Let $A, B$ be Banach spaces, and for any $a\in A$, $B_a\in B$ is a measurable subset. Consider the following optimization problem: $$L(a)=\inf_{b\in B_a}\ell(b),$$ where $\ell(b)$ is a infinite-times differentiable functional of $b$.

Question: prove that $L(a)$ is Frechet differentiable, moreover, $L(a)$ satisfies Taylor expansion $$L(a)-L(a_0)=L'(a_0)(a-a_0)+O(\|a-a_0\|_A^2),$$ where $\|\|_A$ is the norm of $A$.

We can assume that, when $a$ is moving, the set $B_a$ is smoothly moving. I am not sure how to describe such smoothly moving. Intuitively, this can make $L(a)$ smooth in $a$. But I am not sure how to prove this. Ideas or suggestions will be highly appreciated!


1 Answer 1


$\newcommand\R{\mathbb R}$The answer is no.

E.g., let $A=B=\R$, $B_a=[1-a^2,2]$ for $a\in A$, and $\ell(b)=b^3-3b$ for $b\in B$. Then $\ell(b)$ and $B_a$ are perfectly smooth, but $L$ is not differentiable.

Indeed, here $L(a)=-2+a^4\min(0,3-a^2)$ for $a\in A=\R$. The left derivative of $L$ at $\sqrt3$ is $0$ and the right derivative of $L$ at $\sqrt3$ is $-18\sqrt3$.

  • $\begingroup$ @Jeff : I suggest you post this additional question separately, after careful preparation, trying to specify as much as possible the meaning of "interesting scenarios". Then that question would probably attract more attention. For now, can we wrap the matter with your current question? $\endgroup$ Sep 20 at 16:44
  • $\begingroup$ Do you have a response to this answer and my comment? $\endgroup$ Sep 27 at 20:49
  • $\begingroup$ Thanks, Iosif. Sure, please close this question with your counterexample. $\endgroup$
    – Jeff
    Oct 2 at 15:48
  • $\begingroup$ @Jeff : I cannot "close" this question. On the other hand, you can finalize it, in accordance with these guidelines. $\endgroup$ Oct 2 at 16:15

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