A branch-and-cut SDP-based algorithm for minimum sum-of-squares clustering

Abstract

Minimum sum-of-squares clustering (MSSC) consists in partitioning a given set of n points into k clusters in order to minimize the sum of squared distances from the points to the centroid of their cluster. Recently, Peng & Xia (2005) established the equivalence between 0-1 semidefinite programming (SDP) and MSSC. In this paper, we propose a branch-and-cut algorithm for the underlying 0-1 SDP model. The algorithm obtains exact solutions for fairly large data sets with computing times comparable with those of the best exact method found in the literature.Keywords: clustering; sum-of-squares; semidefinite programming. ResumoClusterização por soma mínima de distâncias quadráticas consiste em particionar um dado conjunto de n pontos em k clusters a fim de minimizar a soma das distâncias quadráticas entre os pontos e o centróide de seus respectivos clusters. Recentemente, Peng & Xia (2005) estabeleceram a equivalência entre o problema e programação semidefinida 0-1. Neste artigo, um algoritmo branch-and-cut é proposto para o modelo baseado em programação semidefinida 0-1. O algoritmo obtém soluções exatas para instâncias reais de grande porte em tempos computacionais comparáveis àqueles do melhor método exato proposto na literatura.Palavras-chave: clusterização; soma de distâncias quadráticas; programação semidefinida.Aloise & Hansen -A branch-and-cut SDP-based algorithm for minimum sum-of-squares clustering 504

An improved column generation algorithm for minimum sum-of-squares clustering

“…So, we do not refer to its results in the subsequent tables. As empirically observed in [2], the performance of algorithm bb-sdp deteriorates as k decreases, in contrast with…”

“…The results are also compared to those of two other methods proposed in the literature, i.e., the repetitive branch-and-bound algorithm (rbba) of Brusco [9] and the best branch-and-cut SDP-based algorithm (bb-sdp) of [2]. Tables 2-7 show results for data sets in the plane.…”

“…On the basis of these results, the present authors developed in [2] a branchand-cut SDP-based algorithm for MSSC with lower bounds obtained from the LP relaxation of the 0-1 SDP model. This algorithm obtains exact solutions for fairly large data sets, i.e., n = 202 and k ≥ 9, with computing times comparable with those obtained by the column generation method proposed by du Merle et al [15].…”

Two-Step Semidefinite Programming approach to clustering and dimensionality reduction

“…In such a problem, each object can be considered as a point in a n-dimensional space and each cluster can be identified by its center, called centroid, a non-observable object calculated by taking the mean of all the objects assigned to this cluster [18,32,34]. To express similarity between objects, i.e., homogeneity inside a cluster, several similarity measures have been proposed, such as a metric defined on the data set [2,7]. One of the most used (dis)similarity measures is the squared Euclidean distance [3,4,14,18,34].…”

Improving spectral bounds for clustering problems by Lagrangian relaxation

“…Most of the sophisticated computation on clustering problems has addressed the MSS; see, e.g., the most recent exact methods (Aloise and Hansen, ; Aloise et al, ). We do not discuss these approaches in detail because (i) MSS is computationally rather different from RASS and (ii) we focus here on spectral bounds (for which there is not much computation in the MSS special case).…”