A saddle-point approach to the Monge-Kantorovich optimal transport problem

Christian Léonard

ESAIM: Control, Optimisation and Calculus of Variations (2011)

  • Volume: 17, Issue: 3, page 682-704
  • ISSN: 1292-8119

Abstract

top
The Monge-Kantorovich problem is revisited by means of a variant of the saddle-point method without appealing to c-conjugates. A new abstract characterization of the optimal plans is obtained in the case where the cost function takes infinite values. It leads us to new explicit sufficient and necessary optimality conditions. As by-products, we obtain a new proof of the well-known Kantorovich dual equality and an improvement of the convergence of the minimizing sequences.

How to cite

top

Léonard, Christian. "A saddle-point approach to the Monge-Kantorovich optimal transport problem." ESAIM: Control, Optimisation and Calculus of Variations 17.3 (2011): 682-704. <http://eudml.org/doc/272907>.

@article{Léonard2011,
abstract = {The Monge-Kantorovich problem is revisited by means of a variant of the saddle-point method without appealing to c-conjugates. A new abstract characterization of the optimal plans is obtained in the case where the cost function takes infinite values. It leads us to new explicit sufficient and necessary optimality conditions. As by-products, we obtain a new proof of the well-known Kantorovich dual equality and an improvement of the convergence of the minimizing sequences.},
author = {Léonard, Christian},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {convex optimization; saddle-point; conjugate duality; optimal transport},
language = {eng},
number = {3},
pages = {682-704},
publisher = {EDP-Sciences},
title = {A saddle-point approach to the Monge-Kantorovich optimal transport problem},
url = {http://eudml.org/doc/272907},
volume = {17},
year = {2011},
}

TY - JOUR
AU - Léonard, Christian
TI - A saddle-point approach to the Monge-Kantorovich optimal transport problem
JO - ESAIM: Control, Optimisation and Calculus of Variations
PY - 2011
PB - EDP-Sciences
VL - 17
IS - 3
SP - 682
EP - 704
AB - The Monge-Kantorovich problem is revisited by means of a variant of the saddle-point method without appealing to c-conjugates. A new abstract characterization of the optimal plans is obtained in the case where the cost function takes infinite values. It leads us to new explicit sufficient and necessary optimality conditions. As by-products, we obtain a new proof of the well-known Kantorovich dual equality and an improvement of the convergence of the minimizing sequences.
LA - eng
KW - convex optimization; saddle-point; conjugate duality; optimal transport
UR - http://eudml.org/doc/272907
ER -

References

top
  1. [1] L. Ambrosio and A. Pratelli, Existence and stability results in the L1-theory of optimal transportation – CIME Course, in Lecture Notes in Mathematics 1813. Springer Verlag (2003) 123–160. Zbl1065.49026MR2006307
  2. [2] M. Beiglböck and W. Schachermayer, Duality for Borel measurable cost functions. Trans. Amer. Math. Soc. (to appear). Zbl1228.49046MR2792985
  3. [3] M. Beiglböck, M. Goldstern, G. Maresh and W. Schachermayer, Optimal and better transport plans. J. Funct. Anal.256 (2009) 1907–1927. Zbl1157.49019MR2498564
  4. [4] M. Beiglböck, C. Léonard and W. Schachermayer, A general duality theorem for the Monge-Kantorovich transport problem. Preprint (2009). Zbl1270.49045
  5. [5] J.M. Borwein and A.S. Lewis, Decomposition of multivariate functions. Can. J. Math.44 (1992) 463–482. Zbl0789.54012MR1176365
  6. [6] H. Brezis, Analyse fonctionnelle – Théorie et applications. Masson, Paris (1987). Zbl1147.46300MR697382
  7. [7] G. Dal Maso, An Introduction to Γ-Convergence. Progress in Nonlinear Differential Equations and Their Applications 8. Birkhäuser (1993). Zbl0816.49001MR1201152
  8. [8] L. Decreusefond, Wasserstein distance on configuration space. Potential Anal.28 (2008) 283–300. Zbl1144.60004MR2386101
  9. [9] L. Decreusefond, A. Joulin and N. Savy, Upper bounds on Rubinstein distances on configuration spaces and applications. Communications on Stochastic Analysis (to appear). Zbl1331.60083MR2677197
  10. [10] I. Ekeland and R. Témam, Convex Analysis and Variational Problems, Classics in Applied Mathematics 28. SIAM (1999). Zbl0939.49002MR1727362
  11. [11] D. Feyel and A.S. Üstünel, Monge-Kantorovitch measure transportation and Monge-Ampère equation on Wiener space. Probab. Theory Relat. Fields128 (2004) 347–385. Zbl1055.60052MR2036490
  12. [12] C. Léonard, Convex minimization problems with weak constraint qualifications. Journal of Convex Analysis17 (2010) 312–348. Zbl1193.49042MR2642734
  13. [13] J. Neveu, Bases mathématiques du calcul des probabilités. Masson, Paris (1970). Zbl0137.11203
  14. [14] A. Pratelli, On the sufficiency of the c-cyclical monotonicity for optimality of transport plans. Math. Z.258 (2008) 677–690. Zbl1293.49110MR2369050
  15. [15] S. Rachev and L. Rüschendorf, Mass Transportation Problems. Vol. I: Theory, Vol. II: Applications. Springer-Verlag, New York (1998). Zbl0990.60500MR1619170
  16. [16] L. Rüschendorf, On c-optimal random variables. Statist. Probab. Lett.27 (1996) 267–270. Zbl0847.62046MR1395577
  17. [17] W. Schachermayer and J. Teichman, Characterization of optimal transport plans for the Monge-Kantorovich problem. Proc. Amer. Math. Soc.137 (2009) 519–529. Zbl1165.49015MR2448572
  18. [18] C. Villani, Topics in Optimal Transportation, Graduate Studies in Mathematics 58. American Mathematical Society, Providence (2003). Zbl1106.90001MR1964483
  19. [19] C. Villani, Optimal Transport – Old and New, Grundlehren der mathematischen Wissenschaften 338. Springer (2009). Zbl1156.53003MR2459454

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.