An Algorithm for Finding All Extremal Rays of Polyhedral Convex Cones with Some Complementarity Conditions

In this paper, we show a method for finding all extremal rays of polyhedral convex cones with some complementarity conditions. The polyhedral convex cone is defined as the intersection of half-spaces expressed by linear inequalities. By a complementarity extremal ray, we mean an extremal ray vector that satisfies some complementarity conditions among its elen~nts. Our method is iterative in the sense that, knowing all sUb-complementary extremal rays of the intersection of several half-spaces, we add repeatedly a new half-space to the half-spaces on the foregoing stage and determine all sub-eomplementary extremal rays of the new polyhedral convex cone thus formed, wltil all half-spaces are taken into consideration. Since, in the process of computation, we deal only with subcomplementary extremal rays, we could avoid the exceeding growth of the number of extremal rays. And it is of interest to note that the more complementarities there are, the less amount of computations we need. In the latter part, we apply this method to the general linear complementarity problem, to the nonconvex quadratic programming and to a mathematical programming with control variables.