Hostname: page-component-7dd5485656-kp629 Total loading time: 0 Render date: 2025-10-30T12:12:56.909Z Has data issue: false hasContentIssue false
  • English
  • Français

On c-cyclical monotonicity for optimal transport problem with Coulomb cost

Part of: Existence theories Miscellaneous topics in calculus of variations and optimal control Applied Optimal Transport Optimality conditions

Published online by Cambridge University Press:  23 May 2019

LUIGI DE PASCALE Open the ORCID record for LUIGI DE PASCALE [Opens in a new window]
LUIGI DE PASCALE*
Affiliation:
Universita degli Studi di Firenze, Dipartimento di Matematica ed Informatica, Viale Morgagni, 67/A, Firenze, 50134, Italy email: luigi.depascale@unifi.it
Get access Rights & Permissions [Opens in a new window]

Abstract

It is proved that c-cyclical monotonicity is a sufficient condition for optimality in the multi-marginal optimal transport problem with Coulomb repulsive cost. The notion of c-splitting set and some mild regularity property are the tools. The result may be extended to Coulomb like costs.

Keywords

Multi-marginal optimal transportationMonge–Kantorovich problemoptimality conditionc-cyclical monotonicity

MSC classification

Primary: 49J45: Methods involving semicontinuity and convergence; relaxation
Secondary: 49N15: Duality theory 49K30: Optimal solutions belonging to restricted classes

Information

Type
Papers
Information
European Journal of Applied Mathematics , Volume 30 , Special Issue 6: Applied Optimal Transport , December 2019 , pp. 1210 - 1219
Copyright
© Cambridge University Press 2019 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Annali di Matematica (2015) 194: 307. https://doi.org/10.1007/s10231-013-0376-0CrossRefGoogle Scholar
Beiglböck, M., Léonard, C. & Schachermayer, W. (2012) A general duality theorem for the Monge–Kantorovich transport problem. Stud. Math. 209, 2.CrossRefGoogle Scholar
Bianchini, S. & Caravenna, L. (2010) On optimality of c-cyclically monotone transference plans. C. R. Math. Acad. Sci. Paris 348(11–12), 613–618.CrossRefGoogle Scholar
Bindini, U. (to appear) Marginals with finite repulsive cost. Can. J. Math. DOI: https://dx.doi.org/10.4153/S0008414X1800066410.4153/; arXiv:1702.06301.Google Scholar
Buttazzo, G., De Pascale, L. & Gori-Giorgi, P. (2012) Optimal-transport formulation of electronic density-functional theory. Phys. Rev. A 85(6), 062502.CrossRefGoogle Scholar
Buttazzo, G., Champion, T. & De Pascale, L. (2018) Continuity and estimates for multimarginal optimal transportation problems with singular costs. Appl. Math. Optim. 78(1), 85–200.CrossRefGoogle Scholar
Carlier, G. (2003) On a class of multidimensional optimal transportation problems. J. Convex Anal. 10(2), 517–530.Google Scholar
Carlier, G. & Ekeland, I. (2010) Matching for teams. Econom. Theory 42(2), 397–418.CrossRefGoogle Scholar
Carlier, G. & Nazaret, B. (2008) Optimal transportation for the determinant. ESAIM: Control Optim. Calculus Variations 14(04), 678–698.Google Scholar
Chiapporri, P.-A., McCann, R. & Nesheim, L. (2010) Hedonic price equilibria, stable matching and optimal transport; equivalence, topology and uniqueness. Econom. Theory, 42(2), 317–354.CrossRefGoogle Scholar
Colombo, M. & Di Marino, S. (2013) Equality between Monge and Kantorovich multimarginal problems with coulomb cost. Annali di Matematica Pura ed Applicata, 194, 1–14.Google Scholar
Colombo, M. & Stra, F. (2016) Counterexamples in multi-marginal optimal transport with Coulomb cost and spherically symmetric data. Math. Models Methods Appl. Sci. 26(6), 1025–1049.CrossRefGoogle Scholar
Colombo, M., De Pascale, L. & Di Marino, S. (2015) Multimarginal optimal transport maps for 1-dimensional repulsive costs. Can. J. Math. 67(2), 350–368.CrossRefGoogle Scholar
Colombo, M., Di Marino, S. & Stra, F. (2018) Continuity of multi-marginal optimal transport with repulsive cost. Preprint available at: http://cvgmt.sns.itGoogle Scholar
Cotar, C., Friesecke, G. & Klüppelberg, C. (2013) Density functional theory and optimal transportation with coulomb cost. Commun. Pure Appl. Math. 66(4), 548–599.CrossRefGoogle Scholar
De Pascale, L. (2015) Optimal transport with Coulomb cost. Approximation and duality. ESAIM Math. Model. Numer. Anal. 49(6), 1643–1657.CrossRefGoogle Scholar
Di Marino, S., Gerolin, A. & Nenna, L. (2015) Optimal transportation theory for repulsive costs. In: Bergounioux, M., Oudet, E., Rumpf, M., Carlier, G., Champion, T. and Santambrogio, F. (editors), Topological Optimization and Optimal Transport: In the Applied Sciences, De Gruyter, Berlin, Boston, pp. 204–256.Google Scholar
Friesecke, G., Mendl, C. B., Pass, B., Cotar, C. & Klüppelberg, C. (2013) N-density representability and the optimal transport limit of the Hohenberg-Kohn functional. J. Chem. Phys. 139(16), 164109.CrossRefGoogle ScholarPubMed
Gangbo, W. & Swiech, A. (1998) Optimal maps for the multidimensional Monge-Kantorovich problem. Comm. Pure Appl. Math. 51(1), 23–45.3.0.CO;2-H>CrossRefGoogle Scholar
Ghoussoub, N. & Moameni, A. (2012) A self-dual polar factorization for vector fields. Comm. Pure Appl. Math. doi: 10.1002/cpa.21430.Google Scholar
Gori-Giorgi, P. & Seidl, M. (2010) Density functional theory for strongly-interacting electrons: perspectives for physics and chemistry. Phys. Chem. Chem. Phys. 12(43), 14405–14419.CrossRefGoogle ScholarPubMed
Gori-Giorgi, P., Seidl, M. & Vignale, G. (2009) Density-functional theory for strongly interacting electrons. Phys. Rev. Lett. 103(16), 166402.CrossRefGoogle ScholarPubMed
Griessler, C. (2018) c-cyclical monotonicity as a sufficient criterion for optimality in the multi-marginal Monge-Kantorovich problem Proc. Amer. Math. Soc. 146, 4735–4740.CrossRefGoogle Scholar
Heinich, H. (2002) Problème de Monge pour n-probabilités. Comptes Rendus Mathematique, 334(9), 793–795.CrossRefGoogle Scholar
Kellerer, H. G. (1984) Duality theorems for marginal problems. Prob. Theory Relat. Fields 67(4), 399–432.Google Scholar
Kim, Y. H. & Pass, B. (2014) A general condition for Monge solutions in the multi-marginal optimal transport problem. SIAM J. Math. Anal. 46, 1538–1550.CrossRefGoogle Scholar
Mendl, C. B. & Lin, L. (2013) Kantorovich dual solution for strictly correlated electrons in atoms and molecules. Phys. Rev. B 87(12), 125106.CrossRefGoogle Scholar
Moameni, A. (2016) Invariance properties of the Monge-Kantorovich mass transport problem. Discrete Contin. Dyn. Syst. 36(5), 2653–2671.CrossRefGoogle Scholar
Pass, B. (2011) Uniqueness and Monge solutions in the multimarginal optimal transportation problem. SIAM J. Math. Anal. 43(6), 2758–2775.CrossRefGoogle Scholar
Pass, B. (2012) On the local structure of optimal measures in the multi-marginal optimal transportation problem. Calculus Variations Partial Diff. Equations 43(3–4), 529–536.CrossRefGoogle Scholar
Pratelli, A. (2008) On the sufficiency of c-cyclical monotonicity for optimality of transport plans. Math. Z. 258(3), 677–690.CrossRefGoogle Scholar
Rachev, S. T. & Rüschendorf, L. (1998) Mass transportation problems. Vol. I. Probability and its Applications (New York). Springer-Verlag, New York (Theory).Google Scholar
Santambrogio, F. (2015) Optimal transport for applied mathematicians, Progress in Nonlinear Differential Equations and Their Applications, Vol. 87, Birkhäuser Basel. DOI https://dx.doi.org/10.1007/978-3-319-20828-2CrossRefGoogle Scholar
Schachermayer, W. & Teichmann, J. (2009) Characterization of optimal transport plans for the Monge-Kantorovich problem. Proc. Amer. Math. Soc. 137(2), 519–529.CrossRefGoogle Scholar
Seidl, M. (1999) Strong-interaction limit of density-functional theory. Phys. Rev. A. 60(6), 4387.CrossRefGoogle Scholar
Seidl, M., Perdew, J. P. & Levy, M. (1999) Strictly correlated electrons in density-functional theory. Phys. Rev. A 59(1), 51.CrossRefGoogle Scholar
Seidl, M., Gori-Giorgi, P. & Savin, A. (2007) Strictly correlated electrons in density-functional theory: A general formulation with applications to spherical densities. Phys. Rev. A 75(4), 042511.CrossRefGoogle Scholar
Stra, F. (2018) Classical and multi-marginal optimal transport, with applications PhD thesis, Scuola Normale Superiore, Pisa.Google Scholar
Villani, C. (2003) Topics in optimal transportation. Graduate Studies in Mathematics, Vol. 58, American Mathematical Society, Providence, RI.Google Scholar
Villani, C. (2009) Optimal transport. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Vol. 338, Springer-Verlag, Berlin.Google Scholar