Modulus-based Iteration Method Research Articles

New accelerated modulus-based iteration method for solving large and sparse linear complementarity problem

For the large and sparse linear complementarity problem, we provide a family of new accelerated modulus-based iteration methods in this article. We provide some sufficient criteria for the convergence analysis when the system matrix is a \(P\)-matrix or an \(H_+\)-matrix. In addition, we provide some numerical examples of the different parameters to illustrate the efficacy of our proposed methods. These methods help us reduce the number of iterations and the time required by the CPU, which improves convergence performance.

More on matrix splitting modulus-based iterative methods for solving linear complementarity problem

More on matrix splitting modulus-based iterative methods for solving linear complementarity problem

A Class of Smoothing Modulus-Based Iterative Methods for Solving the Stochastic Mixed Complementarity Problems

In this paper, we present a smoothing modulus-based iterative method for solving the stochastic mixed complementarity problems (SMCP). The main idea is that we firstly transform the expected value model of SMCP into an equivalent nonsmooth system of equations, then obtain an approximation smooth system of equations by using a smoothing function, and finally solve it by the Newton method. We give the convergence analysis, and the numerical results show the effectiveness of the new method for solving the SMCP with symmetry coefficient matrices.

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.

Modulus-based iterative methods for constrained ℓp–ℓq minimization

The need to solve discrete ill-posed problems arises in many areas of science and engineering. Solutions of these problems, if they exist, are very sensitive to perturbations in the available data. Regularization replaces the original problem by a nearby regularized problem, whose solution is less sensitive to the error in the data. The regularized problem contains a fidelity term and a regularization term. Recently, the use of a p-norm to measure the fidelity term and a q-norm to measure the regularization term has received considerable attention. The balance between these terms is determined by a regularization parameter. In many applications, such as in image restoration, the desired solution is known to live in a convex set, such as the nonnegative orthant. It is natural to require the computed solution of the regularized problem to satisfy the same constraint(s). This paper shows that this procedure induces a regularization method and describes a modulus-based iterative method for computing a constrained approximate solution of a smoothed version of the regularized problem. Convergence of the iterative method is shown, and numerical examples that illustrate the performance of the proposed method are presented.

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

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 modulus-based matrix splitting iteration methods, which are obtained by reformulating equivalently SDLCP as an implicit fixed-point matrix equation. The convergence of the proposed modulus-based matrix splitting iteration methods has been analyzed. Numerical experiments have shown that the modulus-based iteration methods are effective for solving SDLCPs.

The Matrix Splitting Iteration Method for Nonlinear Complementarity Problems Associated with Second-Order Cone

For a class of second-order cone nonlinear complementarity problems, abbreviated as SOCNCPs, we establish the modulus-based matrix splitting relaxation iteration methods, which are obtained by reformulating equivalently SOCNCP as an implicit fixed-point equation based on Jordan algebra associated with the second-order cone. The global convergence theorems are given under suitable choices of the involved splitting matrix and parameter matrices. When the splitting matrix is symmetric positive definite, the strategy choice of the parameters is discussed. Numerical experiments have shown that the modulus-based iteration methods are effective for solving SOCNCPs.

An iteration method for nonlinear complementarity problems

In this paper, we propose a preconditioned modulus-based iteration method for nonlinear complementarity problems. The convergence analysis of the proposed method is presented. Numerical examples are given to illustrate the efficiency of the proposed method.

A class of two-step modulus-based matrix splitting iteration methods for quasi-complementarity problems

In this paper, a class of two-step modulus-based matrix splitting (TMMS) iteration methods are proposed to solve the quasi-complementarity problems. In the new TMMS iteration method, we first use the modulus technique to equivalently transform the quasi-complementarity problem into a fixed point equation, and then construct a class of two-step matrix splitting iteration methods to further accelerate the convergence rates. Theoretical results show that the proposed TMMS iteration methods are global convergent under certain conditions when the system matrix is either a positive definite matrix or an $$H_+$$-matrix. Finally, two numerical examples are presented to demonstrate that the TMMS iteration methods are superior to some existing modulus-based iteration methods studied recently for solving the quasi-complementarity problems.

Numerical investigation of Fredholm integral equation of the first kind with noisy data

We consider Fredholm integral equation of the first kind with noisy data and use Landweber-type iterative methods as an iterative solver. We compare regularization property of Tikhonov, truncated singular value decomposition and the iterative methods. Furthermore, we present a necessary and sufficient condition for the convergence analysis of the iterative method. The performance of the iterative method is shown and compared with modulus-based iterative methods for the constrained Tikhonov regularization.

A direct preconditioned modulus-based iteration method for solving nonlinear complementarity problems of H-matrices

In this paper, we establish a direct preconditioned modulus-based iteration method for solving a class of nonlinear complementarity problems with the system matrix being an H-matrix. The convergence theorems of the proposed method are given, which generalize and improve the existing ones. Numerical examples show that the proposed method is efficient.

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

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 proved when the system matrix A is an H+-matrix by using classical and new results from the theory of splitting. Numerical experiments show that the proposed methods are superior to the existing methods in actual implementation.

The general two-sweep modulus-based matrix splitting iteration method for solving linear complementarity problems

In this paper, we present a general two-sweep modulus-based iteration method to solve a class of linear complementarity problems. Convergence analysis shows that the general two-sweep modulus-based matrix splitting iteration method will converge to the exact solution of linear complementarity problem under appropriate conditions. Numerical experiments further show that the proposed methods are superior to the existing methods in actual implementation.

The relaxation modulus-based matrix splitting iteration methods for circular cone nonlinear complementarity problems

In this paper, we study a class of nonlinear complementarity problems associated with circular cone (CCNCP for short), which is a type of non-symmetric cone complementarity problems. Useful properties of the circular cone are investigated, which help to reformulate equivalently CCNCP as an implicit fixed-point equation. Based on the implicit fixed-point equation and splittings of the system matrix, we establish a class of relaxation modulus-based matrix splitting iteration methods for solving such a complementarity problem. The convergence of the proposed modulus-based matrix splitting iteration methods has been analyzed and the strategy choice of the parameters are discussed when the splitting matrix is symmetric positive definite. Numerical experiments have shown that the modulus-based iteration methods are effective for solving CCNCP.

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

For the second-order cone linear complementarity problems, abbreviated as SOCLCPs, we establish two classes of modulus-based matrix splitting iteration methods, which are obtained by reformulating equivalently the SOCLCP as an implicit fixed-point equation based on Jordan algebra associated with the second-order cone. The convergence of these modulus-based matrix splitting iteration methods has been established and the optimal iteration parameters of these methods are discussed when the splitting matrix is symmetric positive definite. Numerical experiments have shown that the modulus-based iteration methods are effective for solving the SOCLCPs.

On the convergence of modulus-based matrix splitting iteration methods for a class of nonlinear complementarity problems with H+-matrices

A class of modulus-based iteration methods for the nonlinear complementarity problem are presented by reformulating the complementarity problem to an implicit fixed-point equation. In order to efficiently implement these methods, practical iteration schemes based on matrix splitting and the choices of the parameters are presented and well studied. Moreover, we extend the convergence theory from the H -compatible splitting to H -splitting when the matrix is an H + -matrix, which results in more choices for the parameters. Numerical experiments are given to verify the theoretical results and illustrate the efficiency of the proposed methods.

Modulus-based iterative methods for constrained Tikhonov regularization

Tikhonov regularization is one of the most popular methods for the solution of linear discrete ill-posed problems. In many applications the desired solution is known to lie in the nonnegative cone. It is then natural to require that the approximate solution determined by Tikhonov regularization also lies in this cone. The present paper describes two iterative methods, that employ modulus-based iterative methods, to compute approximate solutions in the nonnegative cone of large-scale Tikhonov regularization problems. The first method considered consists of two steps: first the given linear discrete ill-posed problem is reduced to a small problem by a Krylov subspace method, and then the reduced Tikhonov regularization problems so obtained is solved. The second method described explores the structure of certain image restoration problems. Computed examples illustrate the performances of these methods.

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.

A preconditioned modulus-based iteration method for solving linear complementarity problems of H-matrices

In this paper, we establish a preconditioned modulus-based iteration method for solving linear complementarity problems with the system matrix being an -matrix. The convergence analysis of the proposed method is given. In particular, we propose a class of special nondiagonal preconditioners with some parameters for the new method, and then give the convergence domain. Numerical examples show that the proposed method is efficient.