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

V. I. Berdyshev

Sbornik. Mathematics, Tome 32 (1977) no. 4, pp. 401-412

Résumé

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.