Timeline for Complexity of this minimization
Current License: CC BY-SA 3.0
6 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 21, 2015 at 21:05 | vote | accept | user66081 | ||
| Dec 21, 2015 at 21:03 | comment | added | user66081 | I was missing the following bit. Write $\alpha = x^b$ and $\beta = x^{1-b}$, and note that $\alpha \beta = {const}$. Then $(\alpha + \beta) \mapsto |\log(\alpha/\beta)|$ is nondecreasing. | |
| Dec 21, 2015 at 18:51 | comment | added | Max Alekseyev | Simpler explanation follows from the AM-GM inequality. | |
| Dec 21, 2015 at 18:39 | comment | added | Linus Hamilton | Because $x^b+x^{1-b}=\left(\sqrt{x^{b}}-\sqrt{x^{1-b}}\right)^{2}+2\sqrt{x^{1}}$. Since $x^1$ does not depend on $b$, minimizing this is equivalent to minimizing $|\sqrt{x^{b}}-\sqrt{x^{1-b}}|$. Which is the same as getting $x^b$ and $x^{1-b}$ as close together as possible. | |
| Dec 21, 2015 at 18:25 | comment | added | user66081 | If the sums of logs are to be as close as possible, the ratio $x^b / x^{1-b}$ should be close to one. Why is this the right condition? | |
| Dec 21, 2015 at 17:39 | history | answered | Linus Hamilton | CC BY-SA 3.0 |