- Multi-step discrete-time Zhang neural networks with application to time-varying nonlinear optimization Sun Min(ziyouxiaodou163.com) Tian Maoying(hfmaoying163.com) Wang Yiju(wangyiju163.com) Abstract: 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 first propose a special two-step Zhang et al. discretization (ZeaD) formula and a general two-step ZeaD formula, whose truncation errors are ${O}(\tau^3)$ and ${O}(\tau^2)$, respectively, and $\tau>0$ denotes the sampling gap. We also propose a general five-step ZeaD formula with truncation error ${O}(\tau^5)$, and prove that the special and general two-step ZdaD formulas is convergent but the general five-step ZeaD formula is not zero-stable, thus is not convergent. 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 proposed in the literature, and thus get a special two-step discrete-time ZNN (DTZNN) model and a general two-step DTZNN model, which contains a free parameter $a_1\in(-1/2,+\infty)$. Theoretical analyses indicate that the sequence generated by the first DTZNN model is not convergent, and for any $a_1\in(-1/2,+\infty)$ and the step-size $h\in(0,(2+4a_1)/(1+a_1))$, the sequence generated by the second DTZNN model converges to zero in an $\mathcal{O}(\tau^2)$ manner, where $\mathcal{O}(\tau^2)$ denotes a vector with every entries being $O(\tau^2)$. Furthermore, we prove that for any fixed $a_1\in(-1/2,+\infty)$, the constant $(2+4a_1)/(1+a_1)$ is the tight upper bound of the step-size $h$ and the constant $(1+2a_1)/(1+a_1)$ is the optimal step-size. Finally, some numerical results and comparisons are provided and analyzed to substantiate the efficacy of the proposed DTZNN models. Keywords: Time-varying nonlinear optimization; Zhang et al. discretization; discrete-time Zhang neural network. Category 1: Applications -- OR and Management Sciences Citation: Download: [PDF]Entry Submitted: 11/30/2018Entry Accepted: 11/30/2018Entry Last Modified: 11/30/2018Modify/Update this entry Visitors Authors More about us Links Subscribe, Unsubscribe Digest Archive Search, Browse the Repository Submit Update Policies Coordinator's Board Classification Scheme Credits Give us feedback Optimization Journals, Sites, Societies Optimization Online is supported by the Mathematical Optmization Society.