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Journal ArticleDOI

On hidden Z-matrices and the linear complementarity problem

01 May 2016-Linear Algebra and its Applications (North-Holland)-Vol. 496, pp 81-100
TL;DR: It is demonstrated how the concept of principal pivot transform can be effectively used to extend many existing results and revisit various results obtained for hidden Z class by Mangasarian, Cottle and Pang in context of solving linear complementarity problems as linear programs.
About: This article is published in Linear Algebra and its Applications.The article was published on 2016-05-01. It has received 5 citations till now. The article focuses on the topics: Complementarity theory & Linear complementarity problem.
Citations
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Journal ArticleDOI
TL;DR: It is shown that for a non-degenerate feasible basis, linear complementarity problem with hidden Z-matrix has unique non- Degenerate solution under some assumptions.
Abstract: In this article, we study linear complementarity problem with hidden Z-matrix. We extend the results of Fiedler and Ptak for the linear system in complementarity problem using game theoretic approa...

18 citations


Cites background from "On hidden Z-matrices and the linear..."

  • ...3 of [13], LCP(q, A) and LCP(q̄, Ā) are equivalent....

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  • ...Now A is a PPT of A and PPT of a hidden Z-matrix is hidden Z [13]....

    [...]

  • ...[13] show that if A is a hidden Z-matrix with v(A) > 0 and some additional assumptions, then A is an E0-matrix....

    [...]

Journal ArticleDOI
TL;DR: In this article, a necessary and sufficient condition is given for the existence of an integer solution of a linear fractional programming problem by using its LCP formulation, where the concept of total dual integrality is utilized to obtain a necessary condition for existence of a integer solution to LCP with a hidden K-matrix.

4 citations

Journal ArticleDOI
TL;DR: In this article, the authors study linear complementarity problem with hidden $Z$-matrix and show that for a non-degenerate feasible basis, the linear system in complementarity with hidden Z$ -matrix has unique nondecent solution under some assumptions.
Abstract: In this article we study linear complementarity problem with hidden $Z$-matrix. We extend the results of Fiedler and Pt{a}k for the linear system in complementarity problem using game theoretic approach. We establish a result related to singular hidden $Z$-matrix. We show that for a non-degenerate feasible basis, linear complementarity problem with hidden $Z$-matrix has unique non-degenerate solution under some assumptions. The purpose of this paper is to study some properties of hidden $Z$-matrix.

2 citations

Journal ArticleDOI
TL;DR: In this article , the authors revisited the class of almost (strictly) semimonotone matrices and partially addressed the conjecture made by Wendler by providing a counter example.

1 citations

Book ChapterDOI
01 Jan 2018
TL;DR: In this paper, the authors study various mathematical programming problems in a common framework known as linear complementarity problem and discuss matrix theoretic properties of some recent matrix classes encountered in linear completeness literature and its processability using Lemke's algorithm.
Abstract: In this chapter, we study various mathematical programming problems in a common framework known as linear complementarity problem. Solving a linear complementarity problem depends on the properties of its underlying matrix class. In this chapter, we discuss matrix theoretic properties of some recent matrix classes encountered in linear complementarity literature and its processability using Lemke’s algorithm.
References
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Book
01 Jan 1944
TL;DR: Theory of games and economic behavior as mentioned in this paper is the classic work upon which modern-day game theory is based, and it has been widely used to analyze a host of real-world phenomena from arms races to optimal policy choices of presidential candidates, from vaccination policy to major league baseball salary negotiations.
Abstract: This is the classic work upon which modern-day game theory is based. What began more than sixty years ago as a modest proposal that a mathematician and an economist write a short paper together blossomed, in 1944, when Princeton University Press published "Theory of Games and Economic Behavior." In it, John von Neumann and Oskar Morgenstern conceived a groundbreaking mathematical theory of economic and social organization, based on a theory of games of strategy. Not only would this revolutionize economics, but the entirely new field of scientific inquiry it yielded--game theory--has since been widely used to analyze a host of real-world phenomena from arms races to optimal policy choices of presidential candidates, from vaccination policy to major league baseball salary negotiations. And it is today established throughout both the social sciences and a wide range of other sciences.

19,337 citations

Book
18 Feb 1992
TL;DR: In this article, the authors present an overview of existing and multiplicity of degree theory and propose pivoting methods and iterative methods for degree analysis, including sensitivity and stability analysis.
Abstract: Introduction. Background. Existence and Multiplicity. Pivoting Methods. Iterative Methods. Geometry and Degree Theory. Sensitivity and Stability Analysis. Chapter Notes and References. Bibliography. Index.

2,897 citations

Journal ArticleDOI
TL;DR: The goal of this documentation is to summarize the essential applications of the nonlinear complementarity problem known to date, to provide a basis for the continued research on the non linear complementarityproblem, and to supply a broad collection of realistic complementarity problems for use in algorithmic experimentation and other studies.
Abstract: This paper gives an extensive documentation of applications of finite-dimensional nonlinear complementarity problems in engineering and equilibrium modeling. For most applications, we describe the problem briefly, state the defining equations of the model, and give functional expressions for the complementarity formulations. The goal of this documentation is threefold: (i) to summarize the essential applications of the nonlinear complementarity problem known to date, (ii) to provide a basis for the continued research on the nonlinear complementarity problem, and (iii) to supply a broad collection of realistic complementarity problems for use in algorithmic experimentation and other studies.

1,016 citations

Journal ArticleDOI
TL;DR: It is shown that the linear complementarity problem of finding az inRn such that Mz + q ⩽ 0, z ⩾ 0 andzT(Mz +q) = 0 can be solved by a single linear program in some important special cases including those whenM or its inverse is a Z-matrix, that is a real square matrix with nonpositive off-diagonal elements.
Abstract: It is shown that the linear complementarity problem of finding az inRn such thatMz + q ? 0, z ? 0 andzT(Mz + q) = 0 can be solved by a single linear program in some important special cases including those whenM or its inverse is a Z-matrix, that is a real square matrix with nonpositive off-diagonal elements. As a consequence certain problems in mechanics, certain problems of finding the least element of a polyhedral set and certain quadratic programming problems, can each be solved by a single linear program.

114 citations