Primal-Dual Method for Searching Equilibrium in Hierarchical Congestion   Population Games
  In this paper, we consider a large class of hierarchical congestion population games. One can show that the equilibrium in a game of such type can be described as a minimum point in a properly constructed multi-level convex optimization problem. We propose a fast primal-dual composite gradient method and apply it to the problem, which is dual to the problem describing the equilibrium in the considered class of games. We prove that this method allows to find an approximate solution of the initial problem without increasing the complexity. 
Pavel Dvurechensky and Alexander Gasnikov and Evgenia Gasnikova and   Sergey Matsievsky and Anton Rodomanov and Inna Usik
math.OC