Apostolos Hadjidimos

On the Solution of the Linear Complementarity Problem by the Generalized Accelerated Overrelaxation Iterative Method

In the present work, we determine intervals of convergence for the various parameters involved for what is known as the generalized accelerated overrelaxation iterative method for the solution of the linear complementarity problem. The convergence intervals found constitute sufficient conditions for the generalized accelerated overrelaxation method to converge and are better than what have been known so far.

On the choice of parameters in MAOR type splitting methods for the linear complementarity problem

In the present work we consider the iterative solution of the Linear Complementarity Problem (LCP), with a nonsingular H+ coefficient matrix A, by using all modulus-based matrix splitting iterative methods that have been around for the last couple of years. A deeper analysis shows that the iterative solution of the LCP by the modified Accelerated Overrelaxation (MAOR) iterative method is the “best”, in a sense made precise in the text, among all those that have been proposed so far regarding the following three issues: i) The positive diagonal matrix-parameter Ω ≥ diag(A) involved in the method is Ω = diag(A), ii) The known convergence intervals for the two AOR parameters, α and β, are the widest possible, and iii) The “best” possible MAOR iterative method is the modified Gauss-Seidel one.

The solution of the linear complementarity problem by the matrix analogue of the accelerated overrelaxation iterative method

The Linear Complementarity Problem (LCP), with an H+−matrix coefficient, is solved by using the new “(Projected) Matrix Analogue of the AOR (MAAOR)” iterative method; this new method constitutes an extension of the “Generalized AOR (GAOR)” iterative method. In this work two sets of convergence intervals of the parameters involved are determined by the theories of “Perron-Frobenius” and of “Regular Splittings”. It is shown that the intervals in question are better than any similar convergence intervals found so far by similar iterative methods. A deeper analysis reveals that the “best” values of the parameters involved are those of the (projected) scalar Gauss-Seidel iterative method. A theoretical comparison of the “best” (projected) Gauss-Seidel and the “best” modulus-based splitting Gauss-Seidel method is in favor of the former method. A number of numerical examples support most of our theoretical findings.

The matrix analogue of the scalar AOR iterative method

The Accelerated Overrelaxation (AOR) and the Generalized AOR (GAOR) iterative methods for the solution of linear systems of algebraic equations ( A x = b , A ? C n i? n , det ( A ) ? 0 , b ? C n ) have been around for about four decades and a plethora of variations of them have been proposed. In this work a novel algorithm is introduced, the Matrix Analogue of the AOR (MAAOR) iterative method, which is analysed and studied. The MAAOR method generalizes both the AOR and the GAOR. Sufficient convergence conditions for the GAOR method are determined when the coefficient matrix A of the linear system to be solved is a Hermitian matrix with positive diagonal elements. Similarly, sufficient convergence conditions for the MAAOR method are determined when A is a nonsingular H -matrix. The new convergence conditions are the most general ones so far. Numerical examples are presented in support of the theory developed.

Comparison of three classes of algorithms for the solution of the linear complementarity problem with an H+-matrix

Abstract There are three main classes of iterative methods for the solution of the linear complementarity problem (LCP). In order of appearance these classes are: the “projected iterative methods”, the “(block) modulus algorithms” and the “modulus-based matrix splitting iterative methods”. Which of the three classes of methods is the “best” one to use for the solution of a certain problem is more or less an “open” question despite the fact that the “best” method within each class is known. It is pointed out that by “best” we mean the minimal upper bound of the norm of the matrix operator of the absolute error vector at any iteration step with respect to the norm of the absolute initial error vector. Note that the first and the third classes of methods are iterative ones while the second one is iterative but needs outer ( ≤ n ) and unknown number of inner iteration steps to terminate. One of the main objectives of this work is to consider the solution of the LCP with an H + -matrix and compare and decide, theoretically if possible otherwise by numerical experiments, as to which of the three “best” methods is the “best” one to use in practice.