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Mat. Sb. (N.S.), 1977, Volume 103(145), Number 4(8), Pages 467–479 (Mi msb2918)  

This article is cited in 5 scientific papers (total in 5 papers)

Stability of a minimization problem under perturbation of the set of admissible elements

V. I. Berdyshev


Abstract: Let $F$ be a continuous real functional on the space $X$. Continuity of the operator $\mathcal F$ from $2^X$ into itself is considered, where $\mathcal F(M)=\{x\in M:F(x)=\inf F(M)\}$ for each $M\in 2^X$. In particular, in the case of a normed space $X$ the following is proved. Write
$$ AB=\sup_{x\in A}\inf_{y\in B}\|x-y\|,\qquad h(A,B)=\max\{AB,BA\},\qquad(A,B\subset X), $$
and let $\mathcal M$ be the totality of all closed convex sets in $X$. A set $M\subset X$ is called approximately compact if every minimizing sequence in $M$ contains a subsequence converging to an element of $M$.
Suppose $X$ is reflexive, $F$ is convex and the set $\{x\in X:F(x)\leqslant r\}$ is bounded for $r>\inf F(X)$ and contains interior points. Then the following assertions are equivalent:
a) $M_\alpha,M\in\mathcal M$, $h(M_\alpha,M)\to0\Rightarrow\mathcal F(M_\alpha)\mathcal F(M)\to0$,
b) every set $M\in\mathcal M$ is approximately compact.
Bibliography: 15 titles.

Full text: PDF file (1328 kB)
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English version:
Mathematics of the USSR-Sbornik, 1977, 32:4, 401–412

Bibliographic databases:

UDC: 519.3
MSC: 49A25, 49A30
Received: 25.10.1976

Citation: V. I. Berdyshev, “Stability of a minimization problem under perturbation of the set of admissible elements”, Mat. Sb. (N.S.), 103(145):4(8) (1977), 467–479; Math. USSR-Sb., 32:4 (1977), 401–412

Citation in format AMSBIB
\Bibitem{Ber77}
\by V.~I.~Berdyshev
\paper Stability of a~minimization problem under perturbation of the set of admissible elements
\jour Mat. Sb. (N.S.)
\yr 1977
\vol 103(145)
\issue 4(8)
\pages 467--479
\mathnet{http://mi.mathnet.ru/msb2918}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=493666}
\zmath{https://zbmath.org/?q=an:0368.90118}
\transl
\jour Math. USSR-Sb.
\yr 1977
\vol 32
\issue 4
\pages 401--412
\crossref{https://doi.org/10.1070/SM1977v032n04ABEH002394}
\isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=A1977GL81400001}


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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. V. I. Berdyshev, “Continuity of a multivalued mapping connected with the problem of minimizing a functional”, Math. USSR-Izv., 16:3 (1981), 431–456  mathnet  crossref  mathscinet  zmath  isi
    2. Roberto Lucchetti, Fioravante Patrone, “Hadamard and Tyhonov well-posedness of a certain class of convex functions”, Journal of Mathematical Analysis and Applications, 88:1 (1982), 204  crossref
    3. Roberto Lucchetti, “On the continuity of the minima for a family of constrained optimization problems∗”, Numerical Functional Analysis and Optimization, 7:4 (1985), 349  crossref
    4. V. S. Balaganskii, L. P. Vlasov, “The problem of convexity of Chebyshev sets”, Russian Math. Surveys, 51:6 (1996), 1127–1190  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi
    5. “Vitalii Ivanovich Berdyshev”, Proc. Steklov Inst. Math. (Suppl.), 265, suppl. 1 (2009), S1–S9  mathnet  crossref  isi
  • Математический сборник (новая серия) - 1964–1988 Sbornik: Mathematics (from 1967)
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