Geodesic
Separation of convex polyhedral sets with column parameters
Kybernetika, Tome 44 (2008) no. 1, pp. 113-130 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

Separation is a famous principle and separation properties are important for optimization theory and various applications. In practice, input data are rarely known exactly and it is advisable to deal with parameters. In this article, we are concerned with the basic characteristics (existence, description, stability etc.) of separating hyperplanes of two convex polyhedral sets depending on parameters. We study the case, when parameters are situated in one column of the constraint matrix from the description of the given convex polyhedral set. We provide also a lot of examples carried out on PC.
Separation is a famous principle and separation properties are important for optimization theory and various applications. In practice, input data are rarely known exactly and it is advisable to deal with parameters. In this article, we are concerned with the basic characteristics (existence, description, stability etc.) of separating hyperplanes of two convex polyhedral sets depending on parameters. We study the case, when parameters are situated in one column of the constraint matrix from the description of the given convex polyhedral set. We provide also a lot of examples carried out on PC.
Classification : 39B82, 52B99, 90C29, 90C31
Keywords: separating hyperplane; parameters; convex polyhedra; solution set; stability set 
@article{KYB_2008_44_1_a8,
     author = {Hlad{\'\i}k, Milan},
     title = {Separation of convex polyhedral sets with column parameters},
     journal = {Kybernetika},
     pages = {113--130},
     year = {2008},
     volume = {44},
     number = {1},
     mrnumber = {2405059},
     zbl = {1142.90034},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/KYB_2008_44_1_a8/}
}
TY  - JOUR
AU  - Hladík, Milan
TI  - Separation of convex polyhedral sets with column parameters
JO  - Kybernetika
PY  - 2008
SP  - 113
EP  - 130
VL  - 44
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/KYB_2008_44_1_a8/
LA  - en
ID  - KYB_2008_44_1_a8
ER  - 
%0 Journal Article
%A Hladík, Milan
%T Separation of convex polyhedral sets with column parameters
%J Kybernetika
%D 2008
%P 113-130
%V 44
%N 1
%U http://geodesic.mathdoc.fr/item/KYB_2008_44_1_a8/
%G en
%F KYB_2008_44_1_a8
Hladík, Milan. Separation of convex polyhedral sets with column parameters. Kybernetika, Tome 44 (2008) no. 1, pp. 113-130. http://geodesic.mathdoc.fr/item/KYB_2008_44_1_a8/

[1] Gal T.: Postoptimal Analyses, Parametric Programming, and Related Topics. McGraw-Hill, New York 1979 | MR | Zbl

[2] Gal T., Greenberg H. J., eds.: Advances in Sensitivity Analysis and Parametric Programming. Kluwer Academic Publishers, Boston 1997 | MR | Zbl

[3] Grünbaum B.: Convex Polytopes. Springer, New York 2003 | MR | Zbl

[4] Grygarová L.: A calculation of all separating hyperplanes of two convex polytopes. Optimization 41 (1997), 57–69 | MR

[5] Grygarová L.: On a calculation of an arbitrary separating hyperplane of convex polyhedral sets. Optimization 43 (1998), 93–112 | MR | Zbl

[6] Hladík M.: Explicit description of all separating hyperplanes of two convex polyhedral sets with RHS-parameters. In: Proc. WDS’04, Part I (J. Šafránková, ed.), Matfyzpress, Praha 2004, pp. 63–70

[7] Kemp M. C., Kimura Y.: Introduction to Mathematical Economics. Springer, New York 1978 | MR | Zbl

[8] Klee V.: Separation and support properties of convex sets – a survey. In: Control Theory and the Calculus of Variations (A. V. Balakrishnan, ed.), Academic Press, New York 1969, pp. 235–303 | MR

[9] Nožička F., Guddat J., Hollatz, H., Bank B.: Theorie der linearen parametrischen Optimierung. Akademie-Verlag, Berlin 1974 | Zbl

[10] Nožička F., Grygarová, L., Lommatzsch K.: Geometrie konvexer Mengen und konvexe Analysis. Akademie-Verlag, Berlin 1988 | MR

[11] Schrijver A.: Theory of Linear and Integer Programming. Wiley, Chichester 1998 | MR | Zbl