Methods For Linear Complementarity Problems Research Articles

On the two-stage multisplitting iteration methods for linear complementarity problems

In this work, the two-stage multisplitting iteration methods based on the equivalent modulus equations are analyzed for solving linear complementarity problems. New convergence results are presented where the convergence domains of the parameter matrices are enlarged compared the existing literatures. Furthermore, the improved domains are validated by numerical examples with parallel computations using OpenACC.

A relaxed two-step modulus-based matrix synchronous multisplitting iteration method for linear complementarity problems

A relaxed two-step modulus-based matrix synchronous multisplitting iteration method for linear complementarity problems

A new matrix splitting generalized iteration method for linear complementarity problems

The linear complementarity problems (LCPs) can be encountered in various scientific computing, management science, and operations research. In this study, we introduce and analyze a new generalized accelerated overrelaxation (NGAOR) method for solving LCPs, in which one special case reduces to a new generalized successive overrelaxation (NGSOR) method. Moreover, we prove the convergence of the proposed methods when the system matrix is an H-matrix (irreducible or strictly diagonally dominant matrix). Numerical results for several experiments are present to show the effectiveness and efficiency of the proposed methods.AMS classification: 65F10, 90C33

Accelerated Relaxation Two-Sweep Modulus-Based Matrix Splitting Iteration Method for Linear Complementarity Problems

In this paper, by applying the accelerated technique and relaxation two-sweep strategy to the modulus-based matrix splitting iteration method, we establish the accelerated relaxation two-sweep modulus-based matrix splitting method. The proposed method contains the accelerated modulus-based matrix splitting and the generalized accelerated modulus-based matrix splitting methods presented recently as special cases. Compared with some existing methods, the proposed method can use more information during each iteration, which may lead to higher computational efficiency. We investigate the convergence properties of the new method, and prove that it is convergent under certain assumptions. Also, for the accelerated relaxation two-sweep modulus-based accelerated overrelaxation method, we analyze the convergence regions of the parameters in detail. Numerical examples are offered to show that the proposed method is efficient and performs better than the generalized accelerated modulus-based matrix splitting one in actual implementation.

The double-relaxation modulus-based matrix splitting iteration method for linear complementarity problems

For the large sparse linear complementarity problems (LCP), by adopting the relaxation and relaxation two-sweep techniques for the modulus-based matrix splitting (MMS) iteration method, we establish a double-relaxation two-sweep modulus-based matrix splitting (DRTMMS) iteration method. This new method contains some known ones developed recently. Some sufficient conditions for guaranteeing the convergence of the DRTMMS method are presented when the system matrices are positive definite matrices and H+-matrices, which generalize some existing results. And the parameter selection strategy of the DRTMMS iteration method is given. Besides, the convergence analyses of the DRTM accelerated overrelaxation (DRTMAOR) method are given. Finally, numerical experiments illustrate that the DRTMMS method is efficient and has better performance than several existing MMS-like methods, and outperforms the projected splitting methods in some cases.

On the new modulus-based matrix splitting method for linear complementarity problem of $$H_{+}$$-matrix

On the new modulus-based matrix splitting method for linear complementarity problem of $$H_{+}$$-matrix

Polyhedral complementarity problem with quasimonotone decreasing mappings

The fixed point problem of piecewise constant mappings in Rn is investigated. This is a polyhedral complementarity problem, which is a generalization of the linear complementarity problem. Such mappings arose in the author?s research on the problem of economic equilibrium in exchange models, where mappings were considered on the price simplex. The author proposed an original approach of polyhedral complementarity, which made it possible to obtain simple algorithms for solving the problem. The present study is a generalization of linear complementarity methods to related problems of a more general nature and reveals a close relationship between linear complementarity and polyhedral complementarity. The investigated method is an analogue of the well-known Lemke method for linear complementarity problems. A class of mappings is described for which the process is monotone, as it is for the linear complementarity problems with positive principal minors of the constraint matrix (class P). It is shown that such a mapping has always unique fixed point.

Accelerated double-relaxation modulus-based matrix splitting iteration method for linear complementarity problems

This paper is concerned with solving linear complementarity problems (LCP) arising in many scientific and engineering fields. We propose an accelerated double-relaxation two-sweep modulus-based matrix splitting (ADRTMMS) iteration method by applying accelerating, relaxation and relaxation two-sweep techniques to the MMS one. This new method contains some known ones developed recently. Some sufficient conditions for guaranteeing the convergence of the ADRTMMS method are presented when the system matrices both are positive definite matrices and H+-matrices, which generalize some existing results. Specially, the convergence of the ADRTM accelerated overrelaxation (ADRTMAOR) method is discussed in details. At last, some numerical examples are provided to show that the ADRTMMS method is efficient and outperforms several existing MMS-like methods.

Two-Step Two-Sweep Modulus-Based Matrix Splitting Iteration Method for Linear Complementarity Problems

Two-Step Two-Sweep Modulus-Based Matrix Splitting Iteration Method for Linear Complementarity Problems

RELAXED MODULUS-BASED SYNCHRONOUS MULTISPLITTING MULTI-PARAMETERS TOR (TWO-PARAMETERS OVER-RELAXATION) METHODS FOR LINEAR COMPLEMENTARITY PROBLEMS

In 2013, Bai and Zhang [<i>Numerical Linear Algebra with Applications</i>, 20(2013):425-439] constructed modulus-based synchronous multisplitting methods by an equivalent reformulation of the linear complementarity problems into a system of fixed-point equations and studied the convergence of them. In 2014, Zhang and Li [<i>Computers and Mathematics with Application</i>, 67(2014):1954-1959] analyzed and obtained the weaker convergence results for linear complementarity problems. In 2008, Zhang et.al. [<i>International Journal of Computer Mathematics</i>, 85(2)(2008):211-224] presented global relaxed non-stationary multisplitting multi-parameter method by introducing some relaxed parameters. In this paper, we generalize Bai and Zhang's methods and study relaxed modulus-based synchronous multisplitting multi-parameters TOR (two-parameters over-relaxation, abbreviated as TOR) methods for linear complementarity problems. Furthermore, the convergence results of our new method in this paper are given when the system matrix is an $H_{+}-$matrix.

Modulus-based inexact non-alternating preconditioned splitting method for linear complementarity problems

For the large sparse linear complementarity problem, by reformulating it as implicit fixed-point equations, we establish a modulus-based iteration method with the aid of an inexact non-alternating preconditioned matrix splitting iteration method used as an inner iteration to solve the module equations rapidly in each iteration step. The convergence properties of this method are carefully demonstrated under certain conditions. Numerical results, including the numbers of iteration steps and the computing times, validate that our method is superior to the other three iteration methods.

A class of new modulus-based matrix splitting methods for linear complementarity problem

In this paper, to economically and fast solve the linear complementarity problem, based on a new equivalent fixed-point form of the linear complementarity problem, we establish a class of new modulus-based matrix splitting methods, which is different from the previously published works. Some sufficient conditions to guarantee the convergence of this new iteration method are presented. Numerical examples are offered to show the efficacy of this new iteration method. Moreover, the comparisons on numerical results show the computational efficiency of this new iteration method advantages over the corresponding modulus method, the modified modulus method and the modulus-based Gauss–Seidel method.

An interior point approach for linear complementarity problem using new parametrized kernel function

In this paper, we propose a primal–dual interior point method for Linear Complementarity Problem (LCP) based on a new parameterized kernel function. The investigation according to it yields the best-known iteration bound for large-update algorithm and thus improves the iteration bound obtained in Bai et al. (SIAM J Optim. 2004;15:101–128) for large-update algorithm. Finally, we present few numerical results to demonstrate the efficiency of the proposed algorithm.

On sufficient properties of sufficient matrices

In this paper we study sufficient matrices, which play an important role in theoretical analysis of interior-point methods for linear complementarity problems. We present new characterisations of these matrices which imply new necessary and sufficient conditions for sufficiency. We use these results to develop an algorithm with exponential iteration complexity which in each iteration solves a simple instance of linear programming problem and is capable to reveal whether given symmetric matrix is sufficient or not. This algorithm demonstrates 100 % accuracy on all tested instances of matrices.

Accelerated Relaxation Modulus-Based Matrix Splitting Iteration Method for Linear Complementarity Problems

For solving large sparse linear complementarity problems effectively, by utilizing the parametric method, the acceleration strategy and the relaxation technique to the modulus-based matrix splitting (MMS) iteration method, we develop the accelerated relaxation MMS (ARMMS) iteration method, which generalizes the generalized accelerated MMS (GAMMS) and the relaxation MMS (RMMS) ones proposed recently. Meanwhile, it is proved that the ARMMS iteration method is convergent under proper restrictions when the system matrix is a positive definite matrix or an $$H_{+}$$ -matrix. In addition, we also study the convergence properties of the ARM accelerated overrelaxation (ARMAOR) method. Numerical experiments show that the proposed iteration method is efficient, and it outperforms some existing ones with suitable choice of the parameter and matrix splitting.

A modified modulus-based multigrid method for linear complementarity problems arising from free boundary problems

The linear complementarity problem arising from a free boundary problem can be equivalently reformulated as a fixed-point equation. We present a modified modulus-based multigrid method to solve this fixed-point equation. This modified method is a full approximation scheme using the modulus-based splitting iteration method as the smoother and avoids the transformation between the auxiliary and the original functions which was necessary in the existing modulus-based multigrid method. We predict its asymptotic convergence factor by applying local Fourier analysis to the corresponding two-grid case. Numerical results show that the W-cycle possesses an h-independent convergence rate and a linear elapsed CPU time, and the convergence rate of the V-cycle can be improved by increasing the smoothing steps. Compared with the existing modulus-based multigrid method, the modified method is more straightforward and is a standard full approximation scheme, which makes it more convenient and efficient in practical applications.

GLOBAL RELAXED MODULUS-BASED SYNCHRONOUS BLOCK MULTISPLITTING MULTI-PARAMETERS METHODS FOR LINEAR COMPLEMENTARITY PROBLEMS

Recently, Bai and Zhang [Numerical Linear Algebra with Applications, 2013, 20, 425Ƀ439] constructed modulus-based synchronous multisplitting methods by an equivalent reformulation of the linear complementarity problem into a system of fixed-point equations and studied the convergence of them; Li et al. [Journal of Nanchang University (Natural Science), 2013, 37, 307Ƀ312] studied synchronous block multisplitting iteration methods; Zhang and Li [Computers and Mathematics with Application, 2014, 67, 1954Ƀ1959] analyzed and obtained the weaker convergence results for linear complementarity problems. In this paper, we generalize their algorithms and further study global relaxed modulus-based synchronous block multisplitting multi-parameters methods for linear complementarity problems. Furthermore, we give the weaker convergence results of our new method in this paper when the system matrix is a block $ H_{+}- $matrix. Therefore, new results provide a guarantee for the optimal relaxation parameters, please refer to [A. Hadjidimos, M. Lapidakis and M. Tzoumas, SIAM Journal on Matrix Analysis and Applications, 2012, 33, 97Ƀ110, (dx.doi.org/10.1137/100811222)], where optimal parameters are determined.

Relaxed Modulus-Based Synchronous Multisplitting Multi-Parameter Methods for Linear Complementarity Problems

In 2013, Bai and Zhang (Numer Linear Algebra Appl, 20:425–439 2013) constructed modulus-based synchronous multisplitting iteration methods by an equivalent reformulation of the linear complementarity problem into a system of fixed-point equations and studied the convergence of them. In this paper, we generalize Bai and Zhang’s method and study relaxed modulus-based synchronous multisplitting multi-parameter methods for linear complementarity problems. Furthermore, when the system matrix is an H+-matrix, we give convergence results for our new methods under weaker conditions than those in Bai and Zhang’s. Finally, numerical experiments are presented to illustrate the efficiency of the proposed methods.

On the equivalence between some projected and modulus-based splitting methods for linear complementarity problems

In this paper, we analyze the relationship between projected and (possibly accelerated) modulus-based matrix splitting methods for linear complementarity problems. In particular, first we show that some well-known projected splitting methods are equivalent, iteration by iteration, to some (accelerated) modulus-based matrix splitting methods with a specific choice of the parameter $${\varOmega }$$. We then generalize this result to any $${\varOmega }$$ by formulating new classes of projected splitting methods and also provide a formal projection-based formulation for general (accelerated) modulus-based matrix splitting methods. Finally, we introduce and solve several test problems to evaluate also numerically the equivalence between the analyzed methods.

A class of modified modulus-based synchronous multisplitting iteration methods for linear complementarity problems

In this paper, we present modified modulus-based synchronous multisplitting iteration methods based on multisplittings of the system matrix for solving the large sparse linear complementarity problems. The proposed methods extend the existing modulus-based synchronous multisplitting iteration methods to a more general case. We establish the convergence theory of these modified modulus-based synchronous multisplitting iteration methods when the system matrix is an H+-matrix. In particular, we investigate the optimal choice of the parameter matrices in theory. Numerical results confirm that the new iteration methods have higher parallel computational efficiency than the existing modulus-based synchronous multisplitting iteration methods. The proposed methods are applied in reconstruction of two-dimensional image data and show the efficiency.