An interior point algorithm for minimum sum-of-squares clustering

Copyright @ 2000 SIAM PublicationsAn exact algorithm is proposed for minimum sum-of-squares nonhierarchical clustering, i.e., for partitioning a given set of points from a Euclidean m-space into a given number of clusters in order to minimize the sum of squared distances from all points to the centroid of the cluster to which they belong. This problem is expressed as a constrained hyperbolic program in 0-1 variables. The resolution method combines an interior point algorithm, i.e., a weighted analytic center column generation method, with branch-and-bound. The auxiliary problem of determining the entering column (i.e., the oracle) is an unconstrained hyperbolic program in 0-1 variables with a quadratic numerator and linear denominator. It is solved through a sequence of unconstrained quadratic programs in 0-1 variables. To accelerate resolution, variable neighborhood search heuristics are used both to get a good initial solution and to solve quickly the auxiliary problem as long as global optimality is not reached. Estimated bounds for the dual variables are deduced from the heuristic solution and used in the resolution process as a trust region. Proved minimum sum-of-squares partitions are determined for the rst time for several fairly large data sets from the literature, including Fisher's 150 iris.This research was supported by the Fonds National de la Recherche Scientifique Suisse, NSERC-Canada, and FCAR-Quebec

The incidence of nominal and real wage rigidity: an individual-based sectoral approach

This paper presents estimates based on individual data of downward nominal and real wage rigidities for thirteen sectors in Belgium, Denmark, Spain and Portugal. Our methodology follows the approach recently developed for the International Wage Flexibility Project, whereby resistance to nominal and real wage cuts is measured through departures of observed individual wage change histograms from an estimated counterfactual wage change distribution that would have prevailed in the absence of rigidity. We evaluate the role of worker and firm characteristics in shaping wage rigidities. We also confront our estimates of wage rigidities to structural features of the labour markets studied, such as the wage bargaining level, variable pay policy and the degree of product market competition. We find that the use of firm-level collective agreements in countries with rather centralized wage formation reduces the degree of real wage rigidity. This finding suggests that some degree of decentralization within highly centralized countries allows firms to adjust wages downwards, when business conditions turn bad. JEL Classification: J31wage rigidity, wage-bargaining institutions

The incidence of nominal and real wage rigidity : An individual-based sectoral approach

This paper presents estimates based on individual data on downward nominal and real wage rigidities for thirteen sectors in Belgium, Denmark, Spain and Portugal. Our methodology follows the approach recently developed for the International Wage Flexibility Project, whereby resistance to nominal and real wage cuts is measured through departures of observed individual wage-change histograms from an estimated counterfactual wage-change distribution that would have prevailed in the absence of any rigidity. We evaluate the role of worker and firm characteristics in shaping wage rigidities. We also confront our estimates of wage rigidities with structural features of the labour markets studied, such as the wage bargaining level, variable pay policy and the degree of product market competition. We find that the use of firm-level collective agreements in countries with rather centralised wage formation reduces the degree of real wage rigidity. This finding suggests that some degree of decentralisation within centralised countries allows firms to adjust wages downwards, when business conditions take a turn for the worsewage rigidity, wage-bargaining institutions

Finite size effects on MπM_\pi in QCD from Chiral Perturbation Theory

We present a determination of the shift $M_\pi(L)-M_\pi$ due to the finite spatial box size $L$ by means of $N_f=2$ Chiral Perturbation Theory and L\"uscher's formula. The range of applicability of the chiral prediction is discussed.Comment: 3 pages, 3 figures, Lattice2002(spectrum

Entropic Upper Bound on Gravitational Binding Energy

We prove that the gravitational binding energy {\Omega} of a self gravitating system described by a mass density distribution {\rho}(x) admits an upper bound B[{\rho}(x)] given by a simple function of an appropriate, non-additive Tsallis' power-law entropic functional Sq evaluated on the density {\rho}. The density distributions that saturate the entropic bound have the form of isotropic q-Gaussian distributions. These maximizer distributions correspond to the Plummer density profile, well known in astrophysics. A heuristic scaling argument is advanced suggesting that the entropic bound B[{\rho}(x)] is unique, in the sense that it is unlikely that exhaustive entropic upper bounds not based on the alluded Sq entropic measure exit. The present findings provide a new link between the physics of self gravitating systems, on the one hand, and the statistical formalism associated with non-additive, power-law entropic measures, on the other hand