Timeline for Dependence of optimization problem on the linear constraints
Current License: CC BY-SA 3.0
8 events
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| Nov 21, 2016 at 19:31 | comment | added | CodeGolf | @Dirk Maybe you are right. Here I just wanna emphasize the perturbation of parameter $(\alpha,\beta)$ for an linear optimization problem under linear constraints, withous mentionning optimal transport. I've already modified my question. | |
| Nov 21, 2016 at 19:29 | history | edited | CodeGolf | CC BY-SA 3.0 |
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| Nov 21, 2016 at 19:28 | comment | added | CodeGolf | @Dirk $c$ is just a matrix. Here we consider the discrete case, so both $I$ and $c$ are fixed, we just make $(\alpha, \beta)$ vary, and wanna study how $P(\alpha, \beta)$ vary w.r.t $(\alpha, \beta)$. | |
| Nov 21, 2016 at 19:07 | comment | added | Dirk | So if you talk about Lipschitz functions in the dual, you probably have fixed $c_{i,j} = d(i,j)$, i.e. $c$ encodes a metric? And isn't $W_1$ actually the same as $P$ if the additional constraints are not there? How does the function $f$ looks like for other $c$ that $c_{i,j} = d(i,j)$? I have to admit that the question is both too vague and not properly worked out. | |
| Nov 21, 2016 at 19:03 | comment | added | CodeGolf | @Dirk Thanks for the reply. Indeed, the duality under the additional constraints is derived. But the optimizers for the dual problem can not be restricted to a class of Lipschitz functions, and that's why the duality can not yield directly the estimation. The matrix $c$ is given and fixed, and of course the function $f$ depend on $c$. | |
| Nov 21, 2016 at 18:18 | comment | added | Dirk | What do you get, when you work out the duality under the additional constraints? Also: Doesn't the estimate depend a lot on the $c_{i,j}$ and don't you need some conditions on these? | |
| Nov 21, 2016 at 14:56 | comment | added | CodeGolf | If there is no constraint $\sum_{j=1}^np_{i,j}x_j=\alpha_ix_i$, $\forall 1\le i\le n$, then it is well known that this estimation can be obtained via the duality of classical optimal transport. | |
| Nov 21, 2016 at 14:53 | history | asked | CodeGolf | CC BY-SA 3.0 |