Stepsize domain confirmation and optimum of ZeaD formula for future optimization

Future optimization, which is also known as discrete-time time-variant optimization problem, is an important issue in scientific fields. Recently, Guo et al. have proposed a new effective three-step discrete-time zeroing dynamics (DTZD) model (Guo et al. Numer. Algorithms 77 (1), 23–36, 2018 ) to solve future optimization problems, which is discretized from continuous-time zeroing dynamics (CTZD) model via utilizing a type of Zhang et al. discretization (ZeaD) formula whose coefficients are proportional to 6, 3, 2 , and 1 (termed as ZeaD formula 6321). In this paper, we mainly focus on the stability of this DTZD model. There is an important parameter that closely relates to the stability of the DTZD model, which is called stepsize. Through theoretical study, we obtain the accurate stepsize domain, which makes the DTZD model stable, and the result, i.e., stepsize h∈ (0,0.8) , confirms Guo et al.’s previous investigation. Furthermore, the optimum of the stepsize, which makes the DTZD model converge fastest to steady state in terms of residual error and also provides the best stability (i.e., most away from unstable state), is discussed and investigated as well on the basis of theoretical derivation. Eventually, numerical experiments are carried out to confirm again the correctness of the stepsize domain and the optimum in the DTZD model for future optimization.