Marianne Akian

Ergodic Control and Polyhedral Approaches to PageRank Optimization

We study a general class of PageRank optimization problems which involve finding an optimal outlink strategy for a web site subject to design constraints. We consider both a continuous problem, in which one can choose the intensity of a link, and a discrete one, in which in each page, there are obligatory links, facultative links and forbidden links. We show that the continuous problem, as well as its discrete variant when there are no constraints coupling different pages, can both be modeled by constrained Markov decision processes with ergodic reward, in which the webmaster determines the transition probabilities of websurfers. Although the number of actions turns out to be exponential, we show that an associated polytope of transition measures has a concise representation, from which we deduce that the continuous problem is solvable in polynomial time, and that the same is true for the discrete problem when there are no coupling constraints. We also provide efficient algorithms, adapted to very large networks. Then, we investigate the qualitative features of optimal outlink strategies, and identify in particular assumptions under which there exists a “master” page to which all controlled pages should point. We report numerical results on fragments of the real web graph.

Convergence analysis of the Max-Plus Finite Element Method for Solving Deterministic Optimal Control Problems

We consider the Max-Plus Finite Element Method for Solving Deterministic Optimal Control Problems, which is a max-plus analogue of the Petrov-Galerkin finite element method. This method, that we introduced in a previous work, relies on a max-plus variational formulation. The error in the sup-norm can be bounded from the difference between the value function and its projections on max-plus and minplus semimodules when the max-plus analogue of the stiffness matrix is exactly known. We derive here a convergence result in arbitrary dimension for approximations of the stiffness matrix relying on the Hamiltonian, and for arbitrary discretization grid. We show that for a class of problems, the error estimate is of order ¿+¿x(¿)-1 or ¿¿+¿x(¿)-1, depending on the choice of the approximation, where ¿ and ¿x are, respectively, the time and space discretization steps. We give numerical examples in dimension 2.