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.     

On convergence of two-stage splitting methods for linear complementarity problems

In this paper, we study the splitting method and two-stage splitting method for the linear complementarity problems. Convergence results for these two methods are presented when the system matrix is an H-matrix and the splittings used are H-splitting. Numerical experiments show that the two-stage splitting method has the same or even better numerical performance than the splitting method in some aspects under certain conditions.     

Modulus-based matrix splitting methods for horizontal linear complementarity problems

Numerical Algorithms - In this paper, we extend modulus-based matrix splitting iteration methods to horizontal linear complementarity problems. We consider both standard and accelerated methods and...     

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 two-step matrix splitting method for the mixed linear complementarity problem

In this paper, on the base of the methodology of the new modulus-based matrix splitting method in [Optim. Lett., (2022) 16:1427-1443], we establish a two-step matrix splitting (TMS) method for solving the mixed linear complementarity problem (MLCP). Two sufficient conditions to ensure the convergence of the proposed method are presented. Numerical examples are provided to illustrate the feasibility and efficiency of the proposed method.     

On the extended linear complementarity problem

For the extended linear complementarity problem (Mangasarian and Pang, 1995), we introduce and characterize column-sufficiency, row-sufficiency andP-properties. These properties are then specialized to the vertical, horizontal and mixed linear complementarity problems. This paper is dedicated to Professor Olvi L. Mangasarian on the occasion of his 60th birthday.     

On the monotone convergence of matrix multisplitting relaxation methods for the linear complementarity problem

The monotone convergence of the parallel matrix multisplittingrelaxation method for linear complementarity problems (see Bai,Z. Z. and Evans, D. J. 1997 Int. J. Comput. Math 63, 309-326)is discussed, and the corresponding comparison theorem aboutthe monotone convergence rate of this method is thoroughly established.     

Two-stage parallel iterative methods for the symmetric linear complementarity problem

In this paper, we propose a two-stage parallel iterative method for solving the symmetric linear complementarity problem. When implemented in a parallel computing environment, the method decomposes the problem into subproblems which are solved by certain iterative procedures concurrently on separate processors. Convergence of the overall method is established under some mild assumptions on how the inner iterations are terminated. Applications of the proposed method to solve strictly convex quadratic programs are discused and numerical results on both a sequential computer (IBM 4381) and a super-computer (CRAYX-MP/24) are reported.This research was based on work supported by the National Science Foundation under grant ECS-8407240 and by a 1986 University Research and Development grant from Cray Research Inc.     

On the modulus algorithm for the linear complementarity problem

Concerning three subclasses of P-matrices the modulus algorithm and the projected successive overrelaxation (PSOR) method solving the linear complementarity problem are compared to each other with respect to convergence. It is shown that the modulus algorithm is convergent for all three subclasses whereas the convergence of the PSOR method is only guaranteed for two of them.     

Modulus‐based matrix splitting iteration methods for linear complementarity problems

For the large sparse linear complementarity problems, by reformulating them as implicit fixed‐point equations based on splittings of the system matrices, we establish a class of modulus‐based matrix splitting iteration methods and prove their convergence when the system matrices are positive‐definite matrices and H₊‐matrices. These results naturally present convergence conditions for the symmetric positive‐definite matrices and the M‐matrices. Numerical results show that the modulus‐based relaxation methods are superior to the projected relaxation methods as well as the modified modulus method in computing efficiency. Copyright © 2009 John Wiley & Sons, Ltd.     

Interior hybrid proximal extragradient methods for the linear monotone complementarity problem

We present new infeasible path-following methods for linear monotone complementarity problems based on Auslender, Teboulle and Ben-Tiba’s log-quadratic barrier functions. The central paths associated with these barriers are always well defined and, for those problems which have a solution, convergent to a pair of complementary solutions. Starting points in these paths are easy to compute. The theoretical iteration-complexity of these new path-following methods is derived and improved by a strategy which uses relaxed hybrid proximal-extragradient steps to control the quadratic term. Encouraging preliminary numerical experiments are presented.     

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.     

On perturbation bounds of the linear complementarity problem

In this paper, new perturbation bounds for linear complementarity problems are presented based on the sign patterns of the solution of the equivalent modulus equations. The new bounds are shown to be the generalization of the existing ones.     

On cone orderings and the linear complementarity problem

This paper first generalizes a characterization of polyhedral sets having least elements, which is obtained by Cottle and Veinott [6], to the situation in which Euclidean space is partially ordered by some general cone ordering (rather than the usual ordering). We then use this generalization to establish the following characterization of the class $\text{C}$ of matrices ($\text{C}$ arises as a generalization of the class of Z-matrices; see [4], [13], [14]): M∈$\text{C}$ if and only if for every vector q for which the linear complementarity problem (q,M) is feasible, the problem (q,M) has a solution which is the least element of the feasible set of (q,M) with respect to a cone ordering induced by some simplicial cone. This latter result generalizes the characterizations of K-and Z-matrices obtained by Cottle and Veinott [6] and Tamir [21], respectively.     

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.     

On the numerical solution of the linear complementarity problem

The well-known linear complementarity problem with definite matrices is considered. It is proposed to solve it using a global optimization algorithm in which one of the basic stages is a special local search. The proposed global search algorithm is tested using a variety of randomly generated problems; a detailed analysis of the computational experiment is given.     

Modified modulus‐based matrix splitting iteration methods for linear complementarity problems

For solving the large sparse linear complementarity problems, we establish modified modulus‐based matrix splitting iteration methods and present the convergence analysis when the system matrices are H₊‐matrices. The optima of parameters involved under some scopes are also analyzed. Numerical results show that in computing efficiency, our new methods are superior to classical modulus‐based matrix splitting iteration methods under suitable conditions. Copyright © 2015 John Wiley & Sons, Ltd.     

Underlying paths in interior point methods for the monotone semidefinite linear complementarity problem

An interior point method defines a search direction at each interior point of the feasible region. The search directions at all interior points together form a direction field, which gives rise to a system of ordinary differential equations (ODEs). Given an initial point in the interior of the feasible region, the unique solution of the ODE system is a curve passing through the point, with tangents parallel to the search directions along the curve. We call such curves off-central paths. We study off-central paths for the monotone semidefinite linear complementarity problem (SDLCP). We show that each off-central path is a well-defined analytic curve with parameter μ ranging over (0, ∞) and any accumulation point of the off-central path is a solution to SDLCP. Through a simple example we show that the off-central paths are not analytic as a function of and have first derivatives which are unbounded as a function of μ at μ = 0 in general. On the other hand, for the same example, we can find a subset of off-central paths which are analytic at μ = 0. These “nice” paths are characterized by some algebraic equations. This research was done during the author’s PhD study at the Department of Mathematics, NUS and as a Research Engineer at the NUS Business School.     

On the gradient-projection method for solving the nonsymmetric linear complementarity problem

The Levitin-Poljak gradient-projection method is applied to solve the linear complementarity problem with a nonsymmetric matrixM, which is either a positive-semidefinite matrix or aP-matrix. Further-more, if the quadratic functionx ^T(Mx + q) is pseudoconvex on the feasible region {x R ⁿ |Mx + q 0,x0}, then the gradient-projection method generates a sequence converging to a solution, provided that the problem has a solution. For the case when the matrixM is aP-matrix and the solution is nondegenerate, the gradient-projection method is finite.This work is based on the author's PhD Dissertation, which was supported by NSF Grant No. MCS-79-01066 at the University of Wisconsin, Madison, Wisconsin.The author would like to thank Professor O. L. Mangasarian for his guidance of the dissertation.