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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 $\mathcal F(M)=\bigl\{x\in M:F(x)=\inf F(M)\bigr\}$ 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
$\bigl\{x\in X:F(x)\leqslant r\bigr\}$ 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.
UDC:
519.3
MSC: 49A25,
49A30 Received: 25.10.1976
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