Izvestiya of Saratov University.

Mathematics. Mechanics. Informatics

ISSN 1816-9791 (Print)
ISSN 2541-9005 (Online)


For citation:

Dudov S. I., Osipcev M. A. On an Approach to Approximate Solving of the Problem for the Best Approximation for Compact Body by a Ball of Fixed Radius. Izv. Sarat. Univ. Math. Mech. Inform., 2014, vol. 14, iss. 3, pp. 267-272. DOI: 10.18500/1816-9791-2014-14-3-267-272

This is an open access article distributed under the terms of Creative Commons Attribution 4.0 International License (CC-BY 4.0).
Published online: 
10.09.2014
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Russian
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UDC: 
519.853

On an Approach to Approximate Solving of the Problem for the Best Approximation for Compact Body by a Ball of Fixed Radius

Autors: 
Dudov Sergey Ivanovitch, Saratov State University
Osipcev Mikhail Anatolievich, Saratov State University
Abstract: 

In this paper, we consider the problem of the best approximation of a compact body by a fixed radius ball with respect to an arbitrary norm in the Hausdorff metric. This problem is reduced to a linear programming problem in the case, when compact body and ball of the norm are polytops.

References: 
  1. Nilol’skii M. S., Silin D. B. On the best approximation of a convex compact set by elements of addial. Proc. Steklov Inst. Math., 1995, vol. 211, pp. 306–321.
  2. Dudov S. I., Zlatorunskaya I. V. Best approximation of compact set by a ball in an arbitrary norm. Sb. Math., 2000, vol. 191, no. 10, pp. 1433–1458. DOI: http:// dx.doi.org/10.1070/SM2000v191n10ABEH000513.
  3. Dudov S. I. Relations between several problems of estimating convex compacta by balls. Sb. Math., 2007, vol. 198, no. 1, pp. 43–58. DOI: http://dx.doi.org/ 10.1070/SM2007v198n01ABEH003828.
  4. Dudov S. I., Meshcheryakova E. A. Method for finding an approximate solution of the asphericity problem for a convex body. Comp. Math. and Math. Physics, 2013, vol. 53, no. 10, pp. 1483–1493. DOI: 10.1134/S0965542513100059.
  5. Pschemichnyi B. N. Vypuklyj analiz i jekstremal’nye zadachi [Convex Analysis and Extremal Problems]. Moscow, Nauka, 1980 (in Russian).
  6. Dem’yanov V. F., Vasil’ev L. V. Nondifferetiable optimization. New York, Optimization software, Inc., Publications Division, 1985.
  7. Dudov S. I. Subdifferentiability and superdifferentiability of distance functions. Math. Notes, 1997, vol. 61, no. 4, pp. 440–450. DOI: 10.1007/BF02354988.
  8. Hiriart-Urruty J. B. Tangent cones, generalized gradients and mathematical programming in Banach spaces. Math. Oper. Research, 1979, vol. 4, no. 1, pp. 79–97.
  9. Vasil’ev F. P. Metody optimizacii [Methods of Optimization]. Moscow, MCSMO, 2011 (in Russian).
  10. Dudov S. I., Zlatorunskaya I. V. Best approximation of compact set by a ball in an arbitrary norm. Adv. Math. Res., 2003, vol. 2, pp. 81–114.
  11. Zuhovickij S. I., Avdeeva L. I. Linejnoe i vypukloe programmirovanie [Linear and convex programming]. Moscow, Nauka, 1964 (in Russian).
  12. Bronstein E. M. Approximation of convex sets by polytopes. J. of Math. Sciences, 2008, vol. 153, no. 6, pp. 727–762. DOI: 10.1007/s10958-008-9144-x.