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I am reading Convex optimization written by Stephen Boyd. In page 640, there is an example said \begin{equation} f(x)=-log(x) \end{equation} is a closed function. But this function seems does not satisfy the definition of closed function in this book. The plot of this function is attached below: enter image description here And the definition of closed function in the book is attached below: enter image description here

Can anybody tell me whether this function is closed or not ?

BioCoder
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2 Answers2

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We have $dom\ f = (0,\infty),$ and for any $\alpha \in\mathbb R$ $$ x\in\ dom\ f\ and\ f(x) \leq \alpha \Leftrightarrow x \ge e^{-\alpha} \Leftrightarrow x \in [e^{-\alpha},\infty). $$ Now, the set $[e^{-\alpha},\infty)$ is closed, so $f$ is a closed function, according to the definition.

jflipp
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  • Is $[e^{-\alpha},\infty)$ closed ?? I think it's a left-closed and right-open interval. – BioCoder Jan 22 '15 at 08:56
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    It's closed because its complement ($-\infty$,$e^{-\alpha}$) is open. – velut luna Jan 22 '15 at 09:05
  • I re-read the definition of closed set in page 638 of Convex Optimization. Although kind of confused, but I will take it as a standard explanation. Thanks a lot for your help! – BioCoder Jan 22 '15 at 09:21
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It's closed. It's level sets are of the form $[b, \infty[$ and hence, closed.

Dirk
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