Class Of Nonlinear Complementarity Problems Research Articles

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.

Improved Inexact Alternating Direction Methods for a Class of Nonlinear Complementarity Problems

Improved Inexact Alternating Direction Methods for a Class of Nonlinear Complementarity Problems

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

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

A generalized modulus-based Newton method for solving a class of non-linear complementarity problems with P-matrices

A generalized modulus-based Newton method for solving a class of non-linear complementarity problems with P-matrices

An index detecting algorithm for a class of TCP([formula omitted]) equipped with nonsingular [formula omitted]-tensors

As a generalization of the well-known linear complementarity problem, tensor complementarity problem (TCP) has been studied extensively in the literature from theoretical perspective. In this paper, we consider a class of TCPs equipped with nonsingular (not necessarily symmetric) M-tensors. The considered TCPs can be regarded as a special class of nonlinear complementarity problems (NCPs), but the underlying mappings in TCPs are not necessarily P0-functions, which preclude the possibility of applying existing Newton-type methods for NCPs to TCPs directly. By introducing a pivot minimum function, we propose an index detecting algorithm, which efficiently exploits the beneficial properties of nonsingular M-tensors and goes beyond algorithmic frameworks designed for general NCPs. Moreover, the proposed algorithm is well-defined in the sense that it generates a nonnegative element-wise nonincreasing sequence converging to a solution of the problem under consideration. Finally, numerical results further support the efficiency and reliability of the proposed algorithm.

Tensor Complementarity Problems—Part I: Basic Theory

Tensors (hypermatrices) are multidimensional analogs of matrices. The tensor complementarity problem is a class of nonlinear complementarity problems with the involved function being defined by a tensor, which is also a direct and natural extension of the linear complementarity problem. In the last few years, the tensor complementarity problem has attracted a lot of attention, and has been studied extensively, from theory to solution methods and applications. This work, with its three parts, aims at contributing to review the state-of-the-art of studies for the tensor complementarity problem and related models. In this part, we describe the theoretical developments for the tensor complementarity problem and related models, including the nonemptiness and compactness of the solution set, global uniqueness and solvability, error bound theory, stability and continuity analysis, and so on. The developments of solution methods and applications for the tensor complementarity problem are given in the second part and the third part, respectively. Some further issues are proposed in all the parts.

Modulus-Based Synchronous Multisplitting Iteration Methods for a Restricted Class of Nonlinear Complementarity Problems

Modulus-Based Synchronous Multisplitting Iteration Methods for a Restricted Class of Nonlinear Complementarity Problems

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.

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 nonsmooth Newton’s method for solving a class of nonlinear complementarity problems of P-matrices

By applying the Newton’s iteration to the equivalent modulus equations of the nonlinear complementarity problems of P-matrices, a modulus-based nonsmooth Newton’s method is established. The nearly quadratic convergence of the new method is proved under some assumptions. The strategy of choosing the initial iteration vector is given, which leads to a modified method. Numerical examples show that the new methods have higher convergence precision and faster convergence rate than the known modulus-based matrix splitting iteration method.

The relaxation modulus-based matrix splitting iteration method for solving a class of nonlinear complementarity problems

ABSTRACTIn this paper, by the relaxation technique, the relaxation modulus-based matrix splitting iteration method is presented for solving a class of nonlinear complementarity problems. The convergence conditions are given when the system matrix is a positive definite matrix or an H-matrix. The strategy of the choice of the relaxation parameters is discussed. Numerical examples show that the proposed method is efficient and accelerates the convergence performance of the modulus-based matrix splitting iteration method and the projection-based matrix splitting iteration method.

The Sign-Based Methods for Solving a Class of Nonlinear Complementarity Problems

In this paper, using the sign patterns of the solution of the equivalent modulus equation, the resolution of the nonlinear complementarity problem shrinks to find the zero of a differentiable nonlinear function. Then, a sign-based Newton’s method is established by applying the Newton’s iteration. The theoretical analysis for the sign patterns of the solution of the equivalent modulus equation is given under the assumption of strictly complementarity. Moreover, by using the known modulus-based matrix splitting iteration method to detect the sign patterns of the solution of the equivalent modulus equation, a practical sign-detection Newton’s method is proposed. Numerical examples show that the new methods are efficient and accelerate the convergence performance with higher precision and less CPU time than the existing modulus-based matrix splitting iteration method and the projection-based matrix splitting iteration method, especially for the large sparse problems.

On the Convergence of Two-Step Modulus-Based Matrix Splitting Iteration Methods for a Restricted Class of Nonlinear Complementarity Problems with H+-Matrices

On the Convergence of Two-Step Modulus-Based Matrix Splitting Iteration Methods for a Restricted Class of Nonlinear Complementarity Problems with H+-Matrices

Modulus-Based Multisplitting Iteration Methods for a Class of Nonlinear Complementarity Problems

Modulus-Based Multisplitting Iteration Methods for a Class of Nonlinear Complementarity Problems

A two-step modulus-based matrix splitting iteration method for solving nonlinear complementarity problems of $$H_+$$ H + -matrices

In this paper, we establish a two-step modulus-based matrix splitting iteration method for solving a class of nonlinear complementarity problems with the system matrix being an $$H_+$$ -matrix. The convergence analysis of the proposed method is given. Numerical examples show that the proposed method is efficient.

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

In this paper, we reformulate a class of nonlinear complementarity problems as the implicit fixed-point equations. We demonstrate accelerated modulus-based matrix splitting iteration method. We show their convergence by assuming that the system matrix is positive definite or the splitting of the system matrix are $$H_+$$ -compatible splitting and discuss the choice of the optimal parameter. Furthermore, we give two-step accelerated modulus-based matrix splitting iteration method, which may achieve higher computing efficiency. Numerical experiments are presented to show the effectiveness of the method.

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.

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

In this paper, we reformulate the nonlinear complementarity problem as an implicit fixed-point equation. We establish a modulus-based matrix splitting iteration method based on the implicit fixed-point equation and prove its convergence theorem under suitable conditions. Furthermore, we propose a two-step modulus-based matrix splitting iteration method, which may achieve higher computing efficiency. We can obtain many matrix splitting iteration methods by suitably choosing the matrix splittings and the parameters. The proposed methods can be regarded as extensions of the methods for linear complementarity problem. Numerical experiments are presented to show the effectiveness of the proposed methods.

Finite algorithms for the numerical solutions of a class of nonlinear complementarity problems

In this paper, we reformulate a nonlinear complementarity problem or a mixed complementarity problem as a system of piecewise almost linear equations. The problems arise, for example, from the obstacle problems with a nonlinear source term or some contact problems. Based on the reformulated systems of the piecewise almost linear equations, we propose a class of semi-iterative algorithms to find the exact solution of the problems. We prove that the semi-iterative algorithms enjoy a nice monotone convergence property in the sense that subsets of the indices consisting of the indices, for which the corresponding components of the iterates violate the constraints, become smaller and smaller. Then the algorithms converge monotonically to the exact solutions of the problems in a finite number of steps. Some numerical experiments are presented to show the effectiveness of the proposed algorithms.

Weighted max-norm estimate of two-stage splitting method for solving a class of nonlinear complementarity problems

In this paper, we consider the splitting method and the two-stage splitting method for solving a class of nonlinear complementarity problems with the coefficient matrix being an $$H$$ H -matrix. Convergence result for the splitting method is presented when the splitting is $$H$$ H -splitting. Moreover, for the two-stage splitting method, we estimate weighted max-norm bounds for iteration errors, and thereby, we show that the sequence generated by the two-stage iteration scheme converges to the unique solution of the nonlinear complementarity problem without any restriction on the initial vector. Numerical results show that both methods are efficient for solving the class of nonlinear complementarity problems.