Timeline for Optimization over space of probability measures
Current License: CC BY-SA 3.0
12 events
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| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
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| Oct 19, 2015 at 7:10 | history | edited | Arash | CC BY-SA 3.0 |
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| Oct 19, 2015 at 6:57 | comment | added | Arash | @fedja, The formulation is general for sure, hopeless I am not so sure; Optimization theory starts with a general formulation and then we have set of very beautiful solutions under different assumptions about the objective functions and constraints. I do not hope, naively, for a general solution but I am rather interested exactly in the answers like those of Tom and Nate. | |
| Oct 19, 2015 at 1:12 | comment | added | fedja | @Arash Yes and no. Most responses you've got were based on the implicit assumption that $f_j$ are fixed functions independent of $\mu$, and my understanding was the same. On the other hand, you are right that it has never been declared, so formally any functional of $\mu$ can be represented this way leading to a hopelessly general question. | |
| Oct 18, 2015 at 20:41 | comment | added | Arash | @fedja, I think I have clearly stated that the motivation behind the problem comes from entropy maximization; Does the current formulation of the problem exclude the possiblity of $f_0$ being something like $\log\frac{\mathrm d\mu}{\mathrm dm}$? I am not so sure. | |
| Oct 18, 2015 at 0:38 | comment | added | fedja | @Arash Differential entropy problem is a terrible example because it is not a problem about a functional of this type: it involves a non-linear expression in the density of the probability measure. | |
| Oct 16, 2015 at 11:30 | comment | added | Arash | @TomSolberg, The answer is insightful but how far we can push the LP analogy; can we use convex optimization analogy? I am not sure because, as stated in the OP, for some cases, the solution is a continuous measure and therefore no hope for a discrete measure to be the solution. | |
| Oct 16, 2015 at 11:24 | comment | added | Arash | @TomSolberg, Thanks a lot; Interestingly I knew but forgot about Miacheal's work on a very related problem to OP. So thanks! However, my question is a little bit more general; here $f_0$ can depend on the measure, hence perturbing the linear programming analogy; For instance take $f_0=\log\frac{\textrm d\mu}{\textrm dm}$ where $m$ is Lebesgue measure. | |
| Oct 15, 2015 at 22:42 | comment | added | Tom Solberg | Isn't this just a semi-infinite linear program? You have a finite constraint space $\mathbb{R}^k$, which means you should be able to formulate a dual program with $k$ variables. See mathoverflow.net/questions/210376/… | |
| Oct 15, 2015 at 21:39 | comment | added | Nate Eldredge | For starters, if $X$ is Polish and the functions $f_i$ are continuous, then existence of a minimizer should follow from the fact that $\mathscr{M}$ is weakly compact. | |
| Oct 15, 2015 at 21:19 | history | edited | Arash | CC BY-SA 3.0 |
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| Oct 15, 2015 at 21:05 | history | asked | Arash | CC BY-SA 3.0 |