Timeline for How to solve such an optimization problem
Current License: CC BY-SA 3.0
13 events
| when toggle format | what | by | license | comment | |
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| Aug 25, 2014 at 0:59 | comment | added | peng | @jjcale: Yes, I have also realized that the optimal $K$ different values are just Fekete points, but I didn't find the correct reference showing the proof of it. | |
| Aug 23, 2014 at 18:35 | comment | added | jjcale | @peng : If the optimal values of $x_i$ take only K different values, then these values are the Fekete points, because then all non zero terms in the sum are maximized. | |
| Aug 22, 2014 at 6:10 | comment | added | peng | I tried differential methods, but I didn't figure it out :( | |
| Aug 22, 2014 at 6:09 | comment | added | peng | Thanks for the update Moritz Firsching. I agree with you that the $K$ values taken by $x_i$ are independent of $N$. I numerical verified this and didn't find any counterexample. However, I don't know how to prove this conjecture. | |
| Aug 21, 2014 at 14:13 | comment | added | Moritz Firsching | @peng did you try using differential methods to prove that if the $x_i$ don't fulfill the condition that $f$ does not have a local maximum at that point? | |
| Aug 21, 2014 at 14:12 | comment | added | Moritz Firsching | @peng: I additionally conjecture that the $K$ values taken by $x_i$ are independent of $N$. However the above answer is only supposed to handle small cases.. | |
| Aug 21, 2014 at 14:09 | comment | added | Moritz Firsching | @peng you are right about $Q$ and the case $N=3$, $K=2$. I corrected it. | |
| Aug 21, 2014 at 14:08 | history | edited | Moritz Firsching | CC BY-SA 3.0 |
added suggestions from comments, additional optimal values for $K=3$
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| Aug 21, 2014 at 7:32 | comment | added | peng | Actually, I think this answer expresses the same conjecture as mine - that for general $N$ and $K$ , the optimal values of $x_i$ should take only $K$ different values, and the numbers of $x_i$ taking the same value should be approximately the same. My difficulty is indeed to prove this conjecture. Therefore, this answer doesnot EXACTLY answer my question. Can you provide me a rigious proof for this conjecture? Thanks a lot! | |
| Aug 21, 2014 at 7:29 | comment | added | peng | The second is about the table, in which the function value for $N = 3$ and $K = 2$ should be $2$, instead of $3$ | |
| Aug 21, 2014 at 7:28 | comment | added | peng | The first one is about the matrix $Q$ in the case of $K = 2$. I think the digaonal entries of $Q$ should be $N-1$, instead of $1$. | |
| Aug 21, 2014 at 7:16 | comment | added | peng | Thanks Moritz Firsching for providing the answer. I have the following concerns about this answer. | |
| Aug 20, 2014 at 15:30 | history | answered | Moritz Firsching | CC BY-SA 3.0 |