American Institute of Mathematical Sciences

Advanced
April  2014, 34(4): 1465-1480. doi: 10.3934/dcds.2014.34.1465

Remarks on multi-marginal symmetric Monge-Kantorovich problems

Nassif Ghoussoub 1, and  Bernard Maurey 2,

1. 

Department of Mathematics, The University of British Columbia, Vancouver BC Canada V6T 1Z2
2. 

Institut de Mathématiques, UMR 7586 - CNRS, Université Paris Diderot - Paris 7, Paris, France

Received  November 2012 Revised  February 2013 Published  October 2013

Symmetric Monge-Kantorovich transport problems involving a cost function given by a family of vector fields were used by Ghoussoub-Moameni to establish polar decompositions of such vector fields into $m$-cyclically monotone maps composed with measure preserving $m$-involutions ($m\geq 2$). In this note, we relate these symmetric transport problems to the Brenier solutions of the Monge and Monge-Kantorovich problem, as well as to the Gangbo-Święch solutions of their multi-marginal counterparts, both of which involving quadratic cost functions.
Keywords: $m$-cyclically antisymmetric functions., Monge-Kantorovich duality, $m$-cyclically monotone vector fields, Mass transport.
Mathematics Subject Classification: Primary: 49J30, 49K30, 49J52; Secondary: 58C35, 58E17, 91B6.
Citation: Nassif Ghoussoub, Bernard Maurey. Remarks on multi-marginal symmetric Monge-Kantorovich problems. Discrete & Continuous Dynamical Systems - A, 2014, 34 (4) : 1465-1480. doi: 10.3934/dcds.2014.34.1465
References:
[1]

M. Agueh and G. Carlier, Barycenters in the Wasserstein space,, SIAM J. Math. Anal., 43 (2011), 904.  doi: 10.1137/100805741.  Google Scholar

[2]

R. S. Burachik and B. F. Svaiter,, Maximal monotonicity, conjugation and the duality product,, Proc. Amer. Math. Soc., 131 (2003), 2379.  doi: 10.1090/S0002-9939-03-07053-9.  Google Scholar

[3]

Y. Brenier, Polar factorization and monotone rearrangement of vector-valued functions,, Comm. Pure Appl. Math., 44 (1991), 375.  doi: 10.1002/cpa.3160440402.  Google Scholar

[4]

G. Carlier and B. Nazaret, Optimal transportation for the determinant,, ESAIM Control Optim. Calc. Var., 14 (2008), 678.  doi: 10.1051/cocv:2008006.  Google Scholar

[5]

S. P. Fitzpatrick, Representing monotone operators by convex functions,, in, 20 (1988), 59.   Google Scholar

[6]

A. Galichon and N. Ghoussoub, Variational representations for N-cyclically monotone vector fields,, , (2012).   Google Scholar

[7]

W. Gangbo and A. Święch, Optimal maps for the multidimensional Monge-Kantorovich problem,, Comm. Pure Appl. Math., 51 (1998), 23.  doi: 10.1002/(SICI)1097-0312(199801)51:1<23::AID-CPA2>3.0.CO;2-H.  Google Scholar

[8]

N. Ghoussoub, Anti-self-dual Lagrangians: Variational resolutions of non self-adjoint equations and dissipative evolutions,, AIHP-Analyse Non Linéaire, 24 (2007), 171.  doi: 10.1016/j.anihpc.2006.02.002.  Google Scholar

[9]

N. Ghoussoub, A variational theory for monotone vector fields,, Journal of Fixed Point Theory and Applications, 4 (2008), 107.  doi: 10.1007/s11784-008-0083-4.  Google Scholar

[10]

N. Ghoussoub, Anti-symmetric Hamiltonians: Variational resolution of Navier-Stokes equations and other nonlinear evolutions,, Comm. Pure & Applied Math., 60 (2007), 619.  doi: 10.1002/cpa.20176.  Google Scholar

[11]

N. Ghoussoub, "Selfdual Partial Differential Systems and their Variational Principles,", Springer Monograph in Mathematics, (2008).   Google Scholar

[12]

N. Ghoussoub and A. Moameni, A self-dual polar factorization for vector fields,, Comm. Pure. Applied. Math., 66 (2013), 905.  doi: 10.1002/cpa.21430.  Google Scholar

[13]

N. Ghoussoub and A. Moameni, Symmetric Monge-Kantorovich problems and polar decompositions of vector fields,, preprint, (2013).   Google Scholar

[14]

P. Millien, "A Functional Analytic Approach to the Selfdual Polar Decomposition,", Masters Thesis, (2011).   Google Scholar

[15]

E. Krauss, A representation of arbitrary maximal monotone operators via subgradients of skew-symmetric saddle functions,, Nonlinear Anal., 9 (1985), 1381.  doi: 10.1016/0362-546X(85)90097-5.  Google Scholar

[16]

B. Pass, Optimal transportation with infinitely many marginals,, J. Funct. Anal., 264 (2013), 947.  doi: 10.1016/j.jfa.2012.12.002.  Google Scholar

[17]

B. Pass, Ph.D thesis,, University of Toronto, (2011).   Google Scholar

[18]

B. Pass, Uniqueness and Monge solutions in the multimarginal optimal transportation problem,, SIAM Journal on Mathematical Analysis, 43 (2011), 2758.  doi: 10.1137/100804917.  Google Scholar

[19]

B. Pass, On the local structure of optimal measures in the multi-marginal optimal transportation problem,, Calculus of Variations and Partial Differential Equations, 43 (2012), 529.  doi: 10.1007/s00526-011-0421-z.  Google Scholar

[20]

R. R. Phelps, "Convex Functions, Monotone Operators and Differentiability,", Second edition, 1364 (1993).   Google Scholar

[21]

T. Rockafellar, "Convex Analysis,", Princeton Mathematical Series, (1970).   Google Scholar

[22]

B. F. Svaiter, Fixed points in the family of convex representations of a maximal monotone operator,, Proc. Amer. Math. Soc., 131 (2003), 3851.  doi: 10.1090/S0002-9939-03-07083-7.  Google Scholar

[23]

C. Villani, "Topics in Optimal Transportation,", Graduate Studies in Mathematics, 58 (2003).  doi: 10.1007/b12016.  Google Scholar

[24]

C. Villani, "Optimal Transport. Old and New,", Grundlehren der Mathematischen Wissenschaften, 338 (2009).  doi: 10.1007/978-3-540-71050-9.  Google Scholar

show all references

References:
[1]

M. Agueh and G. Carlier, Barycenters in the Wasserstein space,, SIAM J. Math. Anal., 43 (2011), 904.  doi: 10.1137/100805741.  Google Scholar

[2]

R. S. Burachik and B. F. Svaiter,, Maximal monotonicity, conjugation and the duality product,, Proc. Amer. Math. Soc., 131 (2003), 2379.  doi: 10.1090/S0002-9939-03-07053-9.  Google Scholar

[3]

Y. Brenier, Polar factorization and monotone rearrangement of vector-valued functions,, Comm. Pure Appl. Math., 44 (1991), 375.  doi: 10.1002/cpa.3160440402.  Google Scholar

[4]

G. Carlier and B. Nazaret, Optimal transportation for the determinant,, ESAIM Control Optim. Calc. Var., 14 (2008), 678.  doi: 10.1051/cocv:2008006.  Google Scholar

[5]

S. P. Fitzpatrick, Representing monotone operators by convex functions,, in, 20 (1988), 59.   Google Scholar

[6]

A. Galichon and N. Ghoussoub, Variational representations for N-cyclically monotone vector fields,, , (2012).   Google Scholar

[7]

W. Gangbo and A. Święch, Optimal maps for the multidimensional Monge-Kantorovich problem,, Comm. Pure Appl. Math., 51 (1998), 23.  doi: 10.1002/(SICI)1097-0312(199801)51:1<23::AID-CPA2>3.0.CO;2-H.  Google Scholar

[8]

N. Ghoussoub, Anti-self-dual Lagrangians: Variational resolutions of non self-adjoint equations and dissipative evolutions,, AIHP-Analyse Non Linéaire, 24 (2007), 171.  doi: 10.1016/j.anihpc.2006.02.002.  Google Scholar

[9]

N. Ghoussoub, A variational theory for monotone vector fields,, Journal of Fixed Point Theory and Applications, 4 (2008), 107.  doi: 10.1007/s11784-008-0083-4.  Google Scholar

[10]

N. Ghoussoub, Anti-symmetric Hamiltonians: Variational resolution of Navier-Stokes equations and other nonlinear evolutions,, Comm. Pure & Applied Math., 60 (2007), 619.  doi: 10.1002/cpa.20176.  Google Scholar

[11]

N. Ghoussoub, "Selfdual Partial Differential Systems and their Variational Principles,", Springer Monograph in Mathematics, (2008).   Google Scholar

[12]

N. Ghoussoub and A. Moameni, A self-dual polar factorization for vector fields,, Comm. Pure. Applied. Math., 66 (2013), 905.  doi: 10.1002/cpa.21430.  Google Scholar

[13]

N. Ghoussoub and A. Moameni, Symmetric Monge-Kantorovich problems and polar decompositions of vector fields,, preprint, (2013).   Google Scholar

[14]

P. Millien, "A Functional Analytic Approach to the Selfdual Polar Decomposition,", Masters Thesis, (2011).   Google Scholar

[15]

E. Krauss, A representation of arbitrary maximal monotone operators via subgradients of skew-symmetric saddle functions,, Nonlinear Anal., 9 (1985), 1381.  doi: 10.1016/0362-546X(85)90097-5.  Google Scholar

[16]

B. Pass, Optimal transportation with infinitely many marginals,, J. Funct. Anal., 264 (2013), 947.  doi: 10.1016/j.jfa.2012.12.002.  Google Scholar

[17]

B. Pass, Ph.D thesis,, University of Toronto, (2011).   Google Scholar

[18]

B. Pass, Uniqueness and Monge solutions in the multimarginal optimal transportation problem,, SIAM Journal on Mathematical Analysis, 43 (2011), 2758.  doi: 10.1137/100804917.  Google Scholar

[19]

B. Pass, On the local structure of optimal measures in the multi-marginal optimal transportation problem,, Calculus of Variations and Partial Differential Equations, 43 (2012), 529.  doi: 10.1007/s00526-011-0421-z.  Google Scholar

[20]

R. R. Phelps, "Convex Functions, Monotone Operators and Differentiability,", Second edition, 1364 (1993).   Google Scholar

[21]

T. Rockafellar, "Convex Analysis,", Princeton Mathematical Series, (1970).   Google Scholar

[22]

B. F. Svaiter, Fixed points in the family of convex representations of a maximal monotone operator,, Proc. Amer. Math. Soc., 131 (2003), 3851.  doi: 10.1090/S0002-9939-03-07083-7.  Google Scholar

[23]

C. Villani, "Topics in Optimal Transportation,", Graduate Studies in Mathematics, 58 (2003).  doi: 10.1007/b12016.  Google Scholar

[24]

C. Villani, "Optimal Transport. Old and New,", Grundlehren der Mathematischen Wissenschaften, 338 (2009).  doi: 10.1007/978-3-540-71050-9.  Google Scholar

[1]

Jesus Garcia Azorero, Juan J. Manfredi, I. Peral, Julio D. Rossi. Limits for Monge-Kantorovich mass transport problems. Communications on Pure & Applied Analysis, 2008, 7 (4) : 853-865. doi: 10.3934/cpaa.2008.7.853
[2]

Abbas Moameni. Invariance properties of the Monge-Kantorovich mass transport problem. Discrete & Continuous Dynamical Systems - A, 2016, 36 (5) : 2653-2671. doi: 10.3934/dcds.2016.36.2653
[3]

Giuseppe Buttazzo, Eugene Stepanov. Transport density in Monge-Kantorovich problems with Dirichlet conditions. Discrete & Continuous Dynamical Systems - A, 2005, 12 (4) : 607-628. doi: 10.3934/dcds.2005.12.607
[4]

Zuo Quan Xu, Jia-An Yan. A note on the Monge-Kantorovich problem in the plane. Communications on Pure & Applied Analysis, 2015, 14 (2) : 517-525. doi: 10.3934/cpaa.2015.14.517
[5]

Yupeng Li, Wuchen Li, Guo Cao. Image segmentation via $ L_1 $ Monge-Kantorovich problem. Inverse Problems & Imaging, 2019, 13 (4) : 805-826. doi: 10.3934/ipi.2019037
[6]

Tuan Phung-Duc, Hiroyuki Masuyama, Shoji Kasahara, Yutaka Takahashi. M/M/3/3 and M/M/4/4 retrial queues. Journal of Industrial & Management Optimization, 2009, 5 (3) : 431-451. doi: 10.3934/jimo.2009.5.431
[7]

Julio C. Rebelo, Ana L. Silva. On the Burnside problem in Diff(M). Discrete & Continuous Dynamical Systems - A, 2007, 17 (2) : 423-439. doi: 10.3934/dcds.2007.17.423
[8]

David Kinderlehrer, Adrian Tudorascu. Transport via mass transportation. Discrete & Continuous Dynamical Systems - B, 2006, 6 (2) : 311-338. doi: 10.3934/dcdsb.2006.6.311
[9]

Dequan Yue, Wuyi Yue, Gang Xu. Analysis of customers' impatience in an M/M/1 queue with working vacations. Journal of Industrial & Management Optimization, 2012, 8 (4) : 895-908. doi: 10.3934/jimo.2012.8.895
[10]

Zsolt Saffer, Wuyi Yue. M/M/c multiple synchronous vacation model with gated discipline. Journal of Industrial & Management Optimization, 2012, 8 (4) : 939-968. doi: 10.3934/jimo.2012.8.939
[11]

Chia-Huang Wu, Kuo-Hsiung Wang, Jau-Chuan Ke, Jyh-Bin Ke. A heuristic algorithm for the optimization of M/M/$s$ queue with multiple working vacations. Journal of Industrial & Management Optimization, 2012, 8 (1) : 1-17. doi: 10.3934/jimo.2012.8.1
[12]

Jun He, Guangjun Xu, Yanmin Liu. Some inequalities for the minimum M-eigenvalue of elasticity M-tensors. Journal of Industrial & Management Optimization, 2020, 16 (6) : 3035-3045. doi: 10.3934/jimo.2019092
[13]

Philipp Reiter. Regularity theory for the Möbius energy. Communications on Pure & Applied Analysis, 2010, 9 (5) : 1463-1471. doi: 10.3934/cpaa.2010.9.1463
[14]

Konovenko Nadiia, Lychagin Valentin. Möbius invariants in image recognition. Journal of Geometric Mechanics, 2017, 9 (2) : 191-206. doi: 10.3934/jgm.2017008
[15]

Hideaki Takagi. Unified and refined analysis of the response time and waiting time in the M/M/m FCFS preemptive-resume priority queue. Journal of Industrial & Management Optimization, 2017, 13 (4) : 1945-1973. doi: 10.3934/jimo.2017026
[16]

Leonardo Câmara, Bruno Scárdua. On the integrability of holomorphic vector fields. Discrete & Continuous Dynamical Systems - A, 2009, 25 (2) : 481-493. doi: 10.3934/dcds.2009.25.481
[17]

Jifeng Chu, Zhaosheng Feng, Ming Li. Periodic shadowing of vector fields. Discrete & Continuous Dynamical Systems - A, 2016, 36 (7) : 3623-3638. doi: 10.3934/dcds.2016.36.3623
[18]

Yubo Yuan. Canonical duality solution for alternating support vector machine. Journal of Industrial & Management Optimization, 2012, 8 (3) : 611-621. doi: 10.3934/jimo.2012.8.611
[19]

Xinmin Yang. On symmetric and self duality in vector optimization problem. Journal of Industrial & Management Optimization, 2011, 7 (3) : 523-529. doi: 10.3934/jimo.2011.7.523
[20]

Hideaki Takagi. Times until service completion and abandonment in an M/M/$ m$ preemptive-resume LCFS queue with impatient customers. Journal of Industrial & Management Optimization, 2018, 14 (4) : 1701-1726. doi: 10.3934/jimo.2018028

2019 Impact Factor: 1.338

Tools

Metrics

  • PDF downloads (48)
  • HTML views (0)
  • Cited by (6)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences