Talk:Kantorovich Problem: Difference between revisions

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==Beginning==
* Instead of ``The Kantorovich's ot problem``, just ``Kantorovich's ot problem``
* ``Leonid Kantorovich, the founder of modern optimization, who was awarded a Nobel...'
* Change the phrase ``Leonid Kantorovich`` to a link to his wikipedia page
* Add a link to Nobel prize: [[https://www.nobelprize.org/prizes/economic-sciences/1975/press-release/]]
==Introduction==
==Introduction==
* ``beside THE Monge Problem``
* Great picture!
* change the phrase ``Monge Problem`` to a link to the OT wiki page.
* Instead of ``The Kantorovich problem allows a non-empty minimization set, a convex constraint set,`` consider ``in the Kantorovich problem, the constraint set is convex and nonempty``.
* While everything you say in the introduction is correct, it seems a little early to get into this level of technicality. Why not you move this section to the bottom and rename it ``Comparison to Monge Problem``?
* Consider referencing the more specific page for ``dual``: https://en.wikipedia.org/wiki/Dual_linear_program
* Instead, your introduction can give an heuristic interpretation of the problem, in terms of ``moving around piles of dirt``, which would be understandable to a non-mathematician. You can end the introduction with  statement of Kantorovich's OT problem, rather than having this in its own separate section.
* More specifically ``it is a linear minimization problem with LINEAR constraints``
* What are <math> X, Y, \mathcal{P}(X), \mathcal{P}(Y)</math>
* Consider removing this discussion of the shipping problem and referring instead to the OT wiki article on the Kantorovich Dual Problem
* Define  <math> \Pi(\mu,\nu) </math>
* Define <math> c(x,y) </math>
 


==Shipping problem==
*It is officially called ``The shipper's problem``
* Cite Villani's Section 1.1.3 and attribute the interpretation to Caffarelli
* This should appear beneath the statement of Kantorovich's Dual Problem. It should be explained how this offers a heuristic interpretation of the dual problem.


==Kantorovich Duality Statement==
==Kantorovich Optimal Transport Problem==
* Change name of section to just ``Kantorovich Duality''
* The problem isn't stated correctly -- it is a minimization problem, right?
* It is not needed to state Kantorovich's problem again
* Remove the last phrase ``or equivalently Linfty dmu Linfty d nu`` since this is already implied by saying it holds for \phi and \psi in L1
*
* The Remarks section is a direct paraphrase of Villani. Unfortunately, it doesn't make sense as written and could be considered copyright infringement. Please remove it.


==References==
==Kantorovich Duality==
* There is an error in the references
* Please remove this section and instead refer to the OT wiki article on the Kantorovich Dual

Revision as of 17:08, 27 May 2020

Introduction

  • Great picture!
  • Instead of ``The Kantorovich problem allows a non-empty minimization set, a convex constraint set,`` consider ``in the Kantorovich problem, the constraint set is convex and nonempty``.
  • Consider referencing the more specific page for ``dual``: https://en.wikipedia.org/wiki/Dual_linear_program
  • More specifically ``it is a linear minimization problem with LINEAR constraints``
  • Consider removing this discussion of the shipping problem and referring instead to the OT wiki article on the Kantorovich Dual Problem


Kantorovich Optimal Transport Problem

  • The problem isn't stated correctly -- it is a minimization problem, right?
  • Remove the last phrase ``or equivalently Linfty dmu Linfty d nu`` since this is already implied by saying it holds for \phi and \psi in L1
  • The Remarks section is a direct paraphrase of Villani. Unfortunately, it doesn't make sense as written and could be considered copyright infringement. Please remove it.

Kantorovich Duality

  • Please remove this section and instead refer to the OT wiki article on the Kantorovich Dual
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