Central paths, generalized proximal point methods, and Cauchy trajectoriesin Riemannian manifolds

We study the relationships between three concepts which arise in connection with variational inequality problems: central paths defined by arbitrary b arriers, generalized proximal point methods (where a Bregman distance subst itutes for the Euclidean one), and Cauchy trajectory in Riemannian manifold s. First we prove that under rather general hypotheses the central path def ined by a general barrier for a monotone variational inequality problem is well defined, bounded, and continuous and converges to the analytic center of the solution set (with respect to the given barrier), thus generalizing results which deal only with complementarity problems and with the logarith mic barrier. Next we prove that a sequence generated by the proximal point method with the Bregman distance naturally induced by the barrier function converges precisely to the same point. Furthermore, for a certain class of problems (including linear programming), such a sequence is contained in th e central path, making the concepts of central path and generalized proxima l point sequence virtually equivalent. Finally we prove that for this class of problems the central path also coincides with the Cauchy trajectory in the Riemannian manifold defined on the positive orthant by a metric given b y the Hessian of the barrier (i.e., a curve whose direction at each point i s the negative gradient of the objective function at that point in the Riem annian metric).