General Third-Order-Accuracy Formulas for Time Discretization Applied to Time-Varying Optimization

Time discretization is an important part of time-varying problems solving that determines convergence, real-time performance and accuracy for solution models. It is a challenging work compared with relatively simple derivative approximation due to unknown future information and stability constraint. To the best of the authors’s knowledge, no effective time discretization was developed other than Euler finite difference formula before recently development of ZeaD formulas. Existing work presents some ZeaD formulas including specific time-discretization formulas having third order accuracy. In this work,
$N$
-instant general third-order-accuracy formula is proposed, and it leads to different general third-order-accuracy formulas when different instant number
$N$
is considered. Stability and convergence are analyzed, and effective domains for parameters in 5-instant and 6-instant general third-order-accuracy formula are given to guarantee effective time discretization. Furthermore,
$N$
-instant general third-order-accuracy formula is employed to solve time-varying optimization, and
$N$
-instant general solution model is proposed. Finally, comparative experimental results are presented to substantiate the effectiveness and superiority of proposed general formulas and models.