The tag has no wiki summary.

learn more… | top users | synonyms

1
vote
1answer
16 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
votes
1answer
40 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
vote
0answers
33 views

PDE-Based Triangle Inequality for Optimal Transportation

Suppose $\Omega$ is a suitably regular domain in $\mathbb{R}^n$ and $\rho_0,\rho_1\in\textrm{Prob}(\Omega)$. Benamou and Brenier showed that the $L_2$ transportation distance between $\rho_0$ and ...
1
vote
1answer
51 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 ...
0
votes
0answers
15 views

Transportation problem with intermediate depots

So I've got the following transportation problem (where I have to find the lowest costs while satisfying the demands) with depots $1,2,3$ and destinations $6,7$. There are also two intermediate depots ...
1
vote
0answers
290 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
votes
1answer
39 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
67 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
votes
0answers
94 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 ...