229 research outputs found
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 Ă©quitable 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
A branch-and-Benders-cut method for nonlinear power design in green wireless local area networks
We consider a problem arising in the design of green wireless local area networks. Decisions on powering-on a set of access points (APs), via the assignment of one power level (PL) to each opened AP, and decisions on the assignment of the user terminals (UTs) to the opened APs, have to be taken simultaneously. The PL assigned to an AP affects, in a nonlinear way, the capacity of the connections between the AP and the UTs that are assigned to it. The objective is to minimize the overall power consumption of the APs, which has two components: location/capacity dimensioning costs of the APs; assignment costs that depend on the total demands assigned to the APs. We develop a branch-and-Benders-cut (BBC) method where, in a non-standard fashion, the master problem includes the variables of the Benders subproblem, but relaxes their integrality. The BBC method has been tested on a large set of instances, and compared to a Benders decomposition algorithm on a subset of instances without assignment costs, where the two approaches can be compared. The computational results show the superiority of BBC in terms of solution quality, scalability and robustness
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