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11 views

$\mathbb{P}(d(X,Y)>\alpha)<\beta$ if $\mathbb{P}(X\in E)\leq \mathbb{P}(Y\in E^{\alpha})+\beta$ for all measurable E

Given two random variables X,Y with measures P,Q (i.e. $P(A)=\mathbb{P}(X\in A)$). Show that if $P(E) \le Q(E^\alpha) + \beta$ for all measurable $E\subset\mathbb{R}$ then ...
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1answer
15 views

Optimisation/ operational research problem classifciation

Hello I am new to operational research and would like help classify the following transport problem. I have a model which simulates a taxi like service, it a has a range of inputs that can be changed ...
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1answer
40 views

Regularity of solutions to a transport equation

Currently I am working on a transport equation and have been able to prove the existence and uniqueness of a weak measurable solution to said equation. I am now working in trying to jot down (with ...
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2answers
23 views

Bivariate optimal density

Consider any feasible $p:[0,1]^2\to [0,1]$ that allows discontinuities and the problem $$\min_{p(.)} \int_0^1\int_0^1 p(x,y)^2 dF(x) dG(y)$$ s.t. $$\int_0^1 p(x,y)dG(y)=k\phantom{0} for \phantom{0} ...
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0answers
29 views

Admissible transference plans, characterizing a condition on measures.

In Villani's, Topics in optimal transportation, pg.18 we find that for a borel measure $\pi\in X\times Y$, where $(X,\mu)$ and $(Y,\nu)$ are probability measure spaces, the condition \begin{equation} ...
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1answer
28 views

The dual of transporting problem

So basically I'm trying to figure out what does a certain variable in dual of transporting problem mean. Transporting problem in matrix form: (We are searching for a min cost of transferring goods ...
1
vote
1answer
27 views

Continuity of subdifferential mapping

I'm reading Cedric Villani's book topics in optimal transportation, and I have a problem on page 53: If $\varphi$ lower semi-continuous, then the subdifferential mapping $\partial\varphi$ is always ...
1
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1answer
35 views

Definition of “deterministic coupling” [Villani]

I'm currently reading through "Optimal transport, old and new" by Cédric Villani. In the first chapter, he defines a coupling of two probability spaces $(\mathcal{X},\mu)$ and $(\mathcal{Y},\nu)$ as a ...
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1answer
25 views

Reference request: Time-optimal trajectories

I am looking for some lecture notes or a textbook for time-optimal trajectories. Any help is greatly appreciated. I am having plenty of trouble with understanding switching $C^+$ and $C^-$ curves.
0
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1answer
77 views

Wasserstein metric: conditions for the existence of minimizer and duality

Let $(X,d)$ be a metric space and let $\mathcal P(X)$ be the set of all Borel probability measures on $(X,d)$. The Wasserstein distance on $\mathcal P(X)$ is given by $$ W_d(\mu,\bar\mu):=\inf_{M\in ...
1
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1answer
109 views

Wasserstein distance from a Dirac measure

http://en.wikipedia.org/wiki/Wasserstein_metric I would like to prove that $$W^1(μ,δx_0)=∫d(x_0,y) μ(dy)$$ let $$γ∈Γ(μ,δx_0)$$ Can we say that it is the product of its marginal distributions ...
1
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0answers
412 views

Transportation problem: optimal solution

So I have an issue with finding the optimal solution (the lowest costs) to a transportation problem. Given the following problem, with $A$ the depots, $B$ the destinations and $C$ the $(i,j)$ matrix ...
4
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1answer
43 views

Why is the shift the optimal plan between $[0,1)$ and $[1,2)$ (with distance-squared cost function)?

Example 1.3 of Optimal and Better Transport Plans reads Consider the task to transport points on the real line (equipped with the Lebesgue measure) from the interval [0, 1) to [1, 2) where the ...
4
votes
0answers
73 views

Spacing nodes by moving the shortest distance possible.

I have a list of N nodes with positions $(x, y)$ each. I want to move each node the shortest possible distance such that every node is placed on the radius $R$ from at least one other node, and is at ...
4
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0answers
120 views

Cyclically monotone sets on four points

A subset of $\mathcal{X} \times \mathcal{Y} \subset \mathbb{R}^d \times \mathbb{R}^d$ is cyclically monotone if $\sum_{i=1}^n \langle x_i,y_i\rangle \ge \sum_{i=1}^n \langle x_i,y_{i+1}\rangle$, where ...