Timeline for Mapping non-convex functions onto convex functions
Current License: CC BY-SA 4.0
17 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jul 6, 2018 at 10:40 | answer | added | user562983 | timeline score: 3 | |
| Jul 6, 2018 at 9:48 | vote | accept | Alex Silva | ||
| Jul 6, 2018 at 9:30 | answer | added | user856 | timeline score: 4 | |
| Jul 6, 2018 at 9:27 | comment | added | Alex Silva | @user550103, I added it up. | |
| Jul 6, 2018 at 9:26 | history | edited | Alex Silva | CC BY-SA 4.0 |
antoher example
|
| Jul 6, 2018 at 9:22 | comment | added | user550103 | @AlexSilva: Ok, I see. Probably, this example should be added as an example in the above problem. | |
| Jul 6, 2018 at 9:22 | comment | added | user562983 | The reason why it can't be true in a meaningful way is the reason why convex optimization is nice: local minima of a convex function are global minima, i.e. if there is some $x$ and some neigbourhood $U$ of $x$ such that $f(y)\ge f(x)$ for all $y\in U$, then $f(y)\ge f(x)$ for all $y$. If $f$ takes distinct values in many distinct local minima, then $f\circ g$ must "forget" that topological obstruction. A convex function covers intervals as a homeomorphism, as a $\cup$-thing or as a $\setminus\underline\quad/$-thing. So $f\circ g$ can't forget about those local minima (unless $g$ avoids them). | |
| Jul 6, 2018 at 9:11 | comment | added | Alex Silva | @user550103, $f(x) = x^3$ and $g(x) = \begin{cases} -x, ~\text{if}~ x < 0\\ x, ~\text{if}~ x \geq 0 \end{cases}$. | |
| Jul 6, 2018 at 9:00 | answer | added | Martin R | timeline score: 6 | |
| Jul 6, 2018 at 8:52 | comment | added | user550103 | @AlexSilva Do you have a better example instead of such a simple concave function? some other non-convex function that is mapped to a convex function? | |
| Jul 6, 2018 at 8:48 | history | edited | Alex Silva | CC BY-SA 4.0 |
clarification
|
| Jul 6, 2018 at 8:44 | comment | added | Alex Silva | @user550103, I agree, but it is just an example. It does not work if $f$ is not concave. | |
| Jul 6, 2018 at 8:42 | comment | added | Alex Silva | @MartinR Thanks for your reply, but I am not interested in the trivial solution $g$ constant. | |
| Jul 6, 2018 at 8:38 | comment | added | Martin R | Any constant function $g$ would do that, but you are probably looking for strictly monotonic (injective) functions. | |
| Jul 6, 2018 at 8:37 | comment | added | user550103 | just a side comment with a clarification. $\log(x)$ is a concave function. So, negated concave should yield convex, agree? | |
| Jul 6, 2018 at 8:37 | comment | added | user562983 | I presume $g$ surjective? | |
| Jul 6, 2018 at 8:31 | history | asked | Alex Silva | CC BY-SA 4.0 |