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 m linear programs for (i) and m convex quadratic programs for (ii), where m is the number of constraints defining the containing set. In (iii), m convex programs need to be solved in order to verify the characterization, where again m is the number of constraints defining the containing set. All polyhedral sets, like the knowledge sets of support vector machine classifiers, are characterized by the intersection of a finite number of closed halfspaces.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Cherkassky, V. and Mulier, F. (1998), Learning from Data — Concepts, Theory and Methods, John Wiley & Sons, New York.
Chung, S.-J. (1989), NP-completeness of the linear complementarity problem, Journal of Optimization Theory and Applications, 60: 393–399.
Dantzig, G. B. (1963), Linear Programming and Extensions, Princeton University Press, Princeton, NJ.
Eaves, B. C. and Freund, R. M. (1982), Optimal scaling of balls and polyhedra, Mathematical Programming 23: 138–147.
Frank, M. and Wolfe, P. (1956), An algorithm for quadratic programming, Naval Research Logistics Quarterly 3: 95–110.
Freund, R. M. and Orlin, J. B. (1985), On the complexity of four polyhedral set containment problems, Mathematical Programming 33: 139–145.
Fung, G., Mangasarian, O. L. and Shavlik, J. (2001), Knowledge-based support vector machine classifiers. Technical Report 01-09, Data Mining Institute, Computer Sciences Department, University of Wisconsin, Madison, WI. ftp://ftp.cs.wisc.edu/pub/dmi/tech-reports/01-09.ps.
Fung, G., Mangasarian, O. L. and Smola, A. (2000), Minimal kernel classifiers. Technical Report 00-08, DataMining Institute, Computer Sciences Department, University ofWisconsin, Madison, WI. ftp://ftp.cs.wisc.edu/pub/dmi/tech-reports/00-08.ps.
Gritzmann, P. and Klee, V. (1992), Inner and outer j-radii of convex bodies in finite-dimensional normed spaces. Discrete and Computational Geometry 7: 255–280.
Kojima, M., Mizuno, S. and Yoshise, A. (1989), A polynomial–time algorithm for a class of linear complementarity problems. Mathematical Programming 44: 1–26.
Mangasarian, O. L. (1978), Characterization of linear complementarity problems as linear programs, Mathematical Programming Study 7: 74–87.
Mangasarian, O. L. (1994), Nonlinear Programming SIAM, Philadelphia, PA.
Mangasarian, O. L. (2000), Generalized support vector machines, in: A. Smola, P. Bartlett, B. Schölkopf, and D. Schuurmans (eds), MIT Press, Cambridge, MA. Advances in Large Margin Classifiers pp. 135–146, ftp://ftp.cs.wisc.edu/math-prog/tech-reports/98-14.ps.
Towell, G. G. and Shavlik, J. W. (1993), The extraction of refined rules from knowledge-based neural networks, Machine Learning 13: 71–101.
Towell, G. G. and Shavlik, J. W. (1994), Knowledge-based artificial neural networks, Artificial Intelligence 70: 119–165.
Vapnik, V. N. (2000), The Nature of Statistical Learning Theory, 2nd edition, Springer, New York.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Mangasarian, O. Set Containment Characterization. Journal of Global Optimization 24, 473–480 (2002). https://doi.org/10.1023/A:1021207718605
Issue Date:
DOI: https://doi.org/10.1023/A:1021207718605