The Linear Complementarity Problem And Related Properties

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 F_A, and further more give the relation between the spectral radius of F_A and the linear complementarity problem for linear transformation S_A. We prove the main results as follows:For an arbitrary given matrix A∈R^n×n and the linear transformation F_A in the space Sⁿ, if A is a positive definite matrix or a negetive definite matrix, then F_A has p₁ property.If the linear transformation F_A in the space S n has p property, then for (?)Q∈Sⁿ, the linear complementarity problemLCP (F_A , S₊ⁿ ,Q) has solutions.If the linear transformation S_A= X-AXA^T and F_A for an arbitrary given matrix A∈R^n×n in the space Sⁿ satisfy the conditionρ( F_A) < 2, then for (?)Q∈Sⁿ, the linear complementarity problem LCP ( S_A, S₊ⁿ ,Q) has solutions.