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19 views

Modified Transportation Problem

Could you point me to some article that tackles a problem similar to this. There are two sets: Sources $A = a_1,a_2,\ldots,a_n$ Destinations $B = b_1,b_2,\ldots,b_n$ The distance between source ...
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1answer
20 views

Optimal Transport PDE formulation

Let $M^{\pm}$ be Polish spaces and $\mu^{\pm}$ be Borel probability measures on $M^+$ and $M^-$ respectively. If $G: M^+ \rightarrow M^-$ with the constraint $G_\#\mu^+ = \mu^-$, why does the ...
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46 views

Need to prove that a transportation problem is never degenerate

This is the question: where ai is the supply from sourcei and bj is the demand at destinationj. I can kind of see why this would happen: say, when we are comparing a1 to b1 + 1/n, regardless of ...
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32 views

Are the set of probability functions with compact support in a fixed closed ball complete under the Wasserstein norm?

Let $B_R$ be a closed ball of radius $R$ in the space $\mathbb{R}^d$. As the title suggests I have this feeling that the set of functions $$S:= \left\lbrace f:\mathbb{R}^d \to \mathbb{R} ...
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87 views

Explanation of formula

Suppose that we have $M$ production stations $A_1, \dots, A_M$ of a product and $N$ destination stations $B_1, \dots, B_N$ of the product. We suppose that $x_{ij}$ units of the product are ...
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30 views

How to solve mass transportation problem with additional constraints?

Let $X_i$ be random variables follwoing distributions $f_i$. Then one would like to solve the following problem: max $\{\mathbb{P}(\Sigma X_i > T) \}$ subject to $X_i \sim f_i$ The solution ...
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14 views

Equivalence of p-Wasserstein metrics in Wiener chaos

Let $\mu_1$ and $\mu_2$ be two probability measures on $\mathbb{R}^d$. Suppose that each measure takes its values in the $2$nd inhomogeneous Wiener chaos. As usual, define the $p$th-Wasserstein ...
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16 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
21 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
88 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
29 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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36 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
30 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 ...
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1answer
44 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 ...
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1answer
59 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
34 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.
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1answer
112 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 ...
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1answer
169 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 ...
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523 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 ...
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1answer
46 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 ...
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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
136 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 ...