We explain and justify a path-following algorithm for solving the equations A(C)(x)=a, where A, is a linear transformation from Rn to Rn, C is a polyhedral convex subset of Rn, and A(C) is the associated normal map. When A(C) is coherently oriented, we are able to prove that the path following method terminates at the unique solution of A(C)(x)=a, which is a generalization of the well known fact that Lemke's method terminates at the unique solution of LCP (q, M) when M is a P = matrix. Otherwise, we identify two classes of matrices which are analogues of the class of copositive-plus and L-matrices in the study of the linear complementarity problem. We then prove that our algorithm processes A(C)(x)=a when A is the linear transformation associated with such matrices. That is, when applied to such a problem, the algorithm will find a solution unless the problem is infeasible in a well specified sense.
