On generalizations of positive subdefinite matrices and the linear complementarity problem
TL;DR: The notion of generalized positive subdefinite matrices of level k is introduced by generalizing the definition of generalized Positive Subdefinite Matrices introduced by Crouzeix and Komlósi in [Applied optimization].
Abstract: In this paper, we introduce the notion of generalized positive subdefinite matrices of level k by generalizing the definition of generalized positive subdefinite matrices introduced by Crouzeix and Komlosi in [Applied optimization. Vol. 59, Dordrecht: Kluwer Academic Publications; 2001. p. 45–63]. The main motivation behind this generalization is to identify new subclasses of row sufficient matrices [Cottle, Pang and Venkateswaran, Linear Algebra Appl. 1989;114/115:231–249] which are not a subclass of copositive matrices. We establish new sufficient conditions for a matrix to be a non-copositive matrix and a non-copositive row sufficient matrix using the notion of generalized positive subdefinite matrices of level k. We present a new termination result for Lemke’s algorithm which supplements Jones’ result [Math Program. 1986;35:239–242].
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TL;DR: By applying the generalized accelerated overrelaxation (GAOR) and the symmetric successive overrelAXation (SSOR) techniques, two class of synchronous matrix multisplitting methods to solve LCP (M,q) are introduced.
12 citations
01 Jan 2001
TL;DR: In this article, it was shown that the linear complementarity problem can be linked with the study of an appropriate quadratic programming problem, and that the solution set to a given LCP always contains the KKT-set of the corresponding QP problem.
Abstract: It is well known that the study of the Linear Complementarity Problem can be linked with the study of an appropriate quadratic programming problem. The solution set to a given LCP always contains the KKT-set of the corresponding quadratic programming problem. The reverse implication holds only for some very special classes of matrices as positive semidefinite, P-, row adequate, row sufficient and positive subdefinite (PSBD)matrices. The paper extends the concept of PSBD matrices and proves that, the above coincidence still holds for this larger class of matrices.
9 citations
"On generalizations of positive subd..." refers background in this paper
...In this paper, we introduce the notion of generalized positive subdefinite matrices of level k by generalizing the definition of generalized positive subdefinite matrices introduced by Crouzeix and Komlósi in [Applied optimization....
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...Crouzeix and Komlósi [7] introduced a generalization of PSBD matrices and named this class as the class of generalized positive subdefinite (GPSBD) matrices....
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TL;DR: This note presents a new termination result for the Lemke linear complementarity algorithm that unifies two previous results.
Abstract: This note presents a new termination result for the Lemke linear complementarity algorithm that unifies two previous results. Special cases include linear complementarity problems satisfying the Evers condition, and those whoseM matrix is copositive-plus.
8 citations
TL;DR: In this paper, two classes of multisplitting chaotic relaxation methods for solving the large sparse linear complementarity problem (LCP) were presented, and the convergence theorems of the methods were investigated.
Abstract: The aim of this paper is to present two classes of multisplitting chaotic relaxation methods for solving the large sparse linear complementarity problem (LCP \((M,q)\)). When the system matrix \(M\) is an \(H\)-matrix with positive diagonal elements, the convergence theorems of the methods are investigated. Also the monotone convergence of the methods is established when \(M\) is an \(L\)-matrix.
4 citations
TL;DR: In this article, Cottle and Verma showed that all row sufficient matrices belong to L. They also established three new characterizations of RSU in terms of the matrix classes L, E0, and Q0 and the structural properties of sign-change invariance, completeness, and fullness.
3 citations