Our purpose here is to present a method for obtaining a global solution to a linearly constrained general quadratic minimization problem. The Kuhn-Tucker necessary conditions for a global solution of the problem are a linear complementarity problem. The author of this paper has proposed a method for a generalized linear complementarity problem, which bases on an algorithm for finding all extremal rays of a polyhedral convex cone with some complementarity conditions. In this paper, we apply the method to the linear complementarity problem of the general quadratic programming. And several devices will be adopted to decrease the amount of computation.