Solution of nonsymmetric linear complementarity problems by iterative methods

Convergence is established for iterative algorithms for the solution of the nonsymmetric linear complementarity problem of findingz such thatMz+q0,z0,z ^T(Mz+q)=0, whereM is a givenn×n real matrix, not necessarily symmeetric, andq is a givenn-vector. It is first shown that, if the spectral radius of a matrix related toM is less than one, then the iterates generated by the general algorithm converge to a solution of the linear complementarity problem. It turns out that convergence properties are quite similar to those of linear systems of equations. As specific cases, two important classes of matrices, Minkowski matrices and quasi-dominant diagonal matrices, are shown to satisfy this convergence condition.The author is grateful to Professor O. L. Mangasarian and the referees for their substantive suggestions and corrections.     

Modulus-based synchronous two-stage multisplitting iteration methods for linear complementarity problems

In order to solve large sparse linear complementarity problems on parallel multiprocessor systems, we construct modulus-based synchronous two-stage multisplitting iteration methods based on two-stage multisplittings of the system matrices. These iteration methods include the multisplitting relaxation methods such as Jacobi, Gauss–Seidel, SOR and AOR of the modulus type as special cases. We establish the convergence theory of these modulus-based synchronous two-stage multisplitting iteration methods and their relaxed variants when the system matrix is an H _?+?-matrix. Numerical results show that in terms of computing time the modulus-based synchronous two-stage multisplitting relaxation methods are more efficient than the modulus-based synchronous multisplitting relaxation methods in actual implementations.     

Modulus‐based matrix splitting iteration methods for linear complementarity problems

For the large sparse linear complementarity problems, by reformulating them as implicit fixed‐point equations based on splittings of the system matrices, we establish a class of modulus‐based matrix splitting iteration methods and prove their convergence when the system matrices are positive‐definite matrices and H₊‐matrices. These results naturally present convergence conditions for the symmetric positive‐definite matrices and the M‐matrices. Numerical results show that the modulus‐based relaxation methods are superior to the projected relaxation methods as well as the modified modulus method in computing efficiency. Copyright © 2009 John Wiley & Sons, Ltd.     

The modulus-based matrix splitting iteration methods for second-order cone linear complementarity problems

This paper presents a new computational technique for solving fractional pantograph differential equations. The fractional derivative is described in the Caputo sense. The main idea is to use Müntz-Legendre wavelet and its operational matrix of fractional-order integration. First, the Müntz-Legendre wavelet is presented. Then a family of piecewise functions is proposed, based on which the fractional order integration of the Müntz-Legendre wavelets are easy to calculate. The proposed approach is used this operational matrix with the collocation points to reduce the under study problem to a system of algebraic equations. An estimation of the error is given in the sense of Sobolev norms. The efficiency and accuracy of the proposed method are illustrated by several numerical examples.     

Modulus‐based synchronous multisplitting iteration methods for linear complementarity problems

By an equivalent reformulation of the linear complementarity problem into a system of fixed‐point equations, we construct modulus‐based synchronous multisplitting iteration methods based on multiple splittings of the system matrix. These iteration methods are suitable to high‐speed parallel multiprocessor systems and include the multisplitting relaxation methods such as Jacobi, Gauss–Seidel, successive overrelaxation, and accelerated overrelaxation of the modulus type as special cases. We establish the convergence theory of these modulus‐based synchronous multisplitting iteration methods and their relaxed variants when the system matrix is an H ₊ ‐matrix. Numerical results show that these new iteration methods can achieve high parallel computational efficiency in actual implementations. Copyright © 2012 John Wiley & Sons, Ltd.     

Modified modulus‐based matrix splitting iteration methods for linear complementarity problems

For solving the large sparse linear complementarity problems, we establish modified modulus‐based matrix splitting iteration methods and present the convergence analysis when the system matrices are H₊‐matrices. The optima of parameters involved under some scopes are also analyzed. Numerical results show that in computing efficiency, our new methods are superior to classical modulus‐based matrix splitting iteration methods under suitable conditions. Copyright © 2015 John Wiley & Sons, Ltd.     

Improved convergence theorems of modulus-based matrix splitting iteration methods for linear complementarity problems

We weaken the convergence conditions of modulus-based matrix splitting and matrix two-stage splitting iteration methods for linear complementarity problems. Thus their applied scopes are further extended.     

Two-step modulus-based matrix splitting iteration method for linear complementarity problems

Bai has recently presented a modulus-based matrix splitting iteration method, which is a powerful alternative for solving the large sparse linear complementarity problems. In this paper, we further present a two-step modulus-based matrix splitting iteration method, which consists of a forward and a backward sweep. Its convergence theory is proved when the system matrix is an H ₊-matrix. Moreover, for the two-step modulus-based relaxation iteration methods, more exact convergence domains are obtained without restriction on the Jacobi matrix associated with the system matrix, which improve the existing convergence theory. Numerical results show that the two-step modulus-based relaxation iteration methods are superior to the modulus-based relaxation iteration methods for solving the large sparse linear complementarity problems.     

Accelerated modulus-based matrix splitting iteration methods for linear complementarity problem

For the large sparse linear complementarity problem, a class of accelerated modulus-based matrix splitting iteration methods is established by reformulating it as a general implicit fixed-point equation, which covers the known modulus-based matrix splitting iteration methods. The convergence conditions are presented when the system matrix is either a positive definite matrix or an H ₊-matrix. Numerical experiments further show that the proposed methods are efficient and accelerate the convergence performance of the modulus-based matrix splitting iteration methods with less iteration steps and CPU time.     

Corrector-predictor methods for sufficient linear complementarity problems

We present a new corrector-predictor method for solving sufficient linear complementarity problems for which a sufficiently centered feasible starting point is available. In contrast with its predictor-corrector counterpart proposed by Miao, the method does not depend on the handicap κ of the problem. The method has \(O((1+\kappa)\sqrt{n}L)\)-iteration complexity, the same as Miao’s method, but our error estimates are sightly better. The algorithm is quadratically convergent for problems having a strictly complementary solution. We also present a family of infeasible higher order corrector-predictor methods that are superlinearly convergent even in the absence of strict complementarity. The algorithms of this class are globally convergent for general positive starting points. They have \(O((1+\kappa)\sqrt{n}L)\)-iteration complexity for feasible, or “almost feasible”, starting points and O((1+κ)² nL)-iteration complexity for “sufficiently large” infeasible starting points.     

Interior point methods for linear complementarity problems

A short survey of interior point methods for linear complementarity problems with emphasis on algorithms that have both polynomial complexity and superlinear convergence is presented. Some recent results obtained by the author and his collaborators are briefly summarized and several directions of future research are proposed. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Inexact multisplitting methods for linear complementarity problems

We present an inexact multisplitting method for solving the linear complementarity problems, which is based on the inexact splitting method and the multisplitting method. This new method provides a specific realization for the multisplitting method and generalizes many existing matrix splitting methods for linear complementarity problems. Convergence for this new method is proved when the coefficient matrix is an _H+$H_{+}$-matrix. Then, two specific iteration forms for this inexact multisplitting method are presented, where the inner iterations are implemented either through a matrix splitting method or through a damped Newton method. Convergence properties for both these specific forms are analyzed, where the system matrix is either an _H+$H_{+}$-matrix or a symmetric matrix.     

Two-step modulus-based matrix splitting iteration methods for implicit complementarity problems

Numerical Algorithms - In this paper, a class of two-step modulus-based matrix splitting (TMMS) iteration methods are proposed to solve the implicit complementarity problems. It is proved that the...     

Convergence of SSOR methods for linear complementarity problems

In this paper, by applying the SSOR splitting, we propose two new iterative methods for solving the linear complementarity problem LCP (M,q). Convergence results for these two methods are presented when M is an H-matrix (and also an M-matrix). Finally, two numerical examples are given to show the efficiency of the presented methods.     

Chaotic iterative methods for the linear complementarity problems

A class of parallel multisplitting chaotic relaxation methods is established for the large sparse linear complementarity problems, and the global and monotone convergence is proved for the H-matrix and the L-matrix classes, respectively. Moreover, comparison theorem is given, which describes the influences of the parameters and the multiple splittings upon the monotone convergence rates of the new methods.     

A relaxation modulus-based matrix splitting iteration method for solving linear complementarity problems

In this paper, a relaxation modulus-based matrix splitting iteration method is established, which covers the known general modulus-based matrix splitting iteration methods. The convergence analysis and the strategy of the choice of the parameters are given. Numerical examples show that the proposed methods are efficient and accelerate the convergence performance with less iteration steps and CPU times.     

A two-step modulus-based matrix splitting iteration method for horizontal linear complementarity problems

Numerical Algorithms - In this paper, for solving horizontal linear complementarity problems, a two-step modulus-based matrix splitting iteration method is established. The convergence analysis of...     

Convergence analysis on matrix splitting iteration algorithm for semidefinite linear complementarity problems

Numerical Algorithms - In this paper, we present some novel observations for the semidefinite linear complementarity problems, abbreviated as SDLCPs. Based on these new results, we establish the...