Timeline for Maximizing sum of homogeneous functions of order one over a polytope
Current License: CC BY-SA 3.0
11 events
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| Sep 26, 2017 at 19:38 | comment | added | Surb | I don't know, it probably depends on $g_i^s$ and $S$, etc. My intuition tells me that it should be possible to construct such a parametrized family where $g_i^s$ is linear for every $i,s$ and differentiable in $s$ such that the optimal solution is positive for every $s$. | |
| Sep 26, 2017 at 18:16 | comment | added | Ozzy | if $f_i(y)$ does not depend on $y(1)$, then by the assumption $f_i=0$. Sorry, it was not clear what I mean by generically. Consider a parameterized family of functions $\{f_i=g_i^s\}$, where $s\in S$ and where $S \subset \mathbb{R}^k$. Let $\mu$ denote the Lebesgue measure, and $Z\subset S$ be the set of parameters for which at optimal solution we have $x_i,x_j>0$ for $i\neq j$. Is it the case that $\mu(Z)=0$? | |
| Sep 26, 2017 at 17:53 | comment | added | Surb | I'm not sure what you exactly mean by generically. Regarding the additional assumption, it seems still possible to build linear cases where for every $i$, $f_i(y)$ does not depend on $y(1)$. | |
| Sep 26, 2017 at 17:07 | history | edited | Ozzy | CC BY-SA 3.0 |
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| Sep 26, 2017 at 17:05 | comment | added | Ozzy | in the linear case, it is true there may exist a solution where all $x_i$ are positive. But this is not true generically. Your second point is correct. Which made me realize that some of the additional structure I have in the problem is necessary for the question to be interesting. I'm updating the question accordingly. | |
| Sep 26, 2017 at 16:55 | comment | added | Surb | What about the case $f_i(z)= z^i$ (the $i$-th coordinate of $z$) for all $i$. Then the optimal solution is $(b^1e_1,\ldots,b^ne_n)$ where $b^i$ are the coordinates of $b$ and $e_1,\ldots,e_n$ the canonical basis. | |
| Sep 26, 2017 at 16:42 | history | edited | Ozzy | CC BY-SA 3.0 |
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| Sep 26, 2017 at 16:41 | comment | added | Ozzy | Correct -- question updated accordingly. | |
| Sep 26, 2017 at 16:24 | history | edited | Ozzy | CC BY-SA 3.0 |
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| Sep 26, 2017 at 16:23 | comment | added | Ozzy | $x_i$ are vectors, see the domain of $f_i$. By increasing, I mean increasing in each coordinate. Question updated to clarify these points. | |
| Sep 26, 2017 at 7:31 | history | asked | Ozzy | CC BY-SA 3.0 |