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IMA Journal of Management Mathematics 2007 18(2):173-189; doi:10.1093/imaman/dpm014
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© The authors 2007. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

Arbitrary-norm hyperplane separation by variable neighbourhood search

Alejandro Karam**,1, Gilles Caporossi***,2 and Pierre Hansen****,2

1 HEC Montréal, 300, ch. de la côte-ste-catherine, H3T 2A7 Montréal, Qúebec, Canada, 2 HEC Montréal and GERAD, 300, ch. de la côte-ste-catherine, H3T 2A7 Montréal, Qúebec, Canada

** Email: alejandro.karam{at}hec.ca

*** Email: gilles.caporossi{at}gerad.ca

**** Email: pierre.hansen{at}gerad.ca


  Abstract

We consider the problem of separating two sets of points in a Euclidean space with a hyperplane that minimizes the sum of p-norm distances to the plane of points lying on the ‘wrong’ side of the plane. A variable neighbourhood search heuristic is used to determine the plane coefficients. For a set of examples with L1-norm, L2-norm and L{infty}-norm, for which the exact solution can be computed, we show that our algorithm finds it in most cases and gets good approximations in the others. The use of our heuristic solutions for problems in these norms can dramatically accelerate exact algorithms. Our method can be applied on very large instances that are intractable by exact algorithms. Since the proposed approach works for truly arbitrary norms (other than the traditional 1, 2 and {infty}), we can explore for the first time the effects of the choice of p on the generalization properties of p-norm hyperplane separation.

Keywords: hyperplane separation; linear discrimination; Lp-norm; VNS

Received on April 2006. accepted on 2 February 2007.


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