Trajetória central, métodos de ponto proximal generalizado e trajetória de Cauchy em variedades riemannianas.

Abstract

In convex otimization problems and, more generally, in variational inequality problems appears concepts of: central paths defined by a barrier function, generalized proximal point algorithm with Bregman’s distances and Cauchy trajectory in Riemannian manifolds. In this work are studed these three concepts and its possible relationships. These relationships are showed principally to linear programming. First is showed, with adequate hypotheses, that a central path is well defined, is bounded, is continuos, have cluster points, these cluster points are solutions of variational inequality problems and converge to the analytic center of the solution set. Next, with adequate hypotheses too, is showed that a sequence generated by the generalized proximal point algorithm converge to someone solution of variational inequality problem. An important fact is when a central path is defined by the Bregman’s distance as a barrier function. In these cases, is showed that a central path and the sequence generated by the generalized proximal point algorithm converges to the same point. Furthermore, to linear programming is showed that the sequence generated by the generalized proximal point algorithm is contained in the central path. Finally, is showed to linear programming that a central path also coincides with a Cauchy trajectory in the Riemannian manifold defined on the open subsets ofIRn with metric given by the hessian of the barrier function.