Trudy Instituta Matematiki i Mekhaniki UrO RAN
RUS  ENG    JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB  
General information
Latest issue
Archive
Impact factor

Search papers
Search references

RSS
Latest issue
Current issues
Archive issues
What is RSS



Trudy Inst. Mat. i Mekh. UrO RAN:
Year:
Volume:
Issue:
Page:
Find






Personal entry:
Login:
Password:
Save password
Enter
Forgotten password?
Register


Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2019, Volume 25, Number 3, Pages 73–85
DOI: https://doi.org/10.21538/0134-4889-2019-25-3-73-85
(Mi timm1648)
 

This article is cited in 4 scientific papers (total in 4 papers)

Abstract Convexity of Functions with Respect to the Set of Lipschitz (Concave) Functions

V. V. Gorokhovika, A. S. Tykounb

a Institute of Mathematics of the National Academy of Sciences of Belarus
b Belarusian State University, Faculty of Mathematics and Mechanics
Full-text PDF (249 kB) Citations (4)
References:
Abstract: The paper is devoted to the abstract ${\mathcal H}$-convexity of functions (where ${\mathcal H}$ is a given set of elementary functions) and its realization in the cases when ${\mathcal H}$ is the space of Lipschitz functions or the set of Lipschitz concave functions. The notion of regular ${\mathcal H}$-convex functions is introduced. These are functions representable as the upper envelopes of the set of their maximal (with respect to the pointwise order) ${\mathcal H}$-minorants. As a generalization of the global subdifferential of a convex function, we introduce the set of maximal support ${\mathcal H}$-minorants at a point and the set of lower ${\mathcal H}$-support points. Using these tools, we formulate both a necessary condition and a sufficient one for global minima of nonsmooth functions. In the second part of the paper, the abstract notions of ${\mathcal H}$-convexity are realized in the specific cases when functions are defined on a metric or normed space $X$ and the set of elementary functions is the space ${\mathcal L}(X,{\mathbb{R}})$ of Lipschitz functions or the set ${\mathcal L}\widehat{C}(X,{\mathbb{R}})$ of Lipschitz concave functions, respectively. An important result of this part of the paper is the proof of the fact that, for a lower semicontinuous function lower bounded by a Lipschitz function, the set of lower ${\mathcal L}$-support points and the set of lower ${\mathcal L}\widehat{C}$-support points coincide and are dense in the effective domain of the function. These results extend the known Brøndsted–Rockafellar theorem on the existence of the subdifferential for convex lower semicontinuous functions to the wider class of lower semicontinuous functions and go back to the Bishop–Phelps theorem on the density of support points in the boundary of a closed convex set, which is one of the most important results of classical convex analysis.
Keywords: abstract convexity, support minorants, support points, global minimum, semicontinuous functions, Lipschitz functions, concave Lipschitz functions, density of support points.
Funding agency Grant number
ГПНИ "Конвергенция-2020" 1.4.01
This work was supported by the National Program for Scientific Research of the Republic of Belarus for 2016–2020 “Convergence 2020” (project no. 1.4.01).
Received: 20.04.2019
Revised: 15.05.2019
Accepted: 20.05.2019
English version:
Proceedings of the Steklov Institute of Mathematics (Supplementary issues), 2020, Volume 309, Issue 1, Pages S36–S46
DOI: https://doi.org/10.1134/S0081543820040057
Bibliographic databases:
Document Type: Article
UDC: 517.27
Language: Russian
Citation: V. V. Gorokhovik, A. S. Tykoun, “Abstract Convexity of Functions with Respect to the Set of Lipschitz (Concave) Functions”, Trudy Inst. Mat. i Mekh. UrO RAN, 25, no. 3, 2019, 73–85; Proc. Steklov Inst. Math. (Suppl.), 309, suppl. 1 (2020), S36–S46
Citation in format AMSBIB
\Bibitem{GorTyk19}
\by V.~V.~Gorokhovik, A.~S.~Tykoun
\paper Abstract Convexity of Functions with Respect to the Set of Lipschitz (Concave) Functions
\serial Trudy Inst. Mat. i Mekh. UrO RAN
\yr 2019
\vol 25
\issue 3
\pages 73--85
\mathnet{http://mi.mathnet.ru/timm1648}
\crossref{https://doi.org/10.21538/0134-4889-2019-25-3-73-85}
\elib{https://elibrary.ru/item.asp?id=39323538}
\transl
\jour Proc. Steklov Inst. Math. (Suppl.)
\yr 2020
\vol 309
\issue , suppl. 1
\pages S36--S46
\crossref{https://doi.org/10.1134/S0081543820040057}
\isi{https://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=Publons&SrcAuth=Publons_CEL&DestLinkType=FullRecord&DestApp=WOS_CPL&KeyUT=000485178300006}
\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85078463998}
Linking options:
  • https://www.mathnet.ru/eng/timm1648
  • https://www.mathnet.ru/eng/timm/v25/i3/p73
  • This publication is cited in the following 4 articles:
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
    Trudy Instituta Matematiki i Mekhaniki UrO RAN
    Statistics & downloads:
    Abstract page:190
    Full-text PDF :57
    References:30
    First page:6
     
      Contact us:
     Terms of Use  Registration to the website  Logotypes © Steklov Mathematical Institute RAS, 2024