Izvestiya of Saratov University.

Mathematics. Mechanics. Informatics

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


For citation:

Abramova V. V., Dudov S. I., Osipcev M. A. The External Estimate of the Compact Set by Lebesgue Set of the Convex Function. Izv. Sarat. Univ. Math. Mech. Inform., 2020, vol. 20, iss. 2, pp. 142-153. DOI: 10.18500/1816-9791-2020-20-2-142-153

This is an open access article distributed under the terms of Creative Commons Attribution 4.0 International License (CC-BY 4.0).
Published online: 
01.06.2020
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Russian
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Article
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519.853

The External Estimate of the Compact Set by Lebesgue Set of the Convex Function

Autors: 
Abramova Veronika V., Saratov State University
Dudov Sergey Ivanovitch, Saratov State University
Osipcev Mikhail Anatolievich, Saratov State University
Abstract: 

The finite-dimensional problem of embedding a given compact D ⊂ p into the lower Lebesgue set G(α) = {y ∈ pf(y) <= α} of the convex function f(·) with the smallest value of α due to the offset of D is considered. Its mathematical formalization leads to the problem of minimizing the function φ(x) = max yD f(y − x) на p. The properties of the function φ(x) are researched, necessary and sufficient conditions and conditions for the uniqueness of the problem solution are obtained. As an important case for applications, the case when f(·) is the Minkowski gauge function of some convex body M is singled out. It is shown that if M is a polyhedron, then the problem reduces to a linear programming problem. The approach to get an approximate solution is proposed in which, having known the approximation of xi to obtain xi+1 it is necessary to solve the simpler problem of embedding the compact set D into the Lebesgue set of the gauge function of the set Mi = G(αi), where αi = f(xi). The rationale for the convergence for a sequence of approximations to the problem solution is given.

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Received: 
12.03.2019
Accepted: 
05.06.2019
Published: 
01.06.2020