A note on solvability of a class of linear complementarity problems

We give a characterization of unique solvability of an infinite family of linear complementarity problems of a special form by means of a finite subset of this family.     

Minimum norm solution to the positive semidefinite linear complementarity problem

In this article, we present an algorithm to compute the minimum norm solution of the positive semidefinite linear complementarity problem. We show that its solution can be obtained using the alternative theorems and a convenient characterization of the solution set of a convex quadratic programming problem. This problem reduces to an unconstrained minimization problem with once differentiable convex objective function. We propose an extension of Newton's method for solving the unconstrained optimization problem. Computational results show that convergence to high accuracy often occurs in just a few iterations.     

A polynomial-time algorithm for a class of linear complementarity problems

Given ann × n matrixM and ann-dimensional vectorq, the problem of findingn-dimensional vectorsx andy satisfyingy = Mx + q, x 0,y 0,x _i y _i = 0 (i = 1, 2,,n) is known as a linear complementarity problem. Under the assumption thatM is positive semidefinite, this paper presents an algorithm that solves the problem in O(n ³ L) arithmetic operations by tracing the path of centers,{(x, y) S: x _i y _i = (i = 1, 2,,n) for some > 0} of the feasible regionS = {(x, y) 0:y = Mx + q}, whereL denotes the size of the input data of the problem.     

Error bounds for linear complementarity problems of DB-matrices

Doubly B-matrices (DB-matrices), which properly contain B-matrices, are introduced by Peña (2003) [2]. In this paper we present error bounds for the linear complementarity problem when the matrix involved is a DB-matrix and a new bound for linear complementarity problem of a B-matrix. The numerical examples show that the bounds are sharp.     

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.     

Ky Fan's N-matrices and linear complementarity problems

We consider the linear complementarity problem (LCP),w=Az + q, w0,z0,w ^T z=0, when all the off-diagonal entries ofA are nonpositive (the class of Z-matrices), all the proper principal minors ofA are positive and the determinant ofA is negative (the class of almost P-matrices). We shall call this the class of F-matrices. We show that ifA is a Z-matrix, thenA is an F-matrix if and only if LCP(q, A) has exactly two solutions for anyq0,q0, and has at most two solutions for any otherq. Research supported by AFOSR-89-0512.     

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.     

Robust solutions to uncertain linear complementarity problems

In this paper,we adopt the robust optimization method to consider linear complementarity problems in which the data is not specified exactly or is uncertain,and it is only known to belong to a prescribed uncertainty set.We propose the notion of the p- robust counterpart and the p-robust solution of uncertain linear complementarity problems.We discuss uncertain linear complementarity problems with three different uncertainty sets,respectively,including an unknown-but-bounded uncertainty set,an ellipsoidal uncertainty set and an intersection-of-ellipsoids uncertainty set,and present some sufficient and necessary(or sufficient) conditions which p- robust solutions satisfy.Some special cases are investigated in this paper.     

Solving more linear complementarity problems with Murty's bard-type algorithm

The linear complementarity problem (M|q) is to findw andz inR ⁿ such thatw–Mz=q,w0,z0,w ^t z=0, givenM inR ^n×n andq in . Murty's Bard-type algorithm for solving LCP is modeled as a digraph.Murty's original convergence proof considered allq inR ⁿ andM inR ^n×n , aP-matrix. We show how to solve more LCP's by restricting the set ofq vectors and enlarging the class ofM matrices beyondP-matrices. The effect is that the graph contains an embedded graph of the type considered by Stickney and Watson wheneverM is a matrix containing a principal submatrix which is aP-matrix. Examples are presented which show what can happen when the hypotheses are further weakened.     

On the connectedness of the solution set to linear complementarity systems

This paper establishes sufficient conditions for the connectedness of nontrivial subsets of the solution set to linear complementarity systems with special structure. Connectedness may be important to investigate stability and sensitivity questions, parametric problems, and for extending a Lemke-type method to a new class of problems. Such a property may help in analyzing the structure of the feasible region by checking the explicitly given matrices of the resulting conditions. From the point of view of geometry, the question is how to analyze the combined geometrical object consisting of a Riemannian manifold, a pointed cone, and level sets determined by linear inequalities.This paper has been mainly prepared while the author was visiting the Department of Mathematics at the University of Pisa. This research was partialy supported by the Hungarian National Research Foundation, Grant No. OTKA-2568.     

Polynomial expected behavior of a pivoting algorithm for linear complementarity and linear programming problems

We show that a particular pivoting algorithm, which we call the lexicographic Lemke algorithm, takes an expected number of steps that is bounded by a quadratic inn, when applied to a random linear complementarity problem of dimensionn. We present two probabilistic models, both requiring some nondegeneracy and sign-invariance properties. The second distribution is concerned with linear complementarity problems that arise from linear programming. In this case we give bounds that are quadratic in the smaller of the two dimensions of the linear programming problem, and independent of the larger. Similar results have been obtained by Adler and Megiddo.Research partially funded by a fellowship from the Alfred Sloan Foundation and by NSF Grant ECS82-15361.     

Investigations of a certain class of linear complementarity problems

A certain class of linear complementarity problems that appeared in an economical study concerning self-employment is investigated. The principal findings for this class of linear complementarity problems are: (i) there is always a solution, which can be found by the Lemke algorithm; (ii) characterizations are found for solutions, some typical for all solutions, some typical for locally nonunique solutions, and some typical for locally unique solutions; (iii) a sufficient condition is found to guarantee a globally unique solution.The research reported in this paper is part of the project Economics of Political Decision Making of the University of Amsterdam. The author acknowledges the financial support obtained from the Ministry of Economic Affairs by way of the Research Institute for Small and Medium-Sized Business as well as the financial support of the Netherlands Organization for Scientific Research. He is grateful to two anonymous referees for many helpful suggestions, to F. A. A. M. Van Winden for stimulating discussions, and to J. J. M. Evers, H. Neudecker, and A. J. J. Talman for comments.     

非对称线性互补问题的并行二级多分裂迭代法

提出了求解非对称线性互补问题的并行二级多分裂迭代算法,并证明了该算法的收敛性,最后通过数值实验验证了算法的有效性和可行性.     

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.     

A CLASS OF GENERALIZED MULTISPLITTING RELAXATION METHODS FOR LINEAR COMPLEMENTARITY PROBLEMS

In this paper,a class of generalized parallel matrix multisplitting relaxation methods for solving linear complementarity problems on the high-speed multiprocessor systems is set up. This class of methods not only includes all the existing relaxation methods for the linear complementarity problems ,but also yields a lot of novel ones in the sense of multisplittlng. We establish the convergence theories of this class of generalized parallel multisplitting relaxation methods under the condition that the system matrix is an H-metrix with positive diagonal elements.     

Existence and uniqueness of solutions for the generalized linear complementarity problem

Cottle and Dantzig (Ref. 1) showed that the generalized linear complementarity problem has a solution for anyqR ^m ifM is a vertical blockP-matrix of type (m ₁,...,m _n ). They also extended known characterizations of squareP-matrices to vertical blockP-matrices.Here we show, using a technique similar to Murty's (Ref. 2), that there exists a unique solution for anyqR ^m if and only ifM is a vertical blockP-matrix of type (m ₁,...,m _n ). To obtain this characterization, we employ a generalization of Tucker's theorem (Ref. 3) and a generalization of a theorem initially introduced by Gale and Nikaido (Ref. 4).     

On computer-assisted proofs for solutions of linear complementarity problems

In this paper we consider the linear complementarity problem where the components of the input data M and q are not exactly known but can be enclosed in intervals. We compare three tests to each other each of which can be used by a computer that supports interval arithmetic to give guaranteed bounds for a solution of the LCP defined by M and q.     

A note on the convergence of the MSMAOR method for linear complementarity problems

Modulus‐based splitting, as well as multisplitting iteration methods, for linear complementarity problems are developed by Zhong‐Zhi Bai. In related papers (see Bai, Z.‐Z., Zhang, L.‐L.: Modulus‐Based Synchronous Multisplitting Iteration Methods for Linear Complementarity Problems. Numerical Linear Algebra with Applications 20 (2013) 425–439, and the references cited therein), the problem of convergence for two‐parameter relaxation methods (accelerated overrelaxation‐type methods) is analyzed under the assumption that one parameter is greater than the other. Here, we will show how we can avoid this assumption and, consequently, improve the convergence area. Copyright © 2013 John Wiley & Sons, Ltd.