New error bounds for the linear complementarity problem with an SB-matrix

Error bounds for SB-matrices linear complementarity problems are given in the paper (Dai et al., Numer Algorithms 61:121–139, 2012). In this paper, new error bounds for the linear complementarity problem when the matrix involved is an SB-matrix are presented and some sufficient conditions that new bounds are sharper than those of the previous paper under certain assumptions are provided. New perturbation bounds of SB-matrices linear complementarity problems are also considered.     

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.     

Error bounds for linear complementarity problems for SB-matrices

S-strictly dominant B-matrices (SB-matrices) are introduced by Li et al. (Numer Linear Algebra Appl 14:391?C405, 2007). In this paper, we give error bounds for the linear complementarity problem when the matrix involved is an SB-matrix, which generalize those of DB-matrix linear complementarity problem and show advantages with respect to the computational cost. Then the perturbation bounds of SB-matrices linear complementarity problems are also provided. The preliminary numerical results show the sharpness of the bounds.     

B-Nekrasov matrices and error bounds for linear complementarity problems

The class of B-Nekrasov matrices is a subclass of P-matrices that contains Nekrasov Z-matrices with positive diagonal entries as well as B-matrices. Error bounds for the linear complementarity problem when the involved matrix is a B-Nekrasov matrix are presented. Numerical examples show the sharpness and applicability of the bounds.     

Weakly chained diagonally dominant <Emphasis Type="Italic">B</Emphasis>-matrices and error bounds for linear complementarity problems

The class of weakly chained diagonally dominant B-matrices, a subclass of P-matrices is introduced. Error bounds for the linear complementarity problem are presented when the involved matrix is a weakly chained diagonally dominant B-matrix. Numerical examples are given to show the sharpness of the proposed bounds.     

New error bounds for the linear complementarity problem of <Emphasis Type="Italic">QN</Emphasis>-matrices

An error bound for the linear complementarity problem (LCP) when the involved matrices are QN-matrices with positive diagonal entries is presented by Dai et al. (Error bounds for the linear complementarity problem of QN-matrices. Calcolo, 53:647-657, 2016), and there are some limitations to this bound because it involves a parameter. In this paper, for LCP with the involved matrix A being a QN-matrix with positive diagonal entries an alternative bound which depends only on the entries of A is given. Numerical examples are given to show that the new bound is better than that provided by Dai et al. in some cases.     

<Emphasis FontCategory="NonProportional">L</Emphasis>-Matrices and solvability of linear complementarity problems by a linear program-Matrices and solvability of linear complementarity problems by a linear program

The classes ofL ₁-matrices,L ₂-matrices,L ₃-matrices andW-matrices are introduced to study solvability of a linear complementarity problem via solving a linear program. Three sufficient conditions are presented to guarantee that a linear complementarity problem is solvable via a linear program. The new sufficient conditions are weaker than the ones introduced by Mangasarian. This fact is also illustrated by an example. Partially supported by NSFC. This author is also with College of Business Administration of Human University as a Lotus chair professor.     

Two class of synchronous matrix multisplitting schemes for solving linear complementarity problems

Many problems in the areas of scientific computing and engineering applications can lead to the solution of the linear complementarity problem LCP (M,q). It is well known that the matrix multisplitting methods have been found very useful for solving LCP (M,q). In this article, by applying the generalized accelerated overrelaxation (GAOR) and the symmetric successive overrelaxation (SSOR) techniques, we introduce two class of synchronous matrix multisplitting methods to solve LCP (M,q). Convergence results for these two methods are presented when M is an H-matrix (and also an M-matrix). Also the monotone convergence of the new methods is established. Finally, the numerical results show that the introduced methods are effective for solving the large and sparse linear complementary problems.     

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.     

On the Lipschitzian property in linear complementarity problems over symmetric cones

Let V be a Euclidean Jordan algebra with symmetric cone K. We show that if a linear transformation L on V has the Lipschitzian property and the linear complementarity problem LCP(L,q) over K has a solution for every invertible q∈V, then 〈L(c),c〉>0 for all primitive idempotents c in V. We show that the converse holds for Lyapunov-like transformations, Stein transformations and quadratic representations. We also show that the Lipschitzian Q-property of the relaxation transformation R_A on V implies that A is a P-matrix.     

Inexact multisplitting methods for linear complementarity problems

We present an inexact multisplitting method for solving the linear complementarity problems, which is based on the inexact splitting method and the multisplitting method. This new method provides a specific realization for the multisplitting method and generalizes many existing matrix splitting methods for linear complementarity problems. Convergence for this new method is proved when the coefficient matrix is an _H+$H_{+}$-matrix. Then, two specific iteration forms for this inexact multisplitting method are presented, where the inner iterations are implemented either through a matrix splitting method or through a damped Newton method. Convergence properties for both these specific forms are analyzed, where the system matrix is either an _H+$H_{+}$-matrix or a symmetric matrix.     

Equivalence of the Generalized Vertical Block Linear Complementarity Problems and Linear Complementarity Problems

In this paper, generalization of a vertical block linear complementarity problem associated with two different types of matrices, one of which is a square matrix and the other is a vertical block matrix, is proposed. The necessary and sufficient conditions for the existence of the solution of the generalized vertical block linear complementarity problem is derived and the relationship between the solution set of the generalized vertical block linear complementarity problem and the linear complementarity problem is established. It is proved that the generalized vertical block linear complementarity problem has the P-property if and only if the vertical block linear complementarity problem has the P-property.     

Improved bounds for the Laplacian energy of Bethe trees

For k, d?2, a Bethe tree is a rooted tree with k levels which the root vertex has degree d, the vertices from level 2 to k-1 have degree d+1 and the vertices at the level k are pendent vertices. So et al, using a theorem by Ky Fan have obtained both upper and lower bounds for the Laplacian energy of bipartite graphs. We shall employ the above mentioned theorem to obtain new and improved bounds for the Laplacian energy in the case of Bethe trees.     

Existence theory and Q-matrix characterization for the generalized linear complementarity problem: Revisited

We provide conditions under which a vertical block matrix is a Q-matrix if one or all representative sub-matrices are Q-matrices and vice versa. It is also shown, by means of counterexamples, that Eq. (3) of [A.A. Ebiefung, Existence theory and Q-matrix characterization for the generalized linear complementarity problem, Linear Algebra Appl. 223/224 (1995) 155-169] is incorrect.     

Improved convergence theorems of the two-step modulus-based matrix splitting and synchronous multisplitting iteration methods for solving linear complementarity problems

Abstract

In this paper, the convergence conditions of the two-step modulus-based matrix splitting and synchronous multisplitting iteration methods for solving linear complementarity problems of H-matrices are weakened. The convergence domain given by the proposed theorems is larger than the existing ones.     

A characterization of Delsarte’s linear programming bound as a ratio bound

It is well known that the ratio bound is an upper bound on the stability number α(G) of a regular graph G. In this note it is proved that, if G is a graph whose edge is a union of classes of a symmetric association scheme, the Delsarte’s linear programming bound can alternatively be stated as the minimum of a set of ratio bounds. This result follows from a recently established relationship between a set of convex quadratic bounds on α(G) and the number ?′(G), a well known variant of the Lovász theta number, which was introduced independently by Schrijver [A. Schrijver, A comparison of the Delsarte and Lovász bounds, IEEE Trans. Inform. Theory 25 (1979) 425-429] and McEliece et al. [R.J. McEliece, E.R. Rodemich, H.C. Rumsey Jr, The Lovász bound and some generalizations, J. Combin. Inform. System Sci. 3 (1978) 134-152].     

Solvability of Newton equations in smoothing-type algorithms for the SOCCP

In this paper, we first investigate the invertibility of a class of matrices. Based on the obtained results, we then discuss the solvability of Newton equations appearing in the smoothing-type algorithm for solving the second-order cone complementarity problem (SOCCP). A condition ensuring the solvability of such a system of Newton equations is given. In addition, our results also show that the assumption that the Jacobian matrix of the function involved in the SOCCP is a P₀-matrix is not enough for ensuring the solvability of such a system of Newton equations, which is different from the one of smoothing-type algorithms for solving many traditional optimization problems in ℜⁿ.     

New bounds for the minimum eigenvalue of the fan product of two M-matrices

In this paper, we mainly use the properties of the minimum eigenvalue of the Fan product of M-matrices and Cauchy-Schwarz inequality, and propose some new bounds for the minimum eigenvalue of the Fan product of two M-matrices. These results involve the maximum absolute value of off-diagonal entries of each row. Hence, the lower bounds for the minimum eigenvalue are easily calculated in the practical examples. In theory, a comparison is given in this paper. Finally, to illustrate our results, a simple example is also considered.     

Error bounds for linear complementarity problems of Nekrasov matrices

We present error bounds for the linear complementarity problem when the involved matrix is a Nekrasov matrix and also when it is a \(\Sigma \) -Nekrasov matrix. The new bounds can improve considerably other previous bounds.     

Q-matrices and spherical geometry

A real n × n matrix M is a Q-matrix if the linear complementarity problem w ? Mz=q, w ? 0, z ? 0, w^tz=0 has a solution for all real n-vectors q. M is nondegenerate if all its principal minors are nonzero. Spherical geometry is applied to the problem of characterizing nondegenerate Q-matrices. The stability of 3 × 3 nondegenerate Q-matrices and a generalization of the partitioning property of P-matrices are rather easily proved using spherical geometry. It is also proved that the set of 4 × 4 nondegenerate Q-matrices is not open.