Theory of linear and integer programming
Theory of linear and integer programming
The complexity and approximability of finding maximum feasible subsystems of linear relations
Theoretical Computer Science
The nature of statistical learning theory
The nature of statistical learning theory
Median hyperplanes in normed spaces — a survey
Discrete Applied Mathematics
ACM Computing Surveys (CSUR)
Projective clustering in high dimensions using core-sets
Proceedings of the eighteenth annual symposium on Computational geometry
Parallel Optimization: Theory, Algorithms and Applications
Parallel Optimization: Theory, Algorithms and Applications
The MIN PFS problem and piecewise linear model estimation
Discrete Applied Mathematics - Special issue: Third ALIO-EURO meeting on applied combinatorial optimization
Journal of Global Optimization
Hardness of Set Cover with Intersection 1
ICALP '00 Proceedings of the 27th International Colloquium on Automata, Languages and Programming
Mathematical Programming for Data Mining: Formulations and Challenges
INFORMS Journal on Computing
Approximation algorithms for projective clustering
Journal of Algorithms
Subspace clustering for high dimensional data: a review
ACM SIGKDD Explorations Newsletter - Special issue on learning from imbalanced datasets
Classification and Regression via Integer Optimization
Operations Research
Randomized relaxation methods for the maximum feasible subsystem problem
IPCO'05 Proceedings of the 11th international conference on Integer Programming and Combinatorial Optimization
Covering a set of points with a minimum number of lines
CIAC'06 Proceedings of the 6th Italian conference on Algorithms and Complexity
A clustering technique for the identification of piecewise affine systems
Automatica (Journal of IFAC)
On the complexity of locating linear facilities in the plane
Operations Research Letters
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Given n points in ℝd and a maximum allowed tolerance ε 0, the minimum hyperplanes clustering problem consists in finding a minimum number of hyperplanes such that the Euclidean distance between each point and the nearest hyperplane is at most ε. We present a column generation approach for this problem based on a mixed integer nonlinear formulation in which the master is a set covering problem and the pricing subproblem is a mixed integer program with a nonconvex normalization constraint. We propose different ways of generating the initial pool of columns and investigate their impact on the overall algorithm. Since the pricing subproblem is substantially complicated by the ℓ2-norm constraint, we consider approximate pricing subproblems involving different norms. Some strategies for refining the solution and speeding-up the overall method are also discussed. The performance of our column generation algorithm is assessed on realistic randomly generated instances as well as on real-world instances.