Algebraic separation and shadowing of arbitrary sets

• Autorzy: Grzybowski J. , Pallaschke D. , Urbański R.

Identyfikatory

DOI

10.14708/cm.v53i2.781

• Języki publikacji: EN

Abstrakty

EN

In this paper we consider a generalization of the separation technique proposed in [10,4,7] for the separation of finitely many compact convex sets Ai, i∈I by another compact convex set S in a locally convex vector space to arbitrary sets in real vector spaces. Then we investigate the notation of shadowing set which is a generalization of the notion of separating set and construct separating sets by means of a generalized Demyanov-difference in locally convex vector spaces.

Słowa kluczowe

EN

separation; geometry of convex sets; Demyanov difference; data classification

• Wydawca: Polskie Towarzystwo Matematyczne
• Czasopismo: Commentationes Mathematicae
• Rocznik: 2013
• Tom: Vol. 53, No. 2
• Strony: 55--64
• Opis fizyczny: Bibliogr. 15 poz., rys.

Twórcy

autor

Grzybowski J.

• Adam Mickiewicz University Poznan, Faculty of Mathematics and Computer Science, Umultowska 87, PL-61-614 Poznan, Poland

autor

Pallaschke D.

• University of Karlsruhe (KIT), Institute of Operations, Kaiserstr. 12, D-76128 Karlsruhe, Germany

autor

Urbański R.

• Adam Mickiewicz University Poznan, Faculty of Mathematics and Computer Science, Umultowska 87, PL-61-614 Poznan, Poland

Bibliografia

• [1] A. Astorino and M. Gaudioso, (2002): Polyhedral separability through successive LP, Journ. of Optimization Theory 112, 265-293.
• [2] F. H. Clarke, (1983): Optimization and Nonsmooth Analysis, J. Wiley Pub. Comp., New York.
• [3] V. F. Demyanov and A. M. Rubinov, (1986): Quasidifferential calculus, Optimization Software Inc., Publications Division, New York.
• [4] M. Gaudioso, E. Gorgone and D. Pallaschke, (2011): Separation of convex sets by Clarke subdifferential, Optimization 59, 1199-1210.
• [5] J. Grzybowski, D. Pallaschke and R. Urbanski, (2012): Demyanov Difference in infinite dimensional Spaces, submitted to the Proceedings of CNSA, St. Petersburg.
• [6] J. Grzybowski, D. Pallaschke and R. Urbanski, (2010): Reduction of finite exhausters, Journ. of Global Optimization 46, 589-601.
• [7] J. Grzybowski, D. Pallaschke and R. Urbanski, (2005): A pre-classification and the separation law for closed bounded convex sets, Optimization Methods and Software 20, 219-229.
• [8] T. Husain and I. Tweddle, (1970): On the extreme points of the sum of two compact convex sets, Math. Ann. 188, 113-122.
• [9] J-E. Martinez-Legaz and A. Martinón, (2012): On the infimum of a quasiconvex vector function over an intersection, TOP 20, 503-516.
• [10] D. Pallaschke and R. Urbanski, (2002): Pairs of Compact Convex Sets —Fractional Arithmetic with Convex Sets, Mathematics and its Applications, Vol. 548, Kluwer Acad. Publ. Dordrecht.
• [11] D. Pallaschke and R. Urbanski, (2002): On the separation and order law of cancellation for bounded sets, Optimization, Vol. 51(3), 487-496.
• [12] A. G. Pinsker, (1966): The space of convex sets of a locally convex space, Trudy Leningrad Engineering-Economic Institute 63, 13-17.
• [13] A. M. Rubinov and I. S. Akhundov, (1992): Differences of compact sets in the sense of Demyanov and its application to non-smooth-analysis, Optimization 23 (1992), 179-189.
• [14] R. Urbanski, (1976): A generalization of the Minkowski-Rådström-Hörmander theorem, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astr. Phys. 24, 709-715.
• [15] R. Zang, Y. Ma and Y. Liu, (2013): (F,K, b)-vex sets and some related properties, International Journal of Computer Mathematics, (2013), 1-7.