Abstract
LetX andY be finite dimensional vector spaces over the real numbers. LetΩ be a binary relation betweenX andY given by a bilinear inequality. TheΩ-polar of a subsetP ofX is the set of all elements ofY which are related byΩ to all elements ofP. TheΩ-polar of a subset ofY is defined similarly. TheΩ-polar of theΩ-polar ofP is called theΩ-closure ofP andP is calledΩ-closed ifP equals itsΩ-closure. We describe theΩ-polar of any finitely generated setP as the solution set of a finite system of linear inequalities and describe theΩ-closure ofP as a finitely generated set. TheΩ-closed polyhedra are characterized in terms of defining systems of linear inequalities and also in terms of the relationship of the polyhedronP with its recessional cone and with certain subsets ofX andY determined by the relationΩ. Six classes of bilinear inequalities are distinguished in the characterization ofΩ-closed polyhedra.
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Griffin, V., Aráoz, J. & Edmonds, J. Polyhedral polarity defined by a general bilinear inequality. Mathematical Programming 23, 117–137 (1982). https://doi.org/10.1007/BF01583783
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DOI: https://doi.org/10.1007/BF01583783