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

Given a set of entities associated with points in Euclidean space, minimum sum-of-squares clustering (MSSC) consists in partitioning this set into clusters such that the sum of squared distances from each point to the centroid of its cluster is minimized. A column generation algorithm for MSSC was g...

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Bibliographic Details
Published in:Mathematical programming Vol. 131; no. 1-2; pp. 195 - 220
Main Authors: Aloise, Daniel, Hansen, Pierre, Liberti, Leo
Format: Journal Article
Language:English
Published: Berlin/Heidelberg Springer-Verlag 01.02.2012
Springer
Springer Nature B.V
Subjects:
Algorithms
Applied sciences
Calculus of Variations and Optimal Control; Optimization
Centroids
Cluster analysis
Clustering
Clusters
Combinatorics
Cost engineering
Exact sciences and technology
Exact solutions
Flows in networks. Combinatorial problems
Full Length Paper
Game theory
Heuristic
Mathematical analysis
Mathematical and Computational Physics
Mathematical Methods in Physics
Mathematical programming
Mathematics
Mathematics and Statistics
Mathematics of Computing
Numerical Analysis
Operational research and scientific management
Operational research. Management science
Optimization
Partitioning
Studies
Theoretical
Column generation
Sum-of-squares
65K05
91C20
90C27
Clustering
ACCPM
Bottleneck
Costs
Mathematical sociology
Cluster
Set partitioning
Combinatorial optimization
Modeling
Cutting plane method
N person game
Exact solution
Center of mass
Classification
Euclidean space
Mathematical programming
ISSN:0025-5610, 1436-4646
Online Access:Get full text
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Description
Summary:Given a set of entities associated with points in Euclidean space, minimum sum-of-squares clustering (MSSC) consists in partitioning this set into clusters such that the sum of squared distances from each point to the centroid of its cluster is minimized. A column generation algorithm for MSSC was given by du Merle et al. in SIAM Journal Scientific Computing 21:1485–1505. The bottleneck of that algorithm is the resolution of the auxiliary problem of finding a column with negative reduced cost. We propose a new way to solve this auxiliary problem based on geometric arguments. This greatly improves the efficiency of the whole algorithm and leads to exact solution of instances with over 2,300 entities, i.e., more than 10 times as much as previously done.
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ISSN:0025-5610
1436-4646
DOI:10.1007/s10107-010-0349-7