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Journal ArticleDOI

On generalizations of positive subdefinite matrices and the linear complementarity problem

03 Oct 2018-Linear & Multilinear Algebra (Taylor & Francis)-Vol. 66, Iss: 10, pp 2024-2035

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].

AbstractIn 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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"On generalizations of positive subd..." refers background in this paper

  • ...The LCP provides a unified framework to study linear and quadratic programming problems as well as bimatrix game problems and it arises in a number of applications in operations research, mathematical economics, engineering and game theory (for details see [1,2])....

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  • ...The significance of matrix classes in the theory of LCP is well-known (see [1,2])....

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Journal ArticleDOI
Abstract: Some simple constructive proofs are given of solutions to the matric system Mz − ω = q; z ≧ 0; ω ≧ 0; and zT ω = 0, for various kinds of data M, q, which embrace the quadratic programming problem and the problem of finding equilibrium points of bimatrix games. The general scheme is, assuming non-degeneracy, to generate an adjacent extreme point path leading to a solution. The scheme does not require that some functional be reduced.

948 citations


"On generalizations of positive subd..." refers background in this paper

  • ...The importance of characterizing non-copositive row sufficient matrices is underscored by the fact that it exemplifies a familiar class of matrices that need not be copositive-plus [14]....

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  • ...Jones [15] proved the following termination criterion for Lemke’s algorithm that unifies Lemke’s result [14] and Evers’ result [24]....

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Journal ArticleDOI
TL;DR: The convergence properties of a variant of the multisplitting methods for solving the large sparse linear complementarity problems presented by Machida, Fukushima, and Ibaraki are further discussed when the system matrices are nonsymmetric and the weightingMatrices are nonnegative and diagonal.
Abstract: The convergence properties of a variant of the multisplitting methods for solving the large sparse linear complementarity problems presented by Machida, Fukushima, and Ibaraki [J. Comput. Appl. Math., 62 (1995), pp. 217--227] are further discussed when the system matrices are nonsymmetric and the weighting matrices are nonnegative and diagonal. This directly results in several novel sufficient conditions for guaranteeing the convergence of these multisplitting methods. Moreover, some applicable parallel multisplitting relaxation methods and their corresponding convergence properties are discussed in detail.

144 citations