A Pivotal Method for Affine Variational Inequalities

We explain and justify a path-following algorithm for solving the equations ACx = a, where A is a linear transformation from Rn to Rn, C is a polyhedral convex subset of Rn, and AC is the associated normal map. When AC is coherently oriented, we are able to prove that the path following method terminates at the unique solution of ACx = a, which is a generalization of the well known fact that Lemke's method terminates at the unique solution of LCPq, M when M is a P = matrix. Otherwise, we identity 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 ACx = 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.

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