A New Heuristic for Feature Selection by Consistent Biclustering

Given a set of data, biclustering aims at finding simultaneous partitions in biclusters of its samples and of the features which are used for representing the samples. Consistent biclusterings allow to obtain correct classifications of the samples from the known classification of the features, and vice versa, and they are very useful for performing supervised classifications. The problem of finding consistent biclusterings can be seen as a feature selection problem, where the features that are not relevant for classification purposes are removed from the set of data, while the total number of features is maximized in order to preserve information. This feature selection problem can be formulated as a linear fractional 0-1 optimization problem. We propose a reformulation of this problem as a bilevel optimization problem, and we present a heuristic algorithm for an efficient solution of the reformulated problem. Computational experiments show that the presented algorithm is able to find better solutions with respect to the ones obtained by employing previously presented heuristic algorithms

Maximizing the number of solved aircraft conflicts through velocity regulation

International audienceWe propose a model for the maximization of the number of aircraft conflicts that can be solved by performing velocity regulation. The model is mixed-integer as binary variables are used to count solved conflicts and to model alternative choices, while nonlinearities appear in the aircraft separation conditions. The main nonlinearities can however be relaxed by standard reformulations. Numerical results show that the model can be satisfactorily applied at least as a preprocessing step in a conflict avoidance procedure in a given airspace

Using mathematical programming to refine heuristic solutions for network clustering

International audienceWe propose mathematical programming based aproaches to refine graph clustering solutions computed by heuristics. Clustering partitions are refined by applying cluster splitting and a combination of merging and splitting actions. A refinement scheme based on iteratively fixing and releasing integer variables of a mixed-integer quadratic optimization formulation appears to be particularly efficient. Computational experiments show the effectiveness and efficiency of the proposed approaches

Edge ratio and community structure in networks

International audienceA hierarchical divisive algorithm is proposed for identifying communities in complex networks. To that effect, the definition of community in the weak sense of Radicchi et al. _Proc. Natl. Acad. Sci. U.S.A. 101, 2658 _2004__ is extended into a criterion for a bipartition to be optimal: one seeks to maximize the minimum for both classes of the bipartition of the ratio of inner edges to cut edges. A mathematical program is used within a dichotomous search to do this in an optimal way for each bipartition. This includes an exact solution of the problem of detecting indivisible communities. The resulting hierarchical divisive algorithm is compared with exact modularity maximization on both artificial and real world data sets. For two problems of the former kind optimal solutions are found; for five problems of the latter kind the edge ratio algorithm always appears to be competitive. Moreover, it provides additional information in several cases, notably through the use of the dendrogram summarizing the resolution. Finally, both algorithms are compared on reduced versions of the data sets of Girvan and Newman _Proc. Natl. Acad. Sci. U.S.A. 99, 7821 _2002__ and of Lancichinetti et al. _Phys. Rev. E 78, 046110 _2008__. Results for these instances appear to be comparable

Hierarchical clustering for the identification of communities in networks

National audienceThe analysis of networks and in particular the identification of communities, or clusters, is a topic of active research and attracts an increasing attention in the operations research as well as the physics communities. Complex systems arising in a variety of fields can be represented as networks, or graphs, where the set of vertices is given by the entities under study and the edges represent relations holding for pairs of vertices. A typical example is given by social networks, modeling interactions among people. Other real-life applications include communicatons networks, such as theWorldWide Web, and transportation networks, representing movements of people or goods

Locally optimal heuristic for modularity maximization of networks

International audienceCommunity detection in networks based on modularity maximization is currently done with hierarchical divisive or agglomerative as well as partitioning heuristics, hybrids, and, in a few papers, exact algorithms. We consider here the case of hierarchical networks in which communities should be detected and propose a divisive heuristic which is locally optimal in the sense that each of the successive bipartitions is done in a provably optimal way. This heuristic is compared with the spectral-based hierarchical divisive heuristic of Newman [Proc. Natl. Acad. Sci. USA 103, 8577 (2006).] and with the hierarchical agglomerative heuristic of Clauset, Newman, and Moore [Phys. Rev. E 70, 066111 (2004).]. Computational results are given for a series of problems of the literature with up to 4941 vertices and 6594 edges. They show that the proposed divisive heuristic gives better results than the divisive heuristic of Newman and than the agglomerative heuristic of Clauset et al

Reformulation of a locally optimal heuristic for modularity maximization

National audienceA network, or graph, G = (V,E) consists of a set of vertices V = {1, . . . , n} and a set of edges E = {1, . . . ,m} connecting vertices. One of the most studied problems in the field of complex systems is to find communities, or clusters, in networks. A community consists of a subset S of the vertices of V where inner edges connecting pairs of vertices of S are more dense than cut edges connecting vertices of S to vertices of V \S. Many criteria have been proposed to evaluate partitions of V into communities

Variable Neighborhood Search for Edge-Ratio Network Clustering

International audienceEdge-ratio clustering was introduced in [Cafieri et al., Phys.Rev. E 81(2):026105, 2010], as a criterion for optimal graph bipartitioning in hierarchical divisive algorithms for cluster identification in networks. Exact algorithms to perform bipartitioning maximizing the edge-ratio were shown to be too time consuming to be applied to large datasets. In this paper, we present a Variable Neighborhood Search (VNS)-based heuristic for hierarchical divisive edge ratio network clustering. We give a full description including the structure of some algorithmic procedures which are used to implement the main steps of the heuristic. Computational results show that the proposed algorithm is very efficient in terms of quality of the bipartitions, moreover the computing time is much smaller than that one for exact algorithms

Hybridizing direct and indirect optimal control approaches for aircraft conflict avoidance

Aircraft conflict avoidance is a crucial issue arising in air traffic management. The problem is to keep a given separation distance for aircraft along their trajectories. We focus on an optimal control model based on speed regulation to achieve aircraft separation. We propose a solution strategy based on the decomposition of the problem and on the hybridization of a direct and an indirect method applied on the obtained subproblems. Numerical results show that the proposed approach is promising in terms of reduction of computing time for conflict avoidance

Optimisation des routes de départ et d’arrivée dans la TMA

National audienceLa forte augmentation du trafic aérien induit une congestion dans la zone proche des aéroports appelée TMA1. Les départs et les arrivées des aéroports se font suivant des routes dites SID 2 et STAR3. L’optimisation de ces routes est un moyen pour réguler le trafic et donc de réduire la congestion autour des aéroports. Dans ce travail nous étudions le problème de la conception de SID et STAR en prenant en compte la configuration et l’environnement autour des aéroports. Il existe beaucoup de travaux consacrés à la conception de trajectoires (voir [1]pour un survol), mais à notre connaissance peu de travaux sont dédiés à la conception de SID et STAR (e.g. [3]). Dans cette dernière, on doit tenir compte de contraintes opérationnelles (la séparation verticale et horizontale entre les routes, l’évitement d’obstacles, la courbure desroutes, la pente des routes tenant compte des taux de montée et de descente des aéronefs) et environnementales (des nuisances sonores, des zones urbaines)