Abstract

An approach to linear programming duality is proposed which relies on quadratic penalization, so that the relation between solutions to the penalized primal and dual problems becomes affine. This yields a new proof of Levin's duality theorem for capacity-constrained optimal transport as an infinite-dimensional application

Author information

Contact details are reproduced from the original publication and may be historical.

Robert J. McCann

Dept. of Mathematics, University of Toronto, Toronto, Ontario, Canada M5S 2E4

mccann@math.toronto.edu

Christian Seis

Department of Mathematics, University of Toronto, Toronto ON Canada M5S 2E4

cseis@math.toronto.edu

References

  1. F. Bethuel, H. Brézis, F. Hélein: Ginzburg-Landau Vortices, Progress in Nonlinear Differential Equations and their Applications 13, Birkhäuser, Boston (1994).https://doi.org/10.1007/978-1-4612-0287-5Google Scholar
  2. L. Kantorovich: On the translocation of masses, C. R. (Dokl.) Acad. Sci. SSSR, n. Ser. 37 (1942) 199–201.Google Scholar
  3. H. G. Kellerer: Marginalprobleme für Funktionen, Math. Ann. 154 (1964) 147–156.https://doi.org/10.1007/BF01360886Google Scholar
  4. H. G. Kellerer: Maßtheoretische Marginalprobleme, Math. Ann. 153 (1964) 168–198.https://doi.org/10.1007/BF01360315Google Scholar
  5. J. Korman, R. J. McCann: Optimal transport with capacity constraints, Trans. Amer. Math. Soc. 367 (2015) 1501–1521.https://doi.org/10.1090/S0002-9947-2014-06032-7Google Scholar
  6. J. Korman, R. J. McCann: Insights into capacity constrained optimal transport, Proc. Natl. Acad. Sci. USA 110 (2013) 10064–10067.https://doi.org/10.1073/pnas.1221333110Google Scholar
  7. J. Korman, R. J. McCann, C. Seis: Dual potentials for capacity constrained optimal transport, Calc. Var. Partial. Differ. Equ., to appear.Google Scholar
  8. V. L. Levin: The problem of mass transfer in a topological space and probability measures with given marginal measures on the product of two spaces, Dokl. Akad. Nauk SSSR 276 (1984) 1059–1064.Google Scholar
  9. D. G. Luenberger, Y. Ye: Linear and Nonlinear Programming, 3rd Ed., International Series in Operations Research & Management Science 116, Springer, New York (2008).https://doi.org/10.1007/978-0-387-74503-9Google Scholar
  10. G. Monge: Mémoire sur la théorie de déblais et de remblais, Histoire de l’Académie Royale des Science de Paris, avec les Mémoires de Mathématique et de Physique pour la même année (1781) 666–704.Google Scholar
  11. S. T. Rachev, L. Rüschendorf: Mass Transportation Problems, Vol. I, Probability and its Applications, Springer, New York (1998).Google Scholar
  12. C. Villani: Topics in Optimal Transportation, Graduate Studies in Mathematics 58, American Mathematical Society, Providence (2003).https://doi.org/10.1090/gsm/058Google Scholar

Suggested citation

J. Korman, R. J. McCann, C. Seis. “An Elementary Approach to Linear Programming Duality with Application to Capacity Constrained Transport.” Journal of Convex Analysis 22 (2015), No. 3, 797–808. https://doi.org/10.68381/jca22040

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Published by Heldermann Verlag, 2015. Rights now held by Banach Press LLC.