Optimum parameter for the SOR-like method for augmented systems

Abstract

Recently, several proposals for the generalization of Young's SOR method to the saddle point problem or the augmented system has been presented. One of the most practical versions is the SOR-like method given by Golub et al., [(2001). SOR-like methods for augmented systems. BIT, 41, 71-85.], where the convergence and the determination of its optimum parameters were given. In this article, a full characterization of the spectral radius of the SOR-like iteration matrix is given, and an explicit expression for the optimum parameter is given in each case. The new results also lead to different results to that of Golub et al. Besides, it is shown that by the choices of the preconditioning matrix, the optimum SOR-like iteration matrix has no complex eigenvalues, therefore, it can be accelerated by semi-iterative methods.

Inner solvers for interior point methods for large scale nonlinear programming

“…and apply the Gauss-Seidel method (SOR-like method in [30] with ω = 1, see also [35]), we obtain the two iterations of the method of multipliers (20-21) (see also [29]). Moreover, we note that, since B t has full row rank, the null space of BB t is equal to the null space of B t and therefore the matrix A is positive definite on the null space of BB t .…”

A Priori Indicator of Convergence Rate for Power Method for Computing the Spectral Radius of Saddle Point Matrices

“…Saddle point problems arise from many scientific research fields and engineering applications, such as mixed finite element methods, constrained least square problems, image processing and so on [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16], and usually generate the following linear system:…”

“…Matrix is symmetric and indefinite [1], and has a peculiar block structure. In order to make better use of the sparsity of in large-scale computation, researchers have developed various efficient algorithms over the past 30 years [1][2][3][4][5][6][7][8]. As far as we know, the focus of current research has shifted to the preconditioning techniques for accelerating the convergence rate of the iteration algorithms.…”