A modified modulus method for symmetric positive‐definite linear complementarity problems

Abstract

By reformulating the linear complementarity problem into a new equivalent fixed-point equation, we deduce a modified modulus method, which is a generalization of the classical one. Convergence for this new method and the optima of the parameter involved are analyzed. Then, an inexact iteration process for this new method is presented, which adopts some kind of iterative methods for determining an approximate solution to each system of linear equations involved in the outer iteration. Global convergence for this inexact modulus method and two specific implementations for the inner iterations are discussed. Numerical results show that our new methods are more efficient than the classical one under suitable conditions.

Two-step modulus-based matrix splitting iteration method for linear complementarity problems

“…We remark that the modulus-based matrix splitting iteration method defined through (4) and (5) includes the modulus iteration method [27], the modified modulus iteration method [18] and the nonstationary extrapolated modulus algorithms [20,21] as its special cases. Also, it can reduce to the nonstationary extrapolated modulus algorithms for the symmetric positive definite matrix with = ( * 1 ) −1 or ( * 2 ) −1 , M = A, N = 0 and γ = 1; see [20,21] for more details.…”

“…This method provides a general framework of the modulus-based matrix splitting iteration methods for solving the LCP(q, A). Besides including the modulus and the modified modulus iteration methods [18,27] as the special cases, it can also yield a series of modulus-based matrix splitting iteration methods by suitably choosing the matrix splittings and the iteration parameters, e.g., the modulus-based Jacobi, Gauss-Seidel, SOR and AOR iteration methods [8]. Numerical experiments have shown that the modulus-based relaxation iteration methods are superior to the projected relaxation iteration methods [16] as well as the modified modulus iteration method [18].…”

“…Besides including the modulus and the modified modulus iteration methods [18,27] as the special cases, it can also yield a series of modulus-based matrix splitting iteration methods by suitably choosing the matrix splittings and the iteration parameters, e.g., the modulus-based Jacobi, Gauss-Seidel, SOR and AOR iteration methods [8]. Numerical experiments have shown that the modulus-based relaxation iteration methods are superior to the projected relaxation iteration methods [16] as well as the modified modulus iteration method [18]. Another advantage of the modulus-based relaxation iteration methods is that neither the projections of the iterates onto the space R n + = {x ∈ R n |x ≥ 0} are required nor the diagonal entries of the matrix A are required to be all nonzeros.…”

Global Modulus-Based Synchronous Multisplitting Multi-Parameters TOR Methods for Linear Complementarity Problems

“…The readers may see references [1][2][3][4] for many problems in scientific computing and engineering applications. When the matrix A is special for LCP(q, A), readers may see the references [5][6][7][8][9][10][11][12][13][14]. Lately, when LCP(q, A) is an algebra system, some scientist have studied it.…”

A wider convergence area for the MSTMAOR iteration methods for LCP

“…In [1] Bai proposed a class of modulus-based splitting iteration methods, which provides a general framework for the modulus iteration [12,13], the modified modulus iteration [9], and the nonstationary extrapolated modulus iteration [10,11]. In order to suit computational requirements of the modern high-speed multiprocessor environments, Bai and Zhang further presented in [5] synchronous parallel counterparts for the modulus-based splitting iteration methods by making use of multiple splittings of the system matrix A [4,14].…”