Ergodic Control and Polyhedral approaches to PageRank Optimization

We study a general class of PageRank optimization problems which consist in 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.Comment: 39 page

The Trust region affine interior point algorithm for convex and nonconvex quadratic programming

Projet PROMATHWe study from a theoretical and numerical point of view an interior point algorithm for quadratic QP using a trust region idea, formulated by Ye and Tse. We show that, under some nondegeneracy hypothesis, the sequence of points converges to a stationary point at a linear rate. We obtain also an asymptotic linear convergence rate for the cost that depends only on the dimension of the problem. Then we show that, provided some modifications are added to the basic algorithm, the method has a good numerical behaviour

Pricing Network Edges to Cross a River.

We consider a Stackelberg pricing problem in directed networks:Tariffs (prices) have to be defined by an operator, the leader, for a subset of the arcs. Clients, the followers, choose paths to route their demand through the network selfishly and independently of each other, on the basis of minimal total cost. The problem is to find tariffs such as to maximize the operator''s revenue. We consider the case where each client takes at most one tariff arc to route the demand.The problem, which we refer to as the river tarification problem, is a special case of problems studied previously in the literature.We prove that the problem is strongly NP-hard.Moreover, we show that the polynomially solvable case of uniform tarification yields an m--approximation algorithm, and this is tight. We suggest a new type of analysis that allows to improve the result to \bigO{\log m}, whenever the input data is polynomially bounded. We furthermore derive an \bigO{m^{1-\varepsilon}}--inapproximability result for problems where the operator must serve all clients, and we discuss some polynomial special cases. Finally, a computational study with instances from France Telecom suggests that uniform pricing performs better in practice than theory would suggest.operations research and management science;

Polynomial cases of the tarification problem

We consider the problem of determining a set of optimal tariffs for an agent in a network, who owns a subset of the arcs of the network, and who wishes to maximize his revenues on this subset from a set of clients that make use of the network.The general variant of this problem is NP-hard, already with a single client. This paper introduces several new polynomially solvable special cases. An important case is the following.For multiple clients, if the number of tariff arcs is bounded from above, we can solve the problem by a polynomial number of linear programs (each of which is of polynomial size). Furthermore, we show that the parametric tarification problem and the single arc fixed charge tarification problem can be solved in polynomial time.Economics ;

Fair Network Resource Allocation and Link Pricing: A Numerical Study

In this paper, we present an analysis of a restricted bi-level optimization model proposed by \cite{pat2} applied to telecommunication network pricing. This framework provides a generalization of telecommunication resource allocation/shadow price-based schemes such as those of \cite{kel8} and \cite{low}, and permits the optimization of link prices with an aim towards revenue maximization. The analysis of the model is based on numerical experiments that we conducted for a variety of networks with varying parameters. We show that in practice the goal of maximizing operator revenue is not always attainable. We present some such cases, as well as valid alternative objectives for these instances

Tariff optimization in Networks

We consider the problem of determining a set of optimal tariffs for an agent in the network, who owns a subset of all the arcs, and who receives revenue by setting the tariffs on the arc he owns. Multiple rational clients are active in the network, who route their demands on the cheapest paths from source to destination. The cost of a path is determined by fixed costs and tariffs on the arcs of the path.We introduce a remodeling of the network, using shortest paths. We develop three algorithms, a path oriented mixed integer program and a known arc oriented mixed integer program. Combined with reduction methods this remodeling enables us to solve the problem to optimality, for quite large instances. We provide computational results for the methods developped and compare them with the results of the arc oriented mixed integer programming formulation of the problem, applied to the original network.Economics ;

Learning a Correlated Equilibrium with Perturbed Regret Minimization

International audienceIn this paper, we consider the problem of learning a correlated equilibrium of a finite non-cooperative game and show a new adaptive heuristic, called Correlated Perturbed Regret Minimization (CPRM) for this purpose. CPRM combines regret minimization to approach the set of correlated equilibria and a simple device suggesting actions to the players to further stabilize the dynamic. Numerical experiments support the hypothesis of the pointwise convergence of the empirical distribution over action profiles to an approximate correlated equilibrium with all players following the devices' suggestions. Additional simulation results suggest that CPRM is adaptive to changes in the game such as departures or arrivals of players

A continuous optimization model for a joint problem of pricing and resource allocation

This paper investigates the problem of maximizing the revenue of a telecommunications operator by simultaneously pricing point-to-point services and allocating bandwidth in its network, while facing competition. Customers are distributed into market segments, i.e., groups of customers with a similar preference for the services. This preference is expressed using utility functions, and customers choose between the offers of the operator and of the competition according to their utility. We model the problem as a leader-follower game between the operator and the customers. This kind of problem has classically been modeled as a bilevel program. A market segmentation is usually defined by a discrete distribution function of the total demand for a service; in this case, the problem can be modeled as a combinatorial optimization problem. In this paper, however, we motivate the use of a continuous distribution function and investigate the nonlinear continuous optimization problem obtained in this case. We analyze the mathematical properties of the problem, and in particular we give a necessary and sufficient condition for its convexity. We introduce methods to solve the problem and we provide encouraging numerical results on realistic telecommunications instances of the problem, showing that it can be solved efficiently