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Consider an optimization problem with real parameters $a$, $b$ and $c$. Let $W^*$ be a solution to the optimization problem. $W^*$ is a finite-dimensional vector. Clearly, $W^*$ is a function of $a$, $b$ and $c$. So I will write $W^*(a,b,c)$.

Assume first that $W^*$ is unique. With that assumption, I was able to show that $W^*$ is continuous as a function of $a$, $b$ and $c$. Thus, I could argue that as a sequence $(a_n, b_n, c_n) \to (a, b, c)$, $W^*(a_n, b_n, c_n) \to W^*(a,b,c)$, with respect to $l^2$ norms for example.

But then I realized that $W^*$ need not be unique. So for a given $(a,b,c)$, there could be two optimal solutions, call them $W_1^*(a,b,c)$ and $W_2^*(a,b,c)$.

I still want to be able to say that if $(a_n, b_n, c_n) \to (a, b, c)$, $W^*(a_n, b_n, c_n)$ should approach either $W_1^*(a,b,c)$ or $W_2^*(a,b,c)$.

Can I make this claim with the standard continuity I already proved assuming uniqueness? Is there some framework which allows this (see title)?

rims
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  • There is a notion of upper semicontinuity for set valued functions. (Well, various notions.) – copper.hat Aug 21 '22 at 16:04
  • copper.hat did you mean hemicontinuity in your previous comment? That's what I found for set-valued functions. https://en.wikipedia.org/wiki/Hemicontinuity – rims Aug 21 '22 at 16:23
  • It seems to be another name for same. – copper.hat Aug 21 '22 at 16:27
  • copper.hat I was able to show that as $(a_n. b_n, c_n) \to (a, b, c)$, $W^(a_n, b_n, c_n)$ must converge to a (not the) solution of the optimization problem with parameters $a$, $b$ and $c$. Given this, what should I call $W^(a, b, c)$ as a function of $a, b$ and $c$. I am having trouble understanding the notion of upper and lower hemicontinuity. Wikipedia is not clear enough for me. – rims Aug 22 '22 at 09:13

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