From Membership to Separation, a Simple Construction

In [3], Martin Grötschel, László Lovász and Alexander Schrijver use a construction of Dmitrii Yudin et Arkadiĭ Nemirovskiĭ to polynomially separate a point x from a centered bounded convex K using a membership oracle. In this note, we present a natural and simple construction which solve the same problem but for the simpler case of polyhedra. Namely, given a well defined polyhedron P with a non-empty interior, a point \( x \notin P \) and a point \( a \in \operatorname{int} {\left( P \right)} \), using a polynomial number of calls of the membership oracle, we find a facet of P whose supporting hyperplane separates x from P.

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Correspondence to Jean François Maurras.

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Maurras, J.F. From Membership to Separation, a Simple Construction. Combinatorica 22, 531–536 (2002). https://doi.org/10.1007/s00493-002-0005-9

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AMS Subject Classification (2000):

  • 05Cxx
  • 11H06
  • 11J13