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...     

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.     

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...     

A generalized variant of two-sweep modulus-based matrix splitting iteration method for solving horizontal linear complementarity problems

Numerical Algorithms - Based on the two-sweep modulus-based matrix splitting iteration (TMMS) method for linear complementarity problems developed by Wu and Li (Comput. Appl. Math. 302:...     

Convergence of modulus-based matrix splitting iteration method for a class of nonlinear complementarity problems

Numerical Algorithms - In this paper, we discuss the modulus-based matrix splitting iteration method for solving a class of nonlinear complementarity problems under a weakened condition, and...     

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.     

New modulus-based matrix splitting methods for implicit complementarity problem

Numerical Algorithms - In this paper, based on the published works by Bai and Tong (J. Univ. Electr. Sci. Tech. China 22:420–424, 1993) and Bai and Huang (J. Univ. Electr. Sci. Tech. China...     

A preconditioned general two-step modulus-based matrix splitting iteration method for linear complementarity problems of H+-matrices

Numerical Algorithms - In this paper, we present a preconditioned general two-step modulus-based iteration method to solve a class of linear complementarity problems. Its convergence theory is...     

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.     

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.     

Accelerated modulus-based matrix splitting iteration methods for a restricted class of nonlinear complementarity problems

To solve a class of nonlinear complementarity problems, accelerated modulus-based matrix splitting iteration methods are presented and analyzed. Convergence analysis and the choice of the parameters are given when the system matrix is either positive definite or an H ₊-matrix. Numerical experiments further demonstrate that the proposed methods are efficient and have better performance than the existing modulus-based iteration method in aspects of the number of iteration steps and CPU time.     

Improved convergence theorems of the two-step modulus-based matrix splitting and synchronous multisplitting iteration methods for solving linear complementarity problems

Abstract

In this paper, the convergence conditions of the two-step modulus-based matrix splitting and synchronous multisplitting iteration methods for solving linear complementarity problems of H-matrices are weakened. The convergence domain given by the proposed theorems is larger than the existing ones.     

Two-step modulus-based matrix splitting iteration methods for retinex problem

Numerical Algorithms - Retinex theory was proposed by Land in the 1970s. Its theoretical basis is that the color of an object is determined by the reflection ability of the object to the light 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...     

A preconditioned general modulus-based matrix splitting iteration method for linear complementarity problems of <Emphasis Type="Italic">H</Emphasis>-matrices

In this paper, we propose a preconditioned general modulus-based matrix splitting iteration method for solving modulus equations arising from linear complementarity problems. Its convergence theory is proved when the system matrix is an H₊-matrix, from which some new convergence conditions can be derived for the (general) modulus-based matrix splitting iteration methods. Numerical results further show that the proposed methods are superior to the existing methods.     

The selection of the optimal parameter in the modulus-based matrix splitting algorithm for linear complementarity problems

The modulus-based matrix splitting (MMS) algorithm is effective to solve linear complementarity problems (Bai in Numer Linear Algebra Appl 17: 917–933, 2010). This algorithm is parameter dependent, and previous studies mainly focus on giving the convergence interval of the iteration parameter. Yet the specific selection approach of the optimal parameter has not been systematically studied due to the nonlinearity of the algorithm. In this work, we first propose a novel and simple strategy for obtaining the optimal parameter of the MMS algorithm by merely solving two quadratic equations in each iteration. Further, we figure out the interval of optimal parameter which is iteration independent and give a practical choice of optimal parameter to avoid iteration-based computations. Compared with the experimental optimal parameter, the numerical results from three problems, including the Signorini problem of the Laplacian, show the feasibility, effectiveness and efficiency of the proposed strategy.

     

A projection-filter method for solving nonlinear complementarity problems

The Josephy-Newton method attacks nonlinear complementarity problems which consists of solving, possibly inexactly, a sequence of linear complementarity problems. Under appropriate regularity assumptions, this method is known to be locally (superlinearly) convergent. Utilizing the filter method, we presented a new globalization strategy for this Newton method applied to nonlinear complementarity problem without any merit function. The strategy is based on the projection-proximal point and filter methodology. Our linesearch procedure uses the regularized Newton direction to force global convergence by means of a projection step which reduces the distance to the solution of the problem. The resulting algorithm is globally convergent to a solution. Under natural assumptions, locally superlinear rate of convergence was established.     

A new method for solving Pareto eigenvalue complementarity problems

In this paper, we introduce a new method, called the Lattice Projection Method (LPM), for solving eigenvalue complementarity problems. The original problem is reformulated to find the roots of a nonsmooth function. A semismooth Newton type method is then applied to approximate the eigenvalues and eigenvectors of the complementarity problems. The LPM is compared to SNM_min and SNM_FB, two methods widely discussed in the literature for solving nonlinear complementarity problems, by using the performance profiles as a comparing tool (Dolan, Moré in Math. Program. 91:201–213, 2002). The performance measures, used to analyze the three solvers on a set of matrices mostly taken from the Matrix Market (Boisvert et al. in The quality of numerical software: assessment and enhancement, pp. 125–137, 1997), are computing time, number of iterations, number of failures and maximum number of solutions found by each solver. The numerical experiments highlight the efficiency of the LPM and show that it is a promising method for solving eigenvalue complementarity problems. Finally, Pareto bi-eigenvalue complementarity problems were solved numerically as an application to confirm the efficiency of our method.