Most patients with SCI require augmented methods for bladder emptying.
Among these methods, the parallel splitting augmented Lagrangian method and the alternating direction method of multipliers are two power tools.
For example, novel split Bregman algorithm [24, 25], dual method [26, 27], and augmented Lagrangian method [21, 22, 28], and these fast algorithms all avoid computing complex curvature associated with TV regularizer term.
Based on the techniques, there are two most noteworthy categories of decomposition methods for the problem (1.1 - 1.2) in the literature: the alternating direction method and the parallel splitting augmented Lagrangian method.
We denote the proposed method in this paper as 'NEW', the alternating projection-based prediction-correction method in [15] as 'HLQ', the inexact parallel splitting augmented Lagrangian method in [12] as 'TY' and the simultaneous method in [13] as 'ZHY'ZHY
However, it should be noted that many algorithms in optimization and variational inequalities, such as penalty-type methods and augmented Lagrangian methods, terminate when the constraint is approximately satisfied.
In fact, NL2SOL is a variation of the Newton's method (augmented Gauss-Newton method), in which a part of the Hessian is computed exactly and a part is approximated by a secant (quasi-Newton) updating method.
In detail, four fast projection methods are proposed, i.e., split Bregman projection method (SBPM), augmented Lagrangian projection method (ALPM), dual split Bregman projection method (DSBPM), and dual augmented Lagrangian projection method (DALPM).
It can be proved to have the property that the modified Bregman method is equivalent to the corresponding accelerated augmented Lagrangian method, and the latter has a rapid convergence rate which can be deemed to be an improvement of [17].
