Modulus-based Matrix Splitting Research Articles

Generalized modulus-based matrix splitting algorithm with Anderson acceleration strategy for vertical linear complementarity problems

The vertical linear complementarity problem (VLCP) with an arbitrary number l of matrices is related to many practical problems, of which the state-of-the-art modulus-based matrix splitting (MMS) method is proved to be an efficient solver. Enlightened by the Anderson acceleration which is a well-established and simple technique for speeding up fixed point iteration solvers with countless applications, we propose an Anderson accelerated generalized modulus-based matrix splitting (AA+GMMS) method for solving the VLCP. We particularly analyze the AA+GMMS method for the problem with l=2 and then generalize the method to any l. More importantly, the convergence theorems and theoretical optimal parameters of the MMS, AA+GMMS methods with any l are obtained in the positive definite case. Eventually, numerical experiments are given to demonstrate the effectiveness of the AA+GMMS method which significantly accelerates the original MMS method. In particular, we explore the parameters involved in the AA+GMMS method, and they have a small extent of impact on the suggested method, reinforcing that the AA+GMMS method is highly efficient.

The simplified modulus-based matrix splitting iteration method for the nonlinear complementarity problem

<abstract><p>In this paper, the simplified modulus-based matrix splitting iteration method was extended to solve the nonlinear complementarity problem, and the convergence conditions were presented from the spectral radius and the matrix norm. Then, for the special cases of this method, we provided some concrete convergence conditions as well as the quasi-optimal parameter matrix. Moreover, some numerical examples were illustrated to show the validity of the convergence results.</p></abstract>

A two-step new modulus-based matrix splitting method for vertical linear complementarity problem

In this paper, for solving the vertical linear complementarity problem (VLCP) effectively, a two-step new modulus-based matrix splitting (TNMMS) iteration method is introduced. Its convergence properties are discussed in depth. The efficiency of the proposed method is confirmed by some numerical experiments from the 2-D boundary problem. Numerical results show that the proposed method is superior to some existing methods, such as the two modulus-based matrix splitting (TMMS) iteration method (Numer. Algor., 2011, 57: 83–99) and the new modulus-based matrix splitting (NMMS) iteration method (Optim., 2023, 72: 2499–2516).

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 nonlinear lopsided PSS-like and HSS-like modulus-based matrix splitting iteration methods for horizontal linear complementarity problem

The nonlinear lopsided PSS-like and HSS-like modulus-based matrix splitting iteration methods for horizontal linear complementarity problem

Modulus-based synchronous multisplitting iteration methods without auxiliary variable for solving vertical linear complementarity problems

In this work, by applying the synchronous multisplitting technique to the non-auxiliary variable modulus equation of the vertical linear complementarity problems, a new parallel method is constructed, which can generalize the existing modulus-based matrix splitting iteration method. The convergence conditions of the proposed method are discussed in the cases of more than two system matrices, and the existing results are improved. Numerical examples under OpenACC framework show that the proposed method can solve the large sparse vertical linear complementarity problems efficiently with high parallel efficiency.

Relaxation modulus-based matrix splitting iteration method for vertical linear complementarity problem

For solving vertical linear complementarity problems (VLCPs), we propose a relaxation modulus-based matrix splitting iteration method and a two-step relaxation modulus-based matrix splitting iteration method. When the system matrices are H+-matrices, we analyze the convergence theory of the proposed methods. Numerical experiments show that relaxation modulus-based matrix splitting iteration method and the two-step relaxation modulus-based matrix splitting iteration method are promising and have great performance in higher dimension.

A modulus-based matrix splitting method for the vertical nonlinear complementarity problem

A modulus-based matrix splitting method for the vertical nonlinear complementarity problem

Improved modulus-based matrix splitting iteration methods for quasi-complementarity problems

In this paper, the improved modulus-based matrix splitting iteration methods are proposed to solve the quasi-complementarity problems. It is proved that the proposed iteration methods are convergent under certain conditions by using some new arguments. Numerical experiments are given to illustrate the high efficiency of the new iteration methods. Numerical results show that the proposed methods have better performance than the existing methods.

An Improved Convergence Condition of the MMS Iteration Method for Horizontal LCP of H+-Matrices

In this paper, inspired by the previous work in (Appl. Math. Comput., 369 (2020) 124890), we focus on the convergence condition of the modulus-based matrix splitting (MMS) iteration method for solving the horizontal linear complementarity problem (HLCP) with H+-matrices. An improved convergence condition of the MMS iteration method is given to improve the range of its applications, in a way which is better than that in the above published article.

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.

Modulus-based matrix splitting methods for complex linear complementarity problem

At present, the numerical solution of the complex linear complementarity problem (CLCP) by iteration methods is scanty. To fill in this gap, the goal of this paper is to establish a class of modulus-based matrix splitting methods for solving the CLCP by expressing the real form of the CLCP as the equivalent absolute value equations. Some sufficient conditions to guarantee the convergence of the proposed methods are provided. Numerical examples are offered to confirm the feasibility and efficiency of the proposed methods.

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

A preconditioned new modulus-based matrix splitting method for solving linear complementarity problem of $ H_+ $-matrices

<abstract><p>For solving the linear complementarity problem (LCP), we propose a preconditioned new modulus-based matrix splitting (PNMMS) iteration method by extending the state-of-the-art new modulus-based matrix splitting (NMMS) iteration method to a more general framework with a customized preconditioner. We devise a generalized preconditioner that is associated with both $ H_+ $-matrix $ A $ and vector $ q $ of the LCP. The convergence analysis is conducted under some mild conditions. In particular, we provide a comparison theorem to theoretically show the PNMMS method accelerates the convergence rate. Numerical experiments further illustrate that the PNMMS method is efficient and has better performance for solving the large and sparse LCP.</p></abstract>

CONVERGENCE OF THE TWO-POINT MODULUS-BASED MATRIX SPLITTING ITERATION METHOD

In this paper, we discuss the convergence of the two-point modulus-based matrix splitting iteration method for solving the linear complementarity problem. Some convergence conditions are presented from the spectral radius and the matrix norm when the system matrix is a <i>P</i>-matrix. Besides, the quasi-optimal cases of the method are studied. Numerical experiments are provided to show the presented results.

Inexact fixed-point iteration method for nonlinear complementarity problems

Based on the modulus decomposition, the structured nonlinear complementarity problem is reformulated as an implicit fixed-point system of nonlinear equations. Distinguishing from some existing modulus-based matrix splitting methods, we present a flexible modulus-based inexact fixed-point iteration method for the resulting system, in which the subproblem can be solved approximately by a linear system-solver. The global convergence of the proposed method is established by assuming that the system matrix is positive definite. Some numerical comparisons are reported to illustrate the applicability of the proposed method, especially for large-scale problems.

The square-Newton iteration for linear complementarity problem

For getting the numerical solution of the linear complementary problem (LCP), there are many methods such as the modulus-based matrix splitting iteration and the modulus-based nonsmooth Newton?s method. We proposed the Square-Newton method to solve the LCP. This method could solve LCP efficiently. We gave the theoretical analysis and numerical experiments in the paper.

THE MODULUS-BASED MATRIX SPLITTING METHOD WITH INNER ITERATION FOR A CLASS OF NONLINEAR COMPLEMENTARITY PROBLEMS

In this paper, we propose a modulus-based matrix splitting iteration method with inner iteration for a class of nonlinear complementarity problems. Convergence conditions of the iteration method are analyzed carefully, which shows that the iteration sequence generated by this method converges to a solution of the NCP under certain conditions. Moreover, the convergence conditions of the proposed method are studied when the system matrix is symmetric positive definite or is an <i>H</i>₊-matrix. Theoretical results are supported by the numerical experiments, which implies that the iteration method with inner iteration is more effective and feasible for solving certain nonlinear complementarity problems.

A new relaxed acceleration two-sweep modulus-based matrix splitting iteration method for solving linear complementarity problems

&lt;abstract&gt;&lt;p&gt;A new relaxed acceleration two-sweep modulus-based matrix splitting (NRATMMS) iteration method is developed to solve linear complementarity problems. The convergence of the NRATMMS method is established with the system matrix $ A $ being an $ H_{+} $-matrix. Numerical experiments show that the proposed method is superior to some existing algorithms under appropriate conditions.&lt;/p&gt;&lt;/abstract&gt;

On estimation of the optimal parameter of the modulus‐based matrix splitting algorithm for linear complementarity problems on second‐order cones

AbstractThere are many studies on the well‐known modulus‐based matrix splitting (MMS) algorithm for solving complementarity problems, but very few studies on its optimal parameter, which is of theoretical and practical importance. Therefore and here, by introducing a novel mapping to explicitly cast the implicit fixed point equation and thus obtain the iteration matrix involved, we first present the estimation approach of the optimal parameter of each step of the MMS algorithm for solving linear complementarity problems on the direct product of second‐order cones (SOCLCPs). It also works on single second‐order cone and the non‐negative orthant. On this basis, we further propose an iteration‐independent optimal parameter selection strategy for practical usage. Finally, the practicability and effectiveness of the new proposal are verified by comparing with the experimental optimal parameter and the diagonal part of system matrix. In addition, with the optimal parameter, the effectiveness of the MMS algorithm can indeed be greatly improved, even better than the state‐of‐the‐art solvers SCS and SuperSCS that solve the equivalent SOC programming.