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Uspekhi Mat. Nauk, 2012, Volume 67, Issue 2(404), Pages 157–186 (Mi umn9466)  

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

Sub- and supergradients of envelopes, semicontinuous closures, and limits of sequences of functions

Yu. S. Ledyaevab, J. S. Treimana

a Western Michigan University, Kalamazoo, USA
b Steklov Mathematical Institute, Russian Academy of Sciences

Abstract: Envelopes $\sup_{\gamma\in\Gamma}f_{\gamma}(x)$ or $\inf_{\gamma\in\Gamma}f_{\gamma}(x)$ of parametric families of functions are typical non-differentiable functions arising in non-smooth analysis, optimization theory, control theory, the theory of generalized solutions of first-order partial differential equations, and other applications. In this survey formulae are obtained for sub- and supergradients of envelopes of lower semicontinuous functions, their corresponding semicontinuous closures, and limits and $\Gamma$-limits of sequences of functions. The unified method of derivation of these formulae for semicontinuous functions is based on the use of multidirectional mean-value inequalities for sets and non-smooth functions. These results are used to prove generalized versions of the Jung and Helly theorems for manifolds of non-positive curvature, to prove uniqueness of solutions of some optimization problems, and to get a new derivation of Stegall's well-known variational principle for smooth Banach spaces. Also, necessary conditions are derived for $\varepsilon$-maximizers of lower semicontinuous functions.
Bibliography: 47 titles.

Keywords: non-linear functional analysis, non-smooth analysis, upper and lower envelopes, generalizations of the Jung, Helly, and Stegall theorems.

DOI: https://doi.org/10.4213/rm9466

Full text: PDF file (764 kB)
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English version:
Russian Mathematical Surveys, 2012, 67:2, 345–373

Bibliographic databases:

Document Type: Article
UDC: 517.988.3
MSC: Primary 49J52; Secondary 52A35, 58C20
Received: 18.01.2012

Citation: Yu. S. Ledyaev, J. S. Treiman, “Sub- and supergradients of envelopes, semicontinuous closures, and limits of sequences of functions”, Uspekhi Mat. Nauk, 67:2(404) (2012), 157–186; Russian Math. Surveys, 67:2 (2012), 345–373

Citation in format AMSBIB
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    1. A. Festa, R. B. Vinter, “Decomposition of differential games with multiple targets”, J. Optim. Theory Appl., 169:3 (2016), 848–875  crossref  mathscinet  zmath  isi  elib  scopus
    2. D. Khlopin, “On boundary conditions at infinity for infinite horizon control problem”, 2017 Constructive Nonsmooth Analysis and Related Topics (Dedicated to the Memory of V. F. Demyanov) (CNSA), IEEE, 2017, 146–148  isi
    3. D. V. Khlopin, “O neobkhodimykh predelnykh gradientakh v zadachakh upravleniya na beskonechnom promezhutke”, Vypusk posvyaschen 70-letnemu yubileyu Aleksandra Georgievicha Chentsova, Tr. IMM UrO RAN, 24, no. 1, 2018, 247–256  mathnet  crossref  elib
    4. Khlopin D.V., “A Maximum Principle For One Infinite Horizon Impulsive Control Problem”, IFAC PAPERSONLINE, 51:32 (2018), 213–218  crossref  isi  scopus
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