order ZeaD formula (24).…”

Section: Appendixmentioning

confidence: 99%

Section: Appendixmentioning

confidence: 99%

See 2 more Smart Citations

General Six-Step Discrete-Time Zhang Neural Network for Time-Varying Tensor Absolute Value Equations

Sun

Liu

2019

Discrete Dynamics in Nature and Society

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“…As a special recurrent neural network, Zhang neural network (ZNN), named after Chinese scholar Zhang Yunong, serves as a unified approach to solve various online timevarying problems, such as time-varying quadratic function minimization [13], future minimization [14], time-varying matrix pseudoinversion [8], and TVNO [12,15,16]. For example, based on ZNN, Jin et al [15] presented a one-step discrete-time ZNN (DTZNN) model for TVNO, whose maximal residual error is theoretically O( 2 ).…”

Section: Introductionmentioning

confidence: 99%

Discrete-Time Zhang Neural Networks for Time-Varying Nonlinear Optimization

Sun

Tian

Wang

2019

Discrete Dynamics in Nature and Society

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As a special kind of recurrent neural networks, Zhang neural network (ZNN) has been successfully applied to various time-variant problems solving. In this paper, we present three Zhang et al. discretization (ZeaD) formulas, including a special two-step ZeaD formula, a general two-step ZeaD formula, and a general five-step ZeaD formula, and prove that the special and general two-step ZeaD formulas are convergent while the general five-step ZeaD formula is not zero-stable and thus is divergent. Then, to solve the time-varying nonlinear optimization (TVNO) in real time, based on the Taylor series expansion and the above two convergent two-step ZeaD formulas, we discrete the continuous-time ZNN (CTZNN) model of TVNO and thus get a special two-step discrete-time ZNN (DTZNN) model and a general two-step DTZNN model. Theoretical analyses indicate that the sequence generated by the first DTZNN model is divergent, while the sequence generated by the second DTZNN model is convergent. Furthermore, for the step-size of the second DTZNN model, its tight upper bound and the optimal step-size are also discussed. Finally, some numerical results and comparisons are provided and analyzed to substantiate the efficacy of the proposed DTZNN models.

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An improved preconditioned conjugate gradient method for unconstrained optimization problem with application in Robot arm control

Muhammad,

Waziri,

Sulaiman

et al. 2024

Engineering Reports

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This work suggests improved conjugate gradient methods for enhancing the efficiency and robustness of the classical conjugate gradient methods. The study modifies the diagonal of the inverse Hessian approximation of the Broyden–Fletcher–Goldfarb–Shanno (BFGS) quasi‐Newton update in order to build a preconditioner for nonlinear conjugate gradient (NCG) methods applied to large‐scale unconstrained optimization problems. Damping techniques were embedded into the algorithm to impose the positive definiteness of the diagonal approximation. This made the methods easy to use and presented a viable way to increase the effectiveness of unconstrained optimization techniques. Experimental findings from a collection of benchmark problems demonstrate the efficiency and robustness of the proposed method when compared to five other existing NCG algorithms. Moreover, the successful application of the algorithm to manipulate robotic planar motion control systems with 3 degrees of freedom has been demonstrated, highlighting the practicality of the proposed approach.

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General four-step discrete-time zeroing and derivative dynamics applied to time-varying nonlinear optimization

Cited by 63 publications

References 29 publications

General Six-Step Discrete-Time Zhang Neural Network for Time-Varying Tensor Absolute Value Equations

General Six-Step Discrete-Time Zhang Neural Network for Time-Varying Tensor Absolute Value Equations

Discrete-Time Zhang Neural Networks for Time-Varying Nonlinear Optimization

An improved preconditioned conjugate gradient method for unconstrained optimization problem with application in Robot arm control

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