Modular Invariant of Quantum Tori II: The Golden Mean

In our first article in this series ("Modular Invariant of Quantum Tori I: Definitions Nonstandard and Standard" arXiv:0909.0143) a modular invariant of quantum tori was defined. In this paper, we consider the case of the quantum torus associated to the golden mean. We show that the modular invariant is approximately 9538.249655644 by producing an explicit formula for it involving weighted versions of the Rogers-Ramanujan functions

Decomposition Methods for Network Design

AbstractNetwork design applications are prevalent in transportation and logistics. We consider the multicommodity capacitated fixed-charge network design problem (MCND), a generic model that captures three important features of network design applications: the interplay between investment and operational costs, the multicommodity aspect, and the presence of capacity constraints. We focus on mathematical programming approaches for the MCND and present three classes of methods that have been used to solve large-scale instances of the MCND: a cutting-plane method, a Benders decomposition algorithm, and Lagrangian relaxation approaches

Commerce équitable : de quelle équité parle-t-on?

Le commerce équitable s’est imposé sur la scène de la consommation responsable. Mais en quoi est-il plus équitable que le commerce conventionnel? Et dans quel sens doit-on entendre dans son cas le concept d’équité? Un détour par la philosophie morale et les théories de la justice d’Aristote, Hume et Rawls et une analyse de ses pratiques telles qu’elles ressortent de nombreuses études de terrain invitent à prendre ses prétentions à l’équité “cum grano salis”

Dynamic smoothness parameter for fast gradient methods

We present and computationally evaluate a variant of the fast gradient method by Nesterov that is capable of exploiting information, even if approximate, about the optimal value of the problem. This information is available in some applications, among which the computation of bounds for hard integer programs. We show that dynamically changing the smoothness parameter of the algorithm using this information results in a better convergence profile of the algorithm in practice

On the Computational Efficiency of Subgradient Methods: a Case Study with Lagrangian Bounds

Subgradient methods (SM) have long been the preferred way to solve the large-scale Nondifferentiable Optimization problems arising from the solution of Lagrangian Duals (LD) of Integer Programs (IP). Although other methods can have better convergence rate in practice, SM have certain advantages that may make them competitive under the right conditions. Furthermore, SM have significantly progressed in recent years, and new versions have been proposed with better theoretical and practical performances in some applications. We computationally evaluate a large class of SM in order to assess if these improvements carry over to the IP setting. For this we build a unified scheme that covers many of the SM proposed in the literature, comprised some often overlooked features like projection and dynamic generation of variables. We fine-tune the many algorithmic parameters of the resulting large class of SM, and we test them on two different Lagrangian duals of the Fixed-Charge Multicommodity Capacitated Network Design problem, in order to assess the impact of the characteristics of the problem on the optimal algorithmic choices. Our results show that, if extensive tuning is performed, SM can be competitive with more sophisticated approaches when the tolerance required for solution is not too tight, which is the case when solving LDs of IPs

Dynamic Smoothness Parameter for Fast Gradient Methods

We present and computationally evaluate a variant of the fast gradient method by Nesterov that is capable of exploiting information, even if approximate, about the optimal value of the problem. This information is available in some applications, among which the computation of bounds for hard integer programs. We show that dynamically changing the smoothness parameter of the algorithm using this information results in a better convergence profile of the algorithm in practice

On the integration of Dantzig-Wolfe and Fenchel decompositions via directional normalizations

The strengthening of linear relaxations and bounds of mixed integer linear programs has been an active research topic for decades. Enumeration-based methods for integer programming like linear programming-based branch-and-bound exploit strong dual bounds to fathom unpromising regions of the feasible space. In this paper, we consider the strengthening of linear programs via a composite of Dantzig-Wolfe and Fenchel decompositions. We provide geometric interpretations of these two classical methods. Motivated by these geometric interpretations, we introduce a novel approach for solving Fenchel sub-problems and introduce a novel decomposition combining Dantzig-Wolfe and Fenchel decompositions in an original manner. We carry out an extensive computational campaign assessing the performance of the novel decomposition on the unsplittable flow problem. Very promising results are obtained when the new approach is compared to classical decomposition methods

Optimal Access Point Power Management for Green IEEE 802.11 Networks

In this paper, we present an approach and an algorithm aimed at minimising the energy consumption of enterprise Wireless Local Area Networks (WLANs) during periods of low user activity. We act on two network management aspects: powering off some Access Points (APs), and choosing the level of transmission power of each AP. An efficient technique to allocate the user terminals to the various APs is the key to achieving this goal. The approach has been formulated as an integer programming problem with nonlinear constraints, which comes from a general but accurate characterisation of the WLAN. This general problem formulation has two implications: the formulation is widely applicable, but the nonlinearity makes it NP-hard. To solve this problem to optimality, we devised an exact algorithm based on a customised version of Benders’ decomposition method. The computational results proved the ability to obtain remarkable power savings. In addition, the good performance of our algorithm in terms of solving times paves the way for its future deployment in real WLANs.publishedVersio