Set Containment Characterization

  • Authors:

  • O. L. Mangasarian

  • Affiliations:

  • Computer Sciences Department, University of Wisconsin, 1210 West Dayton Street, Madison, WI 53706, USA (e-mail: )

  • Venue:
  • Journal of Global Optimization
  • Year:
  • 2002

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Abstract

Characterization of the containment of a polyhedral set in a closed halfspace, a key factor in generating knowledge-based support vector machine classifiers [7], is extended to the following: (i) containment of one polyhedral set in another; (ii) containment of a polyhedral set in a reverse-convex set defined by convex quadratic constraints; (iii) Containment of a general closed convex set, defined by convex constraints, in a reverse-convex set defined by convex nonlinear constraints. The first two characterizations can be determined in polynomial time by solving im linear programs for (i) and im convex quadratic programs for (ii), where im is the number of constraints defining the containing set. In (iii), im convex programs need to be solved in order to verify the characterization, where again im is the number of constraints defining the containing set. All polyhedral sets, like the iknowledge sets of support vector machine classifiers, are characterized by the intersection of a finite number of closed halfspaces.