Abstract
We investigate diverse separation properties of two convex polyhedral sets for the case when there are parameters in one row of the constraint matrix. In particular, we deal with the existence, description and stability properties of the separating hyperplanes of such convex polyhedral sets. We present several examples carried out on PC. We are also interested in supporting separation (separating hyperplanes support both the convex polyhedral sets at given faces) and permanent separation (a hyperplane separates the convex polyhedral sets for all feasible parameters). Finally, we show how the developed theory is applicable in multiobjective linear programming.
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References
N. Cristianini, J. Shawe-Taylor: An Introduction to Support Vector Machines and Other Kernel-Based Learning Methods. Cambridge University Press, Cambridge, 2001. zbl
T. Gal: Postoptimal Analyses, Parametric Programming, and Related Topics. McGraw-Hill, New York, 1979. zbl
T. Gal, H. J. Greenberg, eds.: Advances in Sensitivity Analysis and Parametric Programming. Kluwer Academic Publishers, Dordrecht, 1997. zbl
B. Grünbaum: Convex Polytopes, 2nd edition. Springer, New York, 2003. zbl
L. Grygarová: A calculation of all separating hyperplanes of two convex polytopes. Optimization 41 (1997), 57–69. zbl
L. Grygarová: On a calculation of an arbitrary separating hyperplane of convex polyhedral sets. Optimization 43 (1998), 93–112. zbl
L. Grygarová: Separating support hyperplanes for a pair of convex polyhedral sets. Optimization 43 (1998), 113–143.
L. Grygarová: On a supporting hyperplane for two convex polyhedral sets. Optimization 43 (1998), 235–255. zbl
L. Grygarová: Die Lösbarkeit eines linearen Optimierungsproblems unter Zufügung einer weiteren Restriktionsbedingung. Apl. Mat. 17 (1972), 352–387. (In German.) zbl
M. Hladík: Explicit description of all separating hyperplanes of two convex polyhedral sets with RHS-parameters. Proceedings of WDS’04, Part I (J. Šafránková, ed.). Matfyzpress, Prague, 2004, pp. 63–70.
M. Hladík: Separation of convex polyhedral sets with column parameters. Kybernetika 44 (2008), 113–130. zbl
M.D. Intriligator: Mathematical Optimization and Economic Theory. SIAM, Philadelphia, 2002. zbl
M.C. Kemp, Y. Kimura: Introduction to Mathematical Economics. Springer, New York, 1978. zbl
V. Klee: Separation and support properties of convex sets—a survey. Control Theory and the Calculus of Variations (A.V. Balakrishnan, ed.). Academic Press, New York, 1969, pp. 235–303.
F. Nožička, J. Guddat, H. Hollatz, B. Bank: Theorie der linearen parametrischen Optimierung. Akademie-Verlag, Berlin, 1974. (In German.) zbl
F. Nožička, L. Grygarová, K. Lommatzsch: Geometrie konvexer Mengen und konvexe Analysis. Akademie-Verlag, Berlin, 1988.
M. Padberg: Linear Optimization and Extension. Springer, Berlin, 1999. zbl
R.T. Rockafellar, R. Tyrrel: Convex Analysis. Princeton University Press, Princeton, 1970.
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Hladík, M. On the separation of parametric convex polyhedral sets with application in MOLP. Appl Math 55, 269–289 (2010). https://doi.org/10.1007/s10492-010-0021-9
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DOI: https://doi.org/10.1007/s10492-010-0021-9