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The Linear Complementarity Problem and Related Properties

Author: LiFeng
Tutor: WangXingTao
School: Harbin Institute of Technology
Course: Basic mathematics
Keywords: linear complementarity problem existence of solution p1 property F_A linear transformation
CLC: O151.26
Type: Master's thesis
Year: 2007
Downloads: 140
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Abstract


Theories and algorithms of the linear complementarity problem have been widely used in the field of economics, game theory and mathematical programming. The all conditions of finding stabilization point in nonlinear programming ultimately can be translated into the linear complementarity problem. The linear complementarity problem starts from the field of project physics, mechanics, operational research and economy. It has important applications in the questions of economic balance, noncooperation contest, traffic distribution, and further has compact relations with inequation, double linear programing and nonlinear project group, so it is very important to study this problem both in practical and theory aspects.In this paper we carry on more concrete discussion, study for some properties in the linear complementarity problem, indicate the deduced relations between p1 property and other linear properties, and make originally conclusion more in detail. At the same time we discuss the linear properties of the linear transformation FA, and further more give the relation between the spectral radius of FA and the linear complementarity problem for linear transformation SA. We prove the main results as follows:For an arbitrary given matrix A∈Rn×n and the linear transformation FA in the space Sn, if A is a positive definite matrix or a negetive definite matrix, then FA has p1 property.If the linear transformation FA in the space S n has p property, then for (?)Q∈Sn, the linear complementarity problemLCP (FA , S+n ,Q) has solutions.If the linear transformation SA= X-AXAT and FA for an arbitrary given matrix A∈Rn×n in the space Sn satisfy the conditionρ( FA) < 2, then for (?)Q∈Sn, the linear complementarity problem LCP ( SA, S+n ,Q) has solutions.

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CLC: > Mathematical sciences and chemical > Mathematics > Algebra,number theory, portfolio theory > Theory of algebraic equations,linear algebra > Linear Algebra > The application of linear algebra
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